Problema 68
Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3.
It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.
Total Solution Set
9 4,2,3; 5,3,1; 6,1,2
10 2,3,5; 4,5,1; 6,1,3
By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513.
Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string for a "magic" 5-gon ring?
In the game, Monopoly, the standard board is set up in the following way:
GO A1 CC1 A2 T1 R1 B1 CH1 B2 B3 JAIL
H2 C1
T2 U1
CH3 C3
CC3 CC2
G2J F3 U2 F2 F1 R3 E3 E2 CH2 E1 FP
A player starts on the GO square and adds the scores on two 6-sided dice to determine the number of squares they advance in a clockwise direction. Without any further rules we would expect to visit each square with equal probability: 2.5%. However, landing on G2J (Go To Jail), CC (community chest), and CH (chance) changes this distribution.
In addition to G2J, and one card from each of CC and CH, that orders the player to go directly to jail, if a player rolls three consecutive doubles, they do not advance the result of their 3rd roll. Instead they proceed directly to jail.
At the beginning of the game, the CC and CH cards are shuffled. When a player lands on CC or CH they take a card from the top of the respective pile and, after following the instructions, it is returned to the bottom of the pile. There are sixteen cards in each pile, but for the purpose of this problem we are only concerned with cards that order a movement; any instruction not concerned with movement will be ignored and the player will remain on the CC/CH square.
Community Chest (2/16 cards):
Advance to GO
Go to JAIL
Chance (10/16 cards):
Go to C1
Go to E3
Go to H2
Go to R1
Go to next R (railway company)
Go to next R
Go to next U (utility company)
Go back 3 squares.
The heart of this problem concerns the likelihood of visiting a particular square. That is, the probability of finishing at that square after a roll. For this reason it should be clear that, with the exception of G2J for which the probability of finishing on it is zero, the CH squares will have the lowest probabilities, as 5/8 request a movement to another square, and it is the final square that the player finishes at on each roll that we are interested in. We shall make no distinction between "Just Visiting" and being sent to JAIL, and we shall also ignore the rule about requiring a double to "get out of jail", assuming that they pay to get out on their next turn.
By starting at GO and numbering the squares sequentially from 00 to 39 we can concatenate these two-digit numbers to produce strings that correspond with sets of squares.
Statistically it can be shown that the three most popular squares, in order, are JAIL (6.24%) = Square 10, E3 (3.18%) = Square 24, and GO (3.09%) = Square 00. So these three most popular squares can be listed with the six-digit modal string: 102400.
If, instead of using two 6-sided dice, two 4-sided dice are used, find the six-digit modal string.
A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {a1, a2, ... , ak} is called a product-sum number: N = a1 + a2 + ... + ak = a1 × a2 × ... × ak.
For example, 6 = 1 + 2 + 3 = 1 × 2 × 3.
For a given set of size, k, we shall call the smallest N with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, k = 2, 3, 4, 5, and 6 are as follows.
k=2: 4 = 2 × 2 = 2 + 2
k=3: 6 = 1 × 2 × 3 = 1 + 2 + 3
k=4: 8 = 1 × 1 × 2 × 4 = 1 + 1 + 2 + 4
k=5: 8 = 1 × 1 × 2 × 2 × 2 = 1 + 1 + 2 + 2 + 2
k=6: 12 = 1 × 1 × 1 × 1 × 2 × 6 = 1 + 1 + 1 + 1 + 2 + 6
Hence for 2≤k≤6, the sum of all the minimal product-sum numbers is 4+6+8+12 = 30; note that 8 is only counted once in the sum.
In fact, as the complete set of minimal product-sum numbers for 2≤k≤12 is {4, 6, 8, 12, 15, 16}, the sum is 61.
What is the sum of all the minimal product-sum numbers for 2≤k≤12000?
Each of the six faces on a cube has a different digit (0 to 9) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of 2-digit numbers.
For example, the square number 64 could be formed:
In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below one-hundred: 01, 04, 09, 16, 25, 36, 49, 64, and 81.
For example, one way this can be achieved is by placing {0, 5, 6, 7, 8, 9} on one cube and {1, 2, 3, 4, 8, 9} on the other cube.
However, for this problem we shall allow the 6 or 9 to be turned upside-down so that an arrangement like {0, 5, 6, 7, 8, 9} and {1, 2, 3, 4, 6, 7} allows for all nine square numbers to be displayed; otherwise it would be impossible to obtain 09.
In determining a distinct arrangement we are interested in the digits on each cube, not the order.
{1, 2, 3, 4, 5, 6} is equivalent to {3, 6, 4, 1, 2, 5}
{1, 2, 3, 4, 5, 6} is distinct from {1, 2, 3, 4, 5, 9}
But because we are allowing 6 and 9 to be reversed, the two distinct sets in the last example both represent the extended set {1, 2, 3, 4, 5, 6, 9} for the purpose of forming 2-digit numbers.
How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed?
A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.
44 → 32 → 13 → 10 → 1 → 1
85 → 89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89
Therefore any chain that arrives at 1 or 89 will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at 1 or 89.
How many starting numbers below ten million will arrive at 89?
The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of 28 are 1, 2, 4, 7, and 14. As the sum of these divisors is equal to 28, we call it a perfect number.
Interestingly the sum of the proper divisors of 220 is 284 and the sum of the proper divisors of 284 is 220, forming a chain of two numbers. For this reason, 220 and 284 are called an amicable pair.
Perhaps less well known are longer chains. For example, starting with 12496, we form a chain of five numbers:
12496 → 14288 → 15472 → 14536 → 14264 (→ 12496 → ...)
Since this chain returns to its starting point, it is called an amicable chain.
Find the smallest member of the longest amicable chain with no element exceeding one million.
Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (or zeros) in a 9 by 9 grid in such that each row, column, and 3 by 3 box contains each of the digits 1 to 9. Below is an example of a typical starting puzzle grid and its solution grid.
A well constructed Su Doku puzzle has a unique solution and can be solved by logic, although it may be necessary to employ "guess and test" methods in order to eliminate options (there is much contested opinion over this). The complexity of the search determines the difficulty of the puzzle; the example above is considered easy because it can be solved by straight forward direct deduction.
The 6K text file, sudoku.txt (right click and 'Save Link/Target As...'), contains fifty different Su Doku puzzles ranging in difficulty, but all with unique solutions (the first puzzle in the file is the example above).
By solving all fifty puzzles find the sum of the 3-digit numbers found in the top left corner of each solution grid; for example, 483 is the 3-digit number found in the top left corner of the solution grid above.
By replacing each of the letters in the word CARE with 1, 2, 9, and 6 respectively, we form a square number: 1296 = 362. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: 9216 = 962. We shall call CARE (and RACE) a square anagram word pair and specify further that leading zeroes are not permitted, neither may a different letter have the same digital value as another letter.
Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, find all the square anagram word pairs (a palindromic word is NOT considered to be an anagram of itself).
What is the largest square number formed by any member of such a pair?
NOTE: All anagrams formed must be contained in the given text file.
If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.
As an example, let us consider the sequence of cube numbers. This is defined by the generating function,
un = n3: 1, 8, 27, 64, 125, 216, ...
Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
We shall define OP(k, n) to be the nth term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP).
As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n ≥ 2, OP(1, n) = u1.
Hence we obtain the following OPs for the cubic sequence:
OP(1, n) = 1 1, 1, 1, 1, ...
OP(2, n) = 7n−6 1, 8, 15, ...
OP(3, n) = 6n2−11n+6 1, 8, 27, 58, ...
OP(4, n) = n3 1, 8, 27, 64, 125, ...
Clearly no BOPs exist for k ≥ 4.
By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74.
Consider the following tenth degree polynomial generating function:
un = 1 − n + n2 − n3 + n4 − n5 + n6 − n7 + n8 − n9 + n10
Find the sum of FITs for the BOPs.
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true:
S(B) ≠ S(C); that is, sums of subsets cannot be equal.
If B contains more elements than C then S(B) > S(C).
If S(A) is minimised for a given n, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.
n = 1: {1}
n = 2: {1, 2}
n = 3: {2, 3, 4}
n = 4: {3, 5, 6, 7}
n = 5: {6, 9, 11, 12, 13}
It seems that for a given optimum set, A = {a1, a2, ... , an}, the next optimum set is of the form B = {b, a1+b, a2+b, ... ,an+b}, where b is the "middle" element on the previous row.
By applying this "rule" we would expect the optimum set for n = 6 to be A = {11, 17, 20, 22, 23, 24}, with S(A) = 117. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for n = 6 is A = {11, 18, 19, 20, 22, 25}, with S(A) = 115 and corresponding set string: 111819202225.
Given that A is an optimum special sum set for n = 7, find its set string.
NOTE: This problem is related to problems 105 and 106.
For example, {81, 88, 75, 42, 87, 84, 86, 65} is not a special sum set because 65 + 87 + 88 = 75 + 81 + 84, whereas {157, 150, 164, 119, 79, 159, 161, 139, 158} satisfies both rules for all possible subset pair combinations and S(A) = 1286.
Using sets.txt (right click and "Save Link/Target As..."), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, A1, A2, ..., Ak, and find the value of S(A1) + S(A2) + ... + S(Ak).
For this problem we shall assume that a given set contains n strictly increasing elements and it already satisfies the second rule.
Surprisingly, out of the 25 possible subset pairs that can be obtained from a set for which n = 4, only 1 of these pairs need to be tested for equality (first rule). Similarly, when n = 7, only 70 out of the 966 subset pairs need to be tested.
For n = 12, how many of the 261625 subset pairs that can be obtained need to be tested for equality?
In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.
The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions inside this ring represent single scores. However, the red/green outer ring and middle ring score double and treble scores respectively.
At the centre of the board are two concentric circles called the bull region, or bulls-eye. The outer bull is worth 25 points and the inner bull is a double, worth 50 points.
There are many variations of rules but in the most popular game the players will begin with a score 301 or 501 and the first player to reduce their running total to zero is a winner. However, it is normal to play a "doubles out" system, which means that the player must land a double (including the double bulls-eye at the centre of the board) on their final dart to win; any other dart that would reduce their running total to one or lower means the score for that set of three darts is "bust".
When a player is able to finish on their current score it is called a "checkout" and the highest checkout is 170: T20 T20 D25 (two treble 20s and double bull).
There are exactly eleven distinct ways to checkout on a score of 6:
S1 S1 D2
S1 T1 D1
D1 D1 D1
D1 S2 D1
Note that D1 D2 is considered different to D2 D1 as they finish on different doubles. However, the combination S1 T1 D1 is considered the same as T1 S1 D1.
In addition we shall not include misses in considering combinations; for example, D3 is the same as 0 D3 and 0 0 D3.
Incredibly there are 42336 distinct ways of checking out in total.
How many distinct ways can a player checkout with a score less than 100?
A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square. There are exactly seventeen ways of doing this.
How many ways can a row measuring fifty units in length be filled?
NOTE: Although the example above does not lend itself to the possibility, in general it is permitted to mix block sizes. For example, on a row measuring eight units in length you could use red (3), black (1), and red (4).
NOTE: This is a more difficult version of problem 114.
A row measuring n units in length has red blocks with a minimum length of m units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.
Let the fill-count function, F(m, n), represent the number of ways that a row can be filled.
For example, F(3, 29) = 673135 and F(3, 30) = 1089155.
That is, for m = 3, it can be seen that n = 30 is the smallest value for which the fill-count function first exceeds one million.
In the same way, for m = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so n = 57 is the least value for which the fill-count function first exceeds one million.
For m = 50, find the least value of n for which the fill-count function first exceeds one million.
A row of five black square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (length two), green (length three), or blue (length four).
If red tiles are chosen there are exactly seven ways this can be done.
If green tiles are chosen there are three ways.
And if blue tiles are chosen there are two ways.
Assuming that colours cannot be mixed there are 7 + 3 + 2 = 12 ways of replacing the black tiles in a row measuring five units in length.
How many different ways can the black tiles in a row measuring fifty units in length be replaced if colours cannot be mixed and at least one coloured tile must be used?
NOTE: This is related to problem 117.
Using a combination of black square tiles and oblong tiles chosen from: red tiles measuring two units, green tiles measuring three units, and blue tiles measuring four units, it is possible to tile a row measuring five units in length in exactly fifteen different ways.
How many ways can a row measuring fifty units in length be tiled?
A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted. After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random.
The player pays £1 to play and wins if they have taken more blue discs than red discs at the end of the game.
If the game is played for four turns, the probability of a player winning is exactly 11/120, and so the maximum prize fund the banker should allocate for winning in this game would be £10 before they would expect to incur a loss. Note that any payout will be a whole number of pounds and also includes the original £1 paid to play the game, so in the example given the player actually wins £9.
Find the maximum prize fund that should be allocated to a single game in which fifteen turns are played.
In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around on the mirrors, and eventually works its way back out.
The specific white cell we will be considering is an ellipse with the equation 4x2 + y2 = 100
The section corresponding to −0.01 ≤ x ≤ +0.01 at the top is missing, allowing the light to enter and exit through the hole.
The light beam in this problem starts at the point (0.0,10.1) just outside the white cell, and the beam first impacts the mirror at (1.4,-9.6).
Each time the laser beam hits the surface of the ellipse, it follows the usual law of reflection "angle of incidence equals angle of reflection." That is, both the incident and reflected beams make the same angle with the normal line at the point of incidence.
In the figure on the left, the red line shows the first two points of contact between the laser beam and the wall of the white cell; the blue line shows the line tangent to the ellipse at the point of incidence of the first bounce.
The slope m of the tangent line at any point (x,y) of the given ellipse is: m = −4x/y
The normal line is perpendicular to this tangent line at the point of incidence.
The animation on the right shows the first 10 reflections of the beam.
How many times does the beam hit the internal surface of the white cell before exiting?
In a 3x2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in the sketch.
There are 5 grids smaller than 3x2, vertical and horizontal dimensions being important, i.e. 1x1, 2x1, 3x1, 1x2 and 2x2. If each of them is cross-hatched, the following number of different rectangles could be situated within those smaller grids:
1x1: 1
2x2: 18
Adding those to the 37 of the 3x2 grid, a total of 72 different rectangles could be situated within 3x2 and smaller grids.
How many different rectangles could be situated within 47x43 and smaller grids?
A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of size A5.
Every Monday morning, the foreman opens a new envelope, containing a large sheet of the special paper with size A1.
He proceeds to cut it in half, thus getting two sheets of size A2. Then he cuts one of them in half to get two sheets of size A3 and so on until he obtains the A5-size sheet needed for the first batch of the week.
All the unused sheets are placed back in the envelope.
At the beginning of each subsequent batch, he takes from the envelope one sheet of paper at random. If it is of size A5, he uses it. If it is larger, he repeats the 'cut-in-half' procedure until he has what he needs and any remaining sheets are always placed back in the envelope.
Excluding the first and last batch of the week, find the expected number of times (during each week) that the foreman finds a single sheet of paper in the envelope.
Give your answer rounded to six decimal places using the format x.xxxxxx .
An electric circuit uses exclusively identical capacitors of the same value C.
The capacitors can be connected in series or in parallel to form sub-units, which can then be connected in series or in parallel with other capacitors or other sub-units to form larger sub-units, and so on up to a final circuit.
Using this simple procedure and up to n identical capacitors, we can make circuits having a range of different total capacitances. For example, using up to n=3 capacitors of 60 F each, we can obtain the following 7 distinct total capacitance values:
If we denote by D(n) the number of distinct total capacitance values we can obtain when using up to n equal-valued capacitors and the simple procedure described above, we have: D(1)=1, D(2)=3, D(3)=7 ...
Find D(18).
Reminder : When connecting capacitors C1, C2 etc in parallel, the total capacitance is CT = C1 + C2 +...,
whereas when connecting them in series, the overall capacitance is given by:
Starting from zero the natural numbers are written down in base 10 like this:
0 1 2 3 4 5 6 7 8 9 10 11 12....
Consider the digit d=1. After we write down each number n, we will update the number of ones that have occurred and call this number f(n,1). The first values for f(n,1), then, are as follows:
n f(n,1)
Note that f(n,1) never equals 3.
So the first two solutions of the equation f(n,1)=n are n=0 and n=1. The next solution is n=199981.
In the same manner the function f(n,d) gives the total number of digits d that have been written down after the number n has been written.
In fact, for every digit d ≠ 0, 0 is the first solution of the equation f(n,d)=n.
Let s(d) be the sum of all the solutions for which f(n,d)=n.
You are given that s(1)=22786974071.
Find ∑ s(d) for 1 ≤ d ≤ 9.
Note: if, for some n, f(n,d)=n for more than one value of d this value of n is counted again for every value of d for which f(n,d)=n.
A composite number can be factored many different ways. For instance, not including multiplication by one, 24 can be factored in 7 distinct ways:
24 = 2x2x2x3
24 = 2x3x4
24 = 4x6
24 = 2x12
24 = 24
Recall that the digital root of a number, in base 10, is found by adding together the digits of that number, and repeating that process until a number is arrived at that is less than 10. Thus the digital root of 467 is 8.
We shall call a Digital Root Sum (DRS) the sum of the digital roots of the individual factors of our number.
The chart below demonstrates all of the DRS values for 24.
Digital Root Sum
The maximum Digital Root Sum of 24 is 11.
The function mdrs(n) gives the maximum Digital Root Sum of n. So mdrs(24)=11.
Find ∑mdrs(n) for 1 < n < 1,000,000.
A triomino is a shape consisting of three squares joined via the edges. There are two basic forms:
If all possible orientations are taken into account there are six:
Any n by m grid for which nxm is divisible by 3 can be tiled with triominoes.
If we consider tilings that can be obtained by reflection or rotation from another tiling as different there are 41 ways a 2 by 9 grid can be tiled with triominoes:
In how many ways can a 9 by 12 grid be tiled in this way by triominoes?
A segment is uniquely defined by its two endpoints.
By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.
Moreover when two segments have exactly one point in common it might be the case that that common point is an endpoint of either one of the segments or of both. If a common point of two segments is not an endpoint of either of the segments it is an interior point of both segments.
We will call a common point T of two segments L1 and L2 a true intersection point of L1 and L2 if T is the only common point of L1 and L2 and T is an interior point of both segments.
Consider the three segments L1, L2, and L3:
L1: (27, 44) to (12, 32)
It can be verified that line segments L2 and L3 have a true intersection point. We note that as the one of the end points of L3: (22,40) lies on L1 this is not considered to be a true point of intersection. L1 and L2 have no common point. So among the three line segments, we find one true intersection point.
Now let us do the same for 5000 line segments. To this end, we generate 20000 numbers using the so-called "Blum Blum Shub" pseudo-random number generator.
s0 = 290797
sn+1 = sn×sn (modulo 50515093)
tn = sn (modulo 500)
To create each line segment, we use four consecutive numbers tn. That is, the first line segment is given by:
(t1, t2) to (t3, t4)
The first four numbers computed according to the above generator should be: 27, 144, 12 and 232. The first segment would thus be (27,144) to (12,232).
How many distinct true intersection points are found among the 5000 line segments?
Take the number 6 and multiply it by each of 1273 and 9854:
6 × 1273 = 7638
6 × 9854 = 59124
By concatenating these products we get the 1 to 9 pandigital 763859124. We will call 763859124 the "concatenated product of 6 and (1273,9854)". Notice too, that the concatenation of the input numbers, 612739854, is also 1 to 9 pandigital.
The same can be done for 0 to 9 pandigital numbers.
What is the largest 0 to 9 pandigital 10-digit concatenated product of an integer with two or more other integers, such that the concatenation of the input numbers is also a 0 to 9 pandigital 10-digit number?
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. For example, using exactly thirty-two square tiles we can form two different square laminae:
With one-hundred tiles, and not necessarily using all of the tiles at one time, it is possible to form forty-one different square laminae.
Using up to one million tiles how many different square laminae can be formed?
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.
Given eight tiles it is possible to form a lamina in only one way: 3x3 square with a 1x1 hole in the middle. However, using thirty-two tiles it is possible to form two distinct laminae.
If t represents the number of tiles used, we shall say that t = 8 is type L(1) and t = 32 is type L(2).
Let N(n) be the number of t ≤ 1000000 such that t is type L(n); for example, N(15) = 832.
What is ∑ N(n) for 1 ≤ n ≤ 10?
Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice.
For example, f(10)=5 since there are five different ways to express 10:
10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1
It can be shown that for every fraction p/q (p>0, q>0) there exists at least one integer n such that
f(n)/f(n-1)=p/q.
For instance, the smallest n for which f(n)/f(n-1)=13/17 is 241.
The binary expansion of 241 is 11110001.
Reading this binary number from the most significant bit to the least significant bit there are 4 one's, 3 zeroes and 1 one. We shall call the string 4,3,1 the Shortened Binary Expansion of 241.
Find the Shortened Binary Expansion of the smallest n for which
f(n)/f(n-1)=123456789/987654321.
Give your answer as comma separated integers, without any whitespaces.
Let ABCD be a convex quadrilateral, with diagonals AC and BD. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles.
For example, at vertex A, the two angles are CAD, CAB.
We call such a quadrilateral for which all eight corner angles have integer values when measured in degrees an "integer angled quadrilateral". An example of an integer angled quadrilateral is a square, where all eight corner angles are 45°. Another example is given by DAC = 20°, BAC = 60°, ABD = 50°, CBD = 30°, BCA = 40°, DCA = 30°, CDB = 80°, ADB = 50°.
What is the total number of non-similar integer angled quadrilaterals?
Note: In your calculations you may assume that a calculated angle is integral if it is within a tolerance of 10-9 of an integer value.
For any integer n, consider the three functions
f1,n(x,y,z) = xn+1 + yn+1 − zn+1
f2,n(x,y,z) = (xy + yz + zx)*(xn-1 + yn-1 − zn-1)
f3,n(x,y,z) = xyz*(xn-2 + yn-2 − zn-2)
and their combination
fn(x,y,z) = f1,n(x,y,z) + f2,n(x,y,z) − f3,n(x,y,z)
We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form a / b with
0 < a < b ≤ k and there is (at least) one integer n, so that fn(x,y,z) = 0.
Let s(x,y,z) = x + y + z.
Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples (x,y,z) of order 35.
All the s(x,y,z) and t must be in reduced form.
Find u + v.
The RSA encryption is based on the following procedure:
Generate two distinct primes p and q.
Compute n=pq and φ=(p-1)(q-1).
Find an integer e, 1<e<φ, such that gcd(e,φ)=1.
A message in this system is a number in the interval [0,n-1].
A text to be encrypted is then somehow converted to messages (numbers in the interval [0,n-1]).
To encrypt the text, for each message, m, c=me mod n is calculated.
To decrypt the text, the following procedure is needed: calculate d such that ed=1 mod φ, then for each encrypted message, c, calculate m=cd mod n.
There exist values of e and m such that me mod n=m.
We call messages m for which me mod n=m unconcealed messages.
An issue when choosing e is that there should not be too many unconcealed messages.
For instance, let p=19 and q=37.
Then n=19*37=703 and φ=18*36=648.
If we choose e=181, then, although gcd(181,648)=1 it turns out that all possible messages
m (0≤m≤n-1) are unconcealed when calculating me mod n.
For any valid choice of e there exist some unconcealed messages.
It's important that the number of unconcealed messages is at a minimum.
Choose p=1009 and q=3643.
Find the sum of all values of e, 1<e<φ(1009,3643) and gcd(e,φ)=1, so that the number of unconcealed messages for this value of e is at a minimum.
Let N be a positive integer and let N be split into k equal parts, r = N/k, so that N = r + r + ... + r.
Let P be the product of these parts, P = r × r × ... × r = rk.
For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.25 = 51.53632.
Let M(N) = Pmax for a given value of N.
It turns out that the maximum for N = 11 is found by splitting eleven into four equal parts which leads to Pmax = (11/4)4; that is, M(11) = 14641/256 = 57.19140625, which is a terminating decimal.
However, for N = 8 the maximum is achieved by splitting it into three equal parts, so M(8) = 512/27, which is a non-terminating decimal.
Let D(N) = N if M(N) is a non-terminating decimal and D(N) = -N if M(N) is a terminating decimal.
For example, ΣD(N) for 5 ≤ N ≤ 100 is 2438.
Find ΣD(N) for 5 ≤ N ≤ 10000.
The game Number Mind is a variant of the well known game Master Mind.
Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you'd be told that you have one correct digit; however, you would NOT be told that you also have another digit in the wrong place.
For instance, given the following guesses for a 5-digit secret sequence,
90342 ;2 correct
The correct sequence 39542 is unique.
Based on the following guesses,
5616185650518293 ;2 correct
Find the unique 16-digit secret sequence.
Here are the records from a busy telephone system with one million users:
RecNr Caller Called
The telephone number of the caller and the called number in record n are Caller(n) = S2n-1 and Called(n) = S2n where S1,2,3,... come from the "Lagged Fibonacci Generator":
For 1 ≤ k ≤ 55, Sk = [100003 - 200003k + 300007k3] (modulo 1000000)
For 56 ≤ k, Sk = [Sk-24 + Sk-55] (modulo 1000000)
If Caller(n) = Called(n) then the user is assumed to have misdialled and the call fails; otherwise the call is successful.
From the start of the records, we say that any pair of users X and Y are friends if X calls Y or vice-versa. Similarly, X is a friend of a friend of Z if X is a friend of Y and Y is a friend of Z; and so on for longer chains.
The Prime Minister's phone number is 524287. After how many successful calls, not counting misdials, will 99% of the users (including the PM) be a friend, or a friend of a friend etc., of the Prime Minister?
Let Sm = (x1, x2, ... , xm) be the m-tuple of positive real numbers with x1 + x2 + ... + xm = m for which Pm = x1 * x22 * ... * xmm is maximised.
For example, it can be verified that [P10] = 4112 ([ ] is the integer part function).
Find Σ[Pm] for 2 ≤ m ≤ 15.
A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.
During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (absent).
Although there are eighty-one trinary strings for a 4-day period that can be formed, exactly forty-three strings would lead to a prize:
OOOO OOOA OOOL OOAO OOAA OOAL OOLO OOLA OAOO OAOA
OAOL OAAO OAAL OALO OALA OLOO OLOA OLAO OLAA AOOO
AOOA AOOL AOAO AOAA AOAL AOLO AOLA AAOO AAOA AAOL
AALO AALA ALOO ALOA ALAO ALAA LOOO LOOA LOAO LOAA
LAOO LAOA LAAO
How many "prize" strings exist over a 30-day period?
Let x be a real number.
A best approximation to x for the denominator bound d is a rational number r/s in reduced form, with s ≤ d, such that any rational number which is closer to x than r/s has a denominator larger than d:
|p/q-x| < |r/s-x| q > d
For example, the best approximation to √13 for the denominator bound 20 is 18/5 and the best approximation to √13 for the denominator bound 30 is 101/28.
Find the sum of all denominators of the best approximations to √n for the denominator bound 1012, where n is not a perfect square and 1 < n ≤ 100000.
Build a triangle from all positive integers in the following way:
Each positive integer has up to eight neighbours in the triangle.
A set of three primes is called a prime triplet if one of the three primes has the other two as neighbours in the triangle.
For example, in the second row, the prime numbers 2 and 3 are elements of some prime triplet.
If row 8 is considered, it contains two primes which are elements of some prime triplet, i.e. 29 and 31.
If row 9 is considered, it contains only one prime which is an element of some prime triplet: 37.
Define S(n) as the sum of the primes in row n which are elements of any prime triplet.
Then S(8)=60 and S(9)=37.
You are given that S(10000)=950007619.
Find S(5678027) + S(7208785).
A best approximation to a real number x for the denominator bound d is a rational number r/s (in reduced form) with s ≤ d, so that any rational number p/q which is closer to x than r/s has q > d.
Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. 9/40 has the two best approximations 1/4 and 1/5 for the denominator bound 6. We shall call a real number x ambiguous, if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational.
How many ambiguous numbers x = p/q, 0 < x < 1/100, are there whose denominator q does not exceed 108?
We shall define a sqube to be a number of the form, p2q3, where p and q are distinct primes.
For example, 200 = 5223 or 120072949 = 232613.
The first five squbes are 72, 108, 200, 392, and 500.
Interestingly, 200 is also the first number for which you cannot change any single digit to make a prime; we shall call such numbers, prime-proof. The next prime-proof sqube which contains the contiguous sub-string "200" is 1992008.
Find the 200th prime-proof sqube containing the contiguous sub-string "200".
The binomial coefficients nCk can be arranged in triangular form, Pascal's triangle, like this:
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 21 and 35.
A positive integer n is called squarefree if no square of a prime divides n. Of the twelve distinct numbers in the first eight rows of Pascal's triangle, all except 4 and 20 are squarefree. The sum of the distinct squarefree numbers in the first eight rows is 105.
Find the sum of the distinct squarefree numbers in the first 51 rows of Pascal's triangle.
For some positive integers k, there exists an integer partition of the form 4t = 2t + k,
where 4t, 2t, and k are all positive integers and t is a real number.
The first two such partitions are 41 = 21 + 2 and 41.5849625... = 21.5849625... + 6.
Partitions where t is also an integer are called perfect.
For any m ≥ 1 let P(m) be the proportion of such partitions that are perfect with k ≤ m.
Thus P(6) = 1/2.
In the following table are listed some values of P(m)
P(5) = 1/1
P(10) = 1/2
P(180) = 1/4
P(185) = 3/13
Find the smallest m for which P(m) < 1/12345
A robot moves in a series of one-fifth circular arcs (72°), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot.
One of 70932 possible closed paths of 25 arcs starting northward is
Given that the robot starts facing North, how many journeys of 70 arcs in length can it take that return it, after the final arc, to its starting position?
(Any arc may be traversed multiple times.)
A k-input binary truth table is a map from k input bits (binary digits, 0 [false] or 1 [true]) to 1 output bit. For example, the 2-input binary truth tables for the logical AND and XOR functions are:
x y x AND y
x y x XOR y
How many 6-input binary truth tables, τ, satisfy the formula
τ(a, b, c, d, e, f) AND τ(b, c, d, e, f, a XOR (b AND c)) = 0
for all 6-bit inputs (a, b, c, d, e, f)?
An axis-aligned cuboid, specified by parameters { (x0,y0,z0), (dx,dy,dz) }, consists of all points (X,Y,Z) such that x0 ≤ X ≤ x0+dx, y0 ≤ Y ≤ y0+dy and z0 ≤ Z ≤ z0+dz. The volume of the cuboid is the product, dx × dy × dz. The combined volume of a collection of cuboids is the volume of their union and will be less than the sum of the individual volumes if any cuboids overlap.
Let C1,...,C50000 be a collection of 50000 axis-aligned cuboids such that Cn has parameters
x0 = S6n-5 modulo 10000
y0 = S6n-4 modulo 10000
z0 = S6n-3 modulo 10000
dx = 1 + (S6n-2 modulo 399)
dy = 1 + (S6n-1 modulo 399)
dz = 1 + (S6n modulo 399)
where S1,...,S300000 come from the "Lagged Fibonacci Generator":
Thus, C1 has parameters {(7,53,183),(94,369,56)}, C2 has parameters {(2383,3563,5079),(42,212,344)}, and so on.
The combined volume of the first 100 cuboids, C1,...,C100, is 723581599.
What is the combined volume of all 50000 cuboids, C1,...,C50000 ?
Let φ be Euler's totient function, i.e. for a natural number n, φ(n) is the number of k, 1 ≤ k ≤ n, for which gcd(k,n) = 1.
By iterating φ, each positive integer generates a decreasing chain of numbers ending in 1.
E.g. if we start with 5 the sequence 5,4,2,1 is generated.
Here is a listing of all chains with length 4:
Only two of these chains start with a prime, their sum is 12.
What is the sum of all primes less than 40000000 which generate a chain of length 25?
A positive integer with k (decimal) digits is called balanced if its first k/2 digits sum to the same value as its last k/2 digits, where x , pronounced ceiling of x, is the smallest integer ≥ x, thus π = 4 and 5 = 5.
So, for example, all palindromes are balanced, as is 13722.
Let T(n) be the sum of all balanced numbers less than 10n.
Thus: T(1) = 45, T(2) = 540 and T(5) = 334795890.
Find T(47) mod 315
Let A and B be bit strings (sequences of 0's and 1's).
If A is equal to the leftmost length(A) bits of B, then A is said to be a prefix of B.
For example, 00110 is a prefix of 001101001, but not of 00111 or 100110.
A prefix-free code of size n is a collection of n distinct bit strings such that no string is a prefix of any other. For example, this is a prefix-free code of size 6:
0000, 0001, 001, 01, 10, 11
Now suppose that it costs one penny to transmit a '0' bit, but four pence to transmit a '1'.
Then the total cost of the prefix-free code shown above is 35 pence, which happens to be the cheapest possible for the skewed pricing scheme in question.
In short, we write Cost(6) = 35.
What is Cost(109) ?
Let D0 be the two-letter string "Fa". For n≥1, derive Dn from Dn-1 by the string-rewriting rules:
"a" → "aRbFR"
"b" → "LFaLb"
Thus, D0 = "Fa", D1 = "FaRbFR", D2 = "FaRbFRRLFaLbFR", and so on.
These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit", "L" meaning "turn left 90 degrees", "R" meaning "turn right 90 degrees", and "a" and "b" being ignored. The initial position of the computer cursor is (0,0), pointing up towards (0,1).
Then Dn is an exotic drawing known as the Heighway Dragon of order n. For example, D10 is shown below; counting each "F" as one step, the highlighted spot at (18,16) is the position reached after 500 steps.
What is the position of the cursor after 1012 steps in D50 ?
Give your answer in the form x,y with no spaces.
Let Sn be the regular n-sided polygon – or shape – whose vertices vk (k = 1,2,…,n) have coordinates:
xk = cos( 2k-1/n ×180° )
yk = sin( 2k-1/n ×180° )
Each Sn is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.
The Minkowski sum, S+T, of two shapes S and T is the result of adding every point in S to every point in T, where point addition is performed coordinate-wise: (u, v) + (x, y) = (u+x, v+y).
For example, the sum of S3 and S4 is the six-sided shape shown in pink below:
How many sides does S1864 + S1865 + … + S1909 have?
For any two strings of digits, A and B, we define FA,B to be the sequence (A,B,AB,BAB,ABBAB,...) in which each term is the concatenation of the previous two.
Further, we define DA,B(n) to be the nth digit in the first term of FA,B that contains at least n digits.
Let A=1415926535, B=8979323846. We wish to find DA,B(35), say.
The first few terms of FA,B are:
Then DA,B(35) is the 35th digit in the fifth term, which is 9.
Now we use for A the first 100 digits of π behind the decimal point:
and for B the next hundred digits:
48111745028410270193852110555964462294895493038196 .
Find ∑n = 0,1,...,17 10n× DA,B((127+19n)×7n) .
For an integer n ≥ 4, we define the lower prime square root of n, denoted by lps(n), as the largest prime ≤ √n and the upper prime square root of n, ups(n), as the smallest prime ≥ √n.
So, for example, lps(4) = 2 = ups(4), lps(1000) = 31, ups(1000) = 37.
Let us call an integer n ≥ 4 semidivisible, if one of lps(n) and ups(n) divides n, but not both.
The sum of the semidivisible numbers not exceeding 15 is 30, the numbers are 8, 10 and 12.
15 is not semidivisible because it is a multiple of both lps(15) = 3 and ups(15) = 5.
As a further example, the sum of the 92 semidivisible numbers up to 1000 is 34825.
What is the sum of all semidivisible numbers not exceeding 999966663333 ?
Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market:
Beluga Caviar 5248 640
Christmas Cake 1312 1888
Gammon Joint 2624 3776
Vintage Port 5760 3776
Champagne Truffles 3936 5664
Although the suppliers try very hard to ship their goods in perfect condition, there is inevitably some spoilage - i.e. products gone bad.
The suppliers compare their performance using two types of statistic:
The five per-product spoilage rates for each supplier are equal to the number of products gone bad divided by the number of products supplied, for each of the five products in turn.
The overall spoilage rate for each supplier is equal to the total number of products gone bad divided by the total number of products provided by that supplier.
To their surprise, the suppliers found that each of the five per-product spoilage rates was worse (higher) for 'B' than for 'A' by the same factor (ratio of spoilage rates), m>1; and yet, paradoxically, the overall spoilage rate was worse for 'A' than for 'B', also by a factor of m.
There are thirty-five m>1 for which this surprising result could have occurred, the smallest of which is 1476/1475.
What's the largest possible value of m?
Give your answer as a fraction reduced to its lowest terms, in the form u/v.
A set of disks numbered 1 through 100 are placed in a line in random order.
What is the probability that we have a partial derangement such that exactly 22 prime number discs are found away from their natural positions?
(Any number of non-prime disks may also be found in or out of their natural positions.)
Give your answer rounded to 12 places behind the decimal point in the form 0.abcdefghijkl.
You probably know the game Fifteen Puzzle. Here, instead of numbered tiles, we have seven red tiles and eight blue tiles.
A move is denoted by the uppercase initial of the direction (Left, Right, Up, Down) in which the tile is slid, e.g. starting from configuration (S), by the sequence LULUR we reach the configuration (E):
(S) , (E)
For each path, its checksum is calculated by (pseudocode):
checksum = 0
checksum = (checksum × 243 + m1) mod 100 000 007
checksum = (checksum × 243 + mn) mod 100 000 007
where mk is the ASCII value of the kth letter in the move sequence and the ASCII values for the moves are:
For the sequence LULUR given above, the checksum would be 19761398.
Now, starting from configuration (S), find all shortest ways to reach configuration (T).
(S) , (T)
What is the sum of all checksums for the paths having the minimal length?
We shall call a fraction that cannot be cancelled down a resilient fraction.
Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = 4⁄11.
The resilience of a number d > 1 is then
φ(d)
d - 1
, where φ is Euler's totient function.
We further define the coresilience of a number n > 1 as C(n) =
n - φ(n)
n - 1
The coresilience of a prime p is C(p) =
p - 1
Find the sum of all composite integers 1 < n ≤ 2×1011, for which C(n) is a unit fraction.
A definition for an ellipse is:
Given a circle c with centre M and radius r and a point G such that d(G,M)<r, the locus of the points that are equidistant from c and G form an ellipse.
The construction of the points of the ellipse is shown below.
Given are the points M(-2000,1500) and G(8000,1500).
Given is also the circle c with centre M and radius 15000.
The locus of the points that are equidistant from G and c form an ellipse e.
From a point P outside e the two tangents t1 and t2 to the ellipse are drawn.
Let the points where t1 and t2 touch the ellipse be R and S.
For how many lattice points P is angle RPS greater than 45 degrees?
Consider the region constrained by 1 ≤ x and 0 ≤ y ≤ 1/x.
Let S1 be the largest square that can fit under the curve.
Let S2 be the largest square that fits in the remaining area, and so on.
Let the index of Sn be the pair (left, below) indicating the number of squares to the left of Sn and the number of squares below Sn.
The diagram shows some such squares labelled by number.
S2 has one square to its left and none below, so the index of S2 is (1,0).
It can be seen that the index of S32 is (1,1) as is the index of S50.
50 is the largest n for which the index of Sn is (1,1).
What is the largest n for which the index of Sn is (3,3)?
Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).
As an example, the image below shows a set of twenty points and a few such convex holes. The convex hole shown as a red heptagon has an area equal to 1049694.5 square units, which is the highest possible area for a convex hole on the given set of points.
For our example, we used the first 20 points (T2k−1, T2k), for k = 1,2,…,20, produced with the pseudo-random number generator:
Sn+1 = Sn2 mod 50515093
Tn = ( Sn mod 2000 ) − 1000
i.e. (527, 144), (−488, 732), (−454, −947), …
What is the maximum area for a convex hole on the set containing the first 500 points in the pseudo-random sequence?
Specify your answer including one digit after the decimal point.
A small child has a "number caterpillar" consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers 1 to 40 in order.
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up the pieces at random and places them in the correct order.
As the caterpillar is built up in this way, it forms distinct segments that gradually merge together.
The number of segments starts at zero (no pieces placed), generally increases up to about eleven or twelve, then tends to drop again before finishing at a single segment (all pieces placed).
Piece Placed Segments So Far
Let M be the maximum number of segments encountered during a random tidy-up of the caterpillar.
For a caterpillar of ten pieces, the number of possibilities for each M is
M Possibilities
so the most likely value of M is 3 and the average value is 385643⁄113400 = 3.400732, rounded to six decimal places.
The most likely value of M for a forty-piece caterpillar is 11; but what is the average value of M?
Give your answer rounded to six decimal places.
We define the rounded-square-root of a positive integer n as the square root of n rounded to the nearest integer.
The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of n:
Let d be the number of digits of the number n.
If d is odd, set x0 = 2×10(d-1)⁄2.
If d is even, set x0 = 7×10(d-2)⁄2.
until xk+1 = xk.
As an example, let us find the rounded-square-root of n = 4321.
n has 4 digits, so x0 = 7×10(4-2)⁄2 = 70.
Since x2 = x1, we stop here.
So, after just two iterations, we have found that the rounded-square-root of 4321 is 66 (the actual square root is 65.7343137…).
The number of iterations required when using this method is surprisingly low.
For example, we can find the rounded-square-root of a 5-digit integer (10,000 ≤ n ≤ 99,999) with an average of 3.2102888889 iterations (the average value was rounded to 10 decimal places).
Using the procedure described above, what is the average number of iterations required to find the rounded-square-root of a 14-digit number (1013 ≤ n < 1014)?
Give your answer rounded to 10 decimal places.
Note: The symbols x and x represent the floor function and ceiling function respectively.
Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
Assuming that the only type of available tatami has dimensions 1×2, there are obviously some limitations for the shape and size of the rooms that can be covered.
For this problem, we consider only rectangular rooms with integer dimensions a, b and even size s = a·b.
We use the term 'size' to denote the floor surface area of the room, and — without loss of generality — we add the condition a ≤ b.
There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet.
For example, consider the two arrangements below for a 4×4 room:
The arrangement on the left is acceptable, whereas the one on the right is not: a red "X" in the middle, marks the point where four tatami meet.
Because of this rule, certain even-sized rooms cannot be covered with tatami: we call them tatami-free rooms.
Further, we define T(s) as the number of tatami-free rooms of size s.
The smallest tatami-free room has size s = 70 and dimensions 7×10.
All the other rooms of size s = 70 can be covered with tatami; they are: 1×70, 2×35 and 5×14.
Hence, T(70) = 1.
Similarly, we can verify that T(1320) = 5 because there are exactly 5 tatami-free rooms of size s = 1320:
20×66, 22×60, 24×55, 30×44 and 33×40.
In fact, s = 1320 is the smallest room-size s for which T(s) = 5.
Find the smallest room-size s for which T(s) = 200.
A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:
Uses the digits 1 through 9, in that order and exactly once each.
Any successive digits can be concatenated (for example, using the digits 2, 3 and 4 we obtain the number 234).
Only the four usual binary arithmetic operations (addition, subtraction, multiplication and division) are allowed.
Each operation can be used any number of times, or not at all.
Unary minus is not allowed.
Any number of (possibly nested) parentheses may be used to define the order of operations.
For example, 42 is reachable, since (1/23) * ((4*5)-6) * (78-9) = 42.
What is the sum of all positive reachable integers?
You are given a unique investment opportunity.
Starting with £1 of capital, you can choose a fixed proportion, f, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.
Your return is double your bet for heads and you lose your bet for tails.
For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and if the second toss is tails, you have £1.125.
Choosing f to maximize your chances of having at least £1,000,000,000 after 1,000 flips, what is the chance that you become a billionaire?
All computations are assumed to be exact (no rounding), but give your answer rounded to 12 digits behind the decimal point in the form 0.abcdefghijkl.
For each integer p > 1 coprime to 10 there is a positive divisibility multiplier m < p which preserves divisibility by p for the following function on any positive integer, n:
f(n) = (all but the last digit of n) + (the last digit of n) * m
That is, if m is the divisibility multiplier for p, then f(n) is divisible by p if and only if n is divisible by p.
(When n is much larger than p, f(n) will be less than n and repeated application of f provides a multiplicative divisibility test for p.)
For example, the divisibility multiplier for 113 is 34.
f(76275) = 7627 + 5 * 34 = 7797 : 76275 and 7797 are both divisible by 113
f(12345) = 1234 + 5 * 34 = 1404 : 12345 and 1404 are both not divisible by 113
The sum of the divisibility multipliers for the primes that are coprime to 10 and less than 1000 is 39517. What is the sum of the divisibility multipliers for the primes that are coprime to 10 and less than 107?
Let us define a balanced sculpture of order n as follows:
A polyomino made up of n+1 tiles known as the blocks (n tiles)
and the plinth (remaining tile);
the plinth has its centre at position (x = 0, y = 0);
the blocks have y-coordinates greater than zero (so the plinth is the unique lowest tile);
the centre of mass of all the blocks, combined, has x-coordinate equal to zero.
When counting the sculptures, any arrangements which are simply reflections about the y-axis, are not counted as distinct. For example, the 18 balanced sculptures of order 6 are shown below; note that each pair of mirror images (about the y-axis) is counted as one sculpture:
There are 964 balanced sculptures of order 10 and 360505 of order 15.
How many balanced sculptures are there of order 18?
A modified Collatz sequence of integers is obtained from a starting value a1 in the following way:
an+1 = an/3 if an is divisible by 3. We shall denote this as a large downward step, "D".
an+1 = (4an + 2)/3 if an divided by 3 gives a remainder of 1. We shall denote this as an upward step, "U".
an+1 = (2an - 1)/3 if an divided by 3 gives a remainder of 2. We shall denote this as a small downward step, "d".
The sequence terminates when some an = 1.
Given any integer, we can list out the sequence of steps.
For instance if a1=231, then the sequence {an}={231,77,51,17,11,7,10,14,9,3,1} corresponds to the steps "DdDddUUdDD".
Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....".
For instance, if a1=1004064, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.
In fact, 1004064 is the smallest possible a1 > 106 that begins with the sequence DdDddUUdDD.
What is the smallest a1 > 1015 that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?
You are given a pizza (perfect circle) that has been cut into m·n equal pieces and you want to have exactly one topping on each slice.
Let f(m,n) denote the number of ways you can have toppings on the pizza with m different toppings (m ≥ 2), using each topping on exactly n slices (n ≥ 1).
Reflections are considered distinct, rotations are not.
Thus, for instance, f(2,1) = 1, f(2,2) = f(3,1) = 2 and f(3,2) = 16.
f(3,2) is shown below:
Find the sum of all f(m,n) such that f(m,n) ≤ 1015.
The quadtree encoding allows us to describe a 2N×2N black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:
the first bit deals with the complete 2N×2N region;
"0" denotes a split:
the current 2n×2n region is divided into 4 sub-regions of dimension 2n-1×2n-1,
the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order;
"10" indicates that the current region contains only black pixels;
"11" indicates that the current region contains only white pixels.
Consider the following 4×4 image (colored marks denote places where a split can occur):
This image can be described by several sequences, for example : "001010101001011111011010101010", of length 30, or
"0100101111101110", of length 16, which is the minimal sequence for this image.
For a positive integer N, define DN as the 2N×2N image with the following coloring scheme:
the pixel with coordinates x = 0, y = 0 corresponds to the bottom left pixel,
if (x - 2N-1)2 + (y - 2N-1)2 ≤ 22N-2 then the pixel is black,
otherwise the pixel is white.
What is the length of the minimal sequence describing D24 ?
In a very simplified form, we can consider proteins as strings consisting of hydrophobic (H) and polar (P) elements, e.g. HHPPHHHPHHPH.
For this problem, the orientation of a protein is important; e.g. HPP is considered distinct from PPH. Thus, there are 2n distinct proteins consisting of n elements.
When one encounters these strings in nature, they are always folded in such a way that the number of H-H contact points is as large as possible, since this is energetically advantageous.
As a result, the H-elements tend to accumulate in the inner part, with the P-elements on the outside.
Natural proteins are folded in three dimensions of course, but we will only consider protein folding in two dimensions.
The figure below shows two possible ways that our example protein could be folded (H-H contact points are shown with red dots).
The folding on the left has only six H-H contact points, thus it would never occur naturally.
On the other hand, the folding on the right has nine H-H contact points, which is optimal for this string.
Assuming that H and P elements are equally likely to occur in any position along the string, the average number of H-H contact points in an optimal folding of a random protein string of length 8 turns out to be 850 / 28=3.3203125.
What is the average number of H-H contact points in an optimal folding of a random protein string of length 15?
Give your answer using as many decimal places as necessary for an exact result.
Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.
We'll consider the three-heap normal-play version of Nim, which works as follows:
- At the start of the game there are three heaps of stones.
- On his turn the player removes any positive number of stones from any single heap.
- The first player unable to move (because no stones remain) loses.
If (n1,n2,n3) indicates a Nim position consisting of heaps of size n1, n2 and n3 then there is a simple function X(n1,n2,n3) — that you may look up or attempt to deduce for yourself — that returns:
zero if, with perfect strategy, the player about to move will eventually lose; or
non-zero if, with perfect strategy, the player about to move will eventually win.
For example X(1,2,3) = 0 because, no matter what the current player does, his opponent can respond with a move that leaves two heaps of equal size, at which point every move by the current player can be mirrored by his opponent until no stones remain; so the current player loses. To illustrate:
- current player moves to (1,2,1)
- opponent moves to (1,0,1)
- opponent moves to (0,0,0), and so wins.
For how many positive integers n ≤ 230 does X(n,2n,3n) = 0 ?
The following game is a classic example of Combinatorial Game Theory:
Two players start with a strip of n white squares and they take alternate turns.
On each turn, a player picks two contiguous white squares and paints them black.
The first player who cannot make a move loses.
If n = 1, there are no valid moves, so the first player loses automatically.
If n = 2, there is only one valid move, after which the second player loses.
If n = 3, there are two valid moves, but both leave a situation where the second player loses.
If n = 4, there are three valid moves for the first player; she can win the game by painting the two middle squares.
If n = 5, there are four valid moves for the first player (shown below in red); but no matter what she does, the second player (blue) wins.
So, for 1 ≤ n ≤ 5, there are 3 values of n for which the first player can force a win.
Similarly, for 1 ≤ n ≤ 50, there are 40 values of n for which the first player can force a win.
For 1 ≤ n ≤ 1 000 000, how many values of n are there for which the first player can force a win?
Alice and Bob play the game Nim Square.
Nim Square is just like ordinary three-heap normal play Nim, but the players may only remove a square number of stones from a heap.
The number of stones in the three heaps is represented by the ordered triple (a,b,c).
If 0≤a≤b≤c≤29 then the number of losing positions for the next player is 1160.
Find the number of losing positions for the next player if 0≤a≤b≤c≤100 000.
The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a 500 m by 500 m square area, but they will possess only that area which they wall in. 251001 posts have been placed in a rectangular grid with 1 meter spacing. The wall must be a closed series of straight lines, each line running from post to post.
The bigger countries of course have built a 2000 m wall enclosing the entire 250 000 m2 area. The Duchy of Grand Fenwick, has a tighter budget, and has asked you (their Royal Programmer) to compute what shape would get best maximum enclosed-area/wall-length ratio.
You have done some preliminary calculations on a sheet of paper. For a 2000 meter wall enclosing the 250 000 m2 area the enclosed-area/wall-length ratio is 125.
Although not allowed , but to get an idea if this is anything better: if you place a circle inside the square area touching the four sides the area will be equal to π*2502 m2 and the perimeter will be π*500 m, so the enclosed-area/wall-length ratio will also be 125.
However, if you cut off from the square four triangles with sides 75 m, 75 m and 75√2 m the total area becomes 238750 m2 and the perimeter becomes 1400+300√2 m. So this gives an enclosed-area/wall-length ratio of 130.87, which is significantly better.
Find the maximum enclosed-area/wall-length ratio.
Give your answer rounded to 8 places behind the decimal point in the form abc.defghijk.
Sam and Max are asked to transform two digital clocks into two "digital root" clocks.
A digital root clock is a digital clock that calculates digital roots step by step.
When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.
For example, if the clock is fed the number 137, it will show: "137" → "11" → "2" and then it will go black, waiting for the next number.
Every digital number consists of some light segments: three horizontal (top, middle, bottom) and four vertical (top-left, top-right, bottom-left, bottom-right).
Number "1" is made of vertical top-right and bottom-right, number "4" is made by middle horizontal and vertical top-left, top-right and bottom-right. Number "8" lights them all.
The clocks consume energy only when segments are turned on/off.
To turn on a "2" will cost 5 transitions, while a "7" will cost only 4 transitions.
Sam and Max built two different clocks.
Sam's clock is fed e.g. number 137: the clock shows "137", then the panel is turned off, then the next number ("11") is turned on, then the panel is turned off again and finally the last number ("2") is turned on and, after some time, off.
For the example, with number 137, Sam's clock requires:
"137" : (2 + 5 + 4) × 2 = 22 transitions ("137" on/off).
"11" : (2 + 2) × 2 = 8 transitions ("11" on/off).
"2" : (5) × 2 = 10 transitions ("2" on/off).
For a grand total of 40 transitions.
Max's clock works differently. Instead of turning off the whole panel, it is smart enough to turn off only those segments that won't be needed for the next number.
For number 137, Max's clock requires:
2 + 5 + 4 = 11 transitions ("137" on)
7 transitions (to turn off the segments that are not needed for number "11").
0 transitions (number "11" is already turned on correctly)
3 transitions (to turn off the first "1" and the bottom part of the second "1";
the top part is common with number "2").
4 tansitions (to turn on the remaining segments in order to get a "2")
5 transitions (to turn off number "2").
Of course, Max's clock consumes less power than Sam's one.
The two clocks are fed all the prime numbers between A = 107 and B = 2×107.
Find the difference between the total number of transitions needed by Sam's clock and that needed by Max's one.
A series of three rooms are connected to each other by automatic doors.
Each door is operated by a security card. Once you enter a room the door automatically closes and that security card cannot be used again. A machine at the start will dispense an unlimited number of cards, but each room (including the starting room) contains scanners and if they detect that you are holding more than three security cards or if they detect an unattended security card on the floor, then all the doors will become permanently locked. However, each room contains a box where you may safely store any number of security cards for use at a later stage.
If you simply tried to travel through the rooms one at a time then as you entered room 3 you would have used all three cards and would be trapped in that room forever!
However, if you make use of the storage boxes, then escape is possible. For example, you could enter room 1 using your first card, place one card in the storage box, and use your third card to exit the room back to the start. Then after collecting three more cards from the dispensing machine you could use one to enter room 1 and collect the card you placed in the box a moment ago. You now have three cards again and will be able to travel through the remaining three doors. This method allows you to travel through all three rooms using six security cards in total.
It is possible to travel through six rooms using a total of 123 security cards while carrying a maximum of 3 cards.
Let C be the maximum number of cards which can be carried at any time.
Let R be the number of rooms to travel through.
Let M(C,R) be the minimum number of cards required from the dispensing machine to travel through R rooms carrying up to a maximum of C cards at any time.
For example, M(3,6)=123 and M(4,6)=23.
And, ΣM(C,6)=146 for 3 ≤ C ≤ 4.
You are given that ΣM(C,10)=10382 for 3 ≤ C ≤ 10.
Find ΣM(C,30) for 3 ≤ C ≤ 40.
We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions. Each number (question) we ask, has a cost equal to the number asked and we get one of three possible answers:
"Your guess is lower than the hidden number", or
"Yes, that's it!", or
"Your guess is higher than the hidden number".
Given the value of n, an optimal strategy minimizes the total cost (i.e. the sum of all the questions asked) for the worst possible case. E.g.
If n=3, the best we can do is obviously to ask the number "2". The answer will immediately lead us to find the hidden number (at a total cost = 2).
If n=8, we might decide to use a "binary search" type of strategy: Our first question would be "4" and if the hidden number is higher than 4 we will need one or two additional questions.
Let our second question be "6". If the hidden number is still higher than 6, we will need a third question in order to discriminate between 7 and 8.
Thus, our third question will be "7" and the total cost for this worst-case scenario will be 4+6+7=17.
We can improve considerably the worst-case cost for n=8, by asking "5" as our first question.
If we are told that the hidden number is higher than 5, our second question will be "7", then we'll know for certain what the hidden number is (for a total cost of 5+7=12).
If we are told that the hidden number is lower than 5, our second question will be "3" and if the hidden number is lower than 3 our third question will be "1", giving a total cost of 5+3+1=9.
Since 12>9, the worst-case cost for this strategy is 12. That's better than what we achieved previously with the "binary search" strategy; it is also better than or equal to any other strategy.
So, in fact, we have just described an optimal strategy for n=8.
Let C(n) be the worst-case cost achieved by an optimal strategy for n, as described above.
Thus C(1) = 0, C(2) = 1, C(3) = 2 and C(8) = 12.
Similarly, C(100) = 400 and C(n) = 17575.
Find C(n).
N×N disks are placed on a square game board. Each disk has a black side and white side.
At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus 2×N-1 disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.
It can be proven that 3 is the minimal number of turns to finish this game.
The bottom left disk on the N×N board has coordinates (0,0);
the bottom right disk has coordinates (N-1,0) and the top left disk has coordinates (0,N-1).
Let CN be the following configuration of a board with N×N disks:
A disk at (x,y) satisfying , shows its black side; otherwise, it shows its white side. C5 is shown above.
Let T(N) be the minimal number of turns to finish a game starting from configuration CN or 0 if configuration CN is unsolvable.
We have shown that T(5)=3. You are also given that T(10)=29 and T(1 000)=395253.
Find .
All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2ix3j, where i,j ≥ 0.
Let's consider only those such partitions where none of the terms can divide any of the other terms.
For example, the partition of 17 = 2 + 6 + 9 = (21x30 + 21x31 + 20x32) would not be valid since 2 can divide 6. Neither would the partition 17 = 16 + 1 = (24x30 + 20x30) since 1 can divide 16. The only valid partition of 17 would be 8 + 9 = (23x30 + 20x32).
Many integers have more than one valid partition, the first being 11 having the following two partitions.
11 = 2 + 9 = (21x30 + 20x32)
Let's define P(n) as the number of valid partitions of n. For example, P(11) = 2.
Let's consider only the prime integers q which would have a single valid partition such as P(17).
The sum of the primes q <100 such that P(q)=1 equals 233.
Find the sum of the primes q <1000000 such that P(q)=1.
A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the correct order.
To rearrange the carriages they are all shunted on to a large rotating turntable. After the carriages are uncoupled at a specific point the train moves off the turntable pulling the carriages still attached with it. The remaining carriages are rotated 180 degrees. All of the carriages are then rejoined and this process is repeated as often as necessary in order to obtain the least number of uses of the turntable.
Some arrangements, such as ADCB, can be solved easily: the carriages are separated between A and D, and after DCB are rotated the correct order has been achieved.
However, Simple Simon, the train driver, is not known for his efficiency, so he always solves the problem by initially getting carriage A in the correct place, then carriage B, and so on.
Using four carriages, the worst possible arrangements for Simon, which we shall call maximix arrangements, are DACB and DBAC; each requiring him five rotations (although, using the most efficient approach, they could be solved using just three rotations). The process he uses for DACB is shown below.
It can be verified that there are 24 maximix arrangements for six carriages, of which the tenth lexicographic maximix arrangement is DFAECB.
Find the 2011th lexicographic maximix arrangement for eleven carriages.
A rectangular sheet of grid paper with integer dimensions w × h is given. Its grid spacing is 1.
When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.
For example, from a sheet with dimensions 9 × 4 , we can make rectangles with dimensions 18 × 2, 12 × 3 and 6 × 6 by cutting and rearranging as below:
Similarly, from a sheet with dimensions 9 × 8 , we can make rectangles with dimensions 18 × 4 and 12 × 6 .
For a pair w and h, let F(w,h) be the number of distinct rectangles that can be made from a sheet with dimensions w × h .
For example, F(2,1) = 0, F(2,2) = 1, F(9,4) = 3 and F(9,8) = 2.
Note that rectangles congruent to the initial one are not counted in F(w,h).
Note also that rectangles with dimensions w × h and dimensions h × w are not considered distinct.
For an integer N, let G(N) be the sum of F(w,h) for all pairs w and h which satisfy 0 < h ≤ w ≤ N.
We can verify that G(10) = 55, G(103) = 971745 and G(105) = 9992617687.
Find G(1012). Give your answer modulo 108.
"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep would cross over and become white; and when one of the black sheep bleated, one of the white sheep would cross over and become black."
en.wikisource.org
Initially each flock consists of n sheep. Each sheep (regardless of colour) is equally likely to be the next sheep to bleat. After a sheep has bleated and a sheep from the other flock has crossed over, Peredur may remove a number of white sheep in order to maximize the expected final number of black sheep. Let E(n) be the expected final number of black sheep if Peredur uses an optimal strategy.
You are given that E(5) = 6.871346 rounded to 6 places behind the decimal point.
Find E(10 000) and give your answer rounded to 6 places behind the decimal point.
One variant of N.G. de Bruijn's silver dollar game can be described as follows:
On a strip of squares a number of coins are placed, at most one coin per square. Only one coin, called the silver dollar, has any value. Two players take turns making moves. At each turn a player must make either a regular or a special move.
A regular move consists of selecting one coin and moving it one or more squares to the left. The coin cannot move out of the strip or jump on or over another coin.
Alternatively, the player can choose to make the special move of pocketing the leftmost coin rather than making a regular move. If no regular moves are possible, the player is forced to pocket the leftmost coin.
The winner is the player who pockets the silver dollar.
A winning configuration is an arrangement of coins on the strip where the first player can force a win no matter what the second player does.
Let W(n,c) be the number of winning configurations for a strip of n squares, c worthless coins and one silver dollar.
You are given that W(10,2) = 324 and W(100,10) = 1514704946113500.
Find W(1 000 000, 100) modulo the semiprime 1000 036 000 099 (= 1 000 003 · 1 000 033).
We define the Matrix Sum of a matrix as the maximum sum of matrix elements with each element being the only one in his row and column. For example, the Matrix Sum of the matrix below equals 3315 ( = 863 + 383 + 343 + 959 + 767):
497 383 563 79 973
287 63 343 169 583
Find the Matrix Sum of:
7 53 183 439 863 497 383 563 79 973 287 63 343 169 583
447 283 463 29 23 487 463 993 119 883 327 493 423 159 743
217 623 3 399 853 407 103 983 89 463 290 516 212 462 350
870 456 192 162 593 473 915 45 989 873 823 965 425 329 803
322 148 972 962 286 255 941 541 265 323 925 281 601 95 973
445 721 11 525 473 65 511 164 138 672 18 428 154 448 848
821 461 843 513 17 901 711 993 293 157 274 94 192 156 574
34 124 4 878 450 476 712 914 838 669 875 299 823 329 699
813 883 451 509 615 77 281 613 459 205 380 274 302 35 805
Each one of the 25 sheep in a flock must be tested for a rare virus, known to affect 2% of the sheep population. An accurate and extremely sensitive PCR test exists for blood samples, producing a clear positive / negative result, but it is very time-consuming and expensive.
Because of the high cost, the vet-in-charge suggests that instead of performing 25 separate tests, the following procedure can be used instead:
The sheep are split into 5 groups of 5 sheep in each group. For each group, the 5 samples are mixed together and a single test is performed. Then,
If the result is negative, all the sheep in that group are deemed to be virus-free.
If the result is positive, 5 additional tests will be performed (a separate test for each animal) to determine the affected individual(s).
Since the probability of infection for any specific animal is only 0.02, the first test (on the pooled samples) for each group will be:
Negative (and no more tests needed) with probability 0.985 = 0.9039207968.
Positive (5 additional tests needed) with probability 1 - 0.9039207968 = 0.0960792032.
Thus, the expected number of tests for each group is 1 + 0.0960792032 × 5 = 1.480396016.
Consequently, all 5 groups can be screened using an average of only 1.480396016 × 5 = 7.40198008 tests, which represents a huge saving of more than 70% !
Although the scheme we have just described seems to be very efficient, it can still be improved considerably (always assuming that the test is sufficiently sensitive and that there are no adverse effects caused by mixing different samples). E.g.:
We may start by running a test on a mixture of all the 25 samples. It can be verified that in about 60.35% of the cases this test will be negative, thus no more tests will be needed. Further testing will only be required for the remaining 39.65% of the cases.
If we know that at least one animal in a group of 5 is infected and the first 4 individual tests come out negative, there is no need to run a test on the fifth animal (we know that it must be infected).
We can try a different number of groups / different number of animals in each group, adjusting those numbers at each level so that the total expected number of tests will be minimised.
To simplify the very wide range of possibilities, there is one restriction we place when devising the most cost-efficient testing scheme: whenever we start with a mixed sample, all the sheep contributing to that sample must be fully screened (i.e. a verdict of infected / virus-free must be reached for all of them) before we start examining any other animals.
For the current example, it turns out that the most cost-efficient testing scheme (we'll call it the optimal strategy) requires an average of just 4.155452 tests!
Using the optimal strategy, let T(s,p) represent the average number of tests needed to screen a flock of s sheep for a virus having probability p to be present in any individual.
Thus, rounded to six decimal places, T(25, 0.02) = 4.155452 and T(25, 0.10) = 12.702124.
Find ΣT(10000, p) for p=0.01, 0.02, 0.03, ... 0.50.
A moon could be described by the sphere C(r) with centre (0,0,0) and radius r.
There are stations on the moon at the points on the surface of C(r) with integer coordinates. The station at (0,0,r) is called North Pole station, the station at (0,0,-r) is called South Pole station.
All stations are connected with each other via the shortest road on the great arc through the stations. A journey between two stations is risky. If d is the length of the road between two stations, (d/(π r))2 is a measure for the risk of the journey (let us call it the risk of the road). If the journey includes more than two stations, the risk of the journey is the sum of risks of the used roads.
A direct journey from the North Pole station to the South Pole station has the length πr and risk 1. The journey from the North Pole station to the South Pole station via (0,r,0) has the same length, but a smaller risk: (½πr/(πr))2+(½πr/(πr))2=0.5.
The minimal risk of a journey from the North Pole station to the South Pole station on C(r) is M(r).
You are given that M(7)=0.1784943998 rounded to 10 digits behind the decimal point.
Find ∑M(2n-1) for 1≤n≤15.
Give your answer rounded to 10 digits behind the decimal point in the form a.bcdefghijk.
A cyclic number with n digits has a very interesting property:
When it is multiplied by 1, 2, 3, 4, ... n, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!
The smallest cyclic number is the 6-digit number 142857 :
142857 × 1 = 142857
The next cyclic number is 0588235294117647 with 16 digits :
0588235294117647 × 1 = 0588235294117647
0588235294117647 × 16 = 9411764705882352
Note that for cyclic numbers, leading zeros are important.
There is only one cyclic number for which, the eleven leftmost digits are 00000000137 and the five rightmost digits are 56789 (i.e., it has the form 00000000137...56789 with an unknown number of digits in the middle). Find the sum of all its digits.
An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered 1, 2, 3, etc.), and each floor contains an infinite number of rooms (numbered 1, 2, 3, etc.).
Initially the hotel is empty. Hilbert declares a rule on how the nth person is assigned a room: person n gets the first vacant room in the lowest numbered floor satisfying either of the following:
the floor is empty
the floor is not empty, and if the latest person taking a room in that floor is person m, then m + n is a perfect square
Person 1 gets room 1 in floor 1 since floor 1 is empty.
Person 2 does not get room 2 in floor 1 since 1 + 2 = 3 is not a perfect square.
Person 2 instead gets room 1 in floor 2 since floor 2 is empty.
Person 3 gets room 2 in floor 1 since 1 + 3 = 4 is a perfect square.
Eventually, every person in the line gets a room in the hotel.
Define P(f, r) to be n if person n occupies room r in floor f, and 0 if no person occupies the room. Here are a few examples:
P(1, 1) = 1
P(10, 20) = 440
P(25, 75) = 4863
P(99, 100) = 19454
Find the sum of all P(f, r) for all positive f and r such that f × r = 71328803586048 and give the last 8 digits as your answer.
Consider the number 54.
54 can be factored in 7 distinct ways into one or more factors larger than 1:
54, 2×27, 3×18, 6×9, 3×3×6, 2×3×9 and 2×3×3×3.
If we require that the factors are all squarefree only two ways remain: 3×3×6 and 2×3×3×3.
Let's call Fsf(n) the number of ways n can be factored into one or more squarefree factors larger than 1, so Fsf(54)=2.
Let S(n) be ∑Fsf(k) for k=2 to n.
S(100)=193.
Find S(10 000 000 000).
A cubic Bézier curve is defined by four points: P0, P1, P2 and P3.
The curve is constructed as follows:
On the segments P0P1, P1P2 and P2P3 the points Q0,Q1 and Q2 are drawn such that P0Q0/P0P1=P1Q1/P1P2=P2Q2/P2P3=t (t in [0,1]).
On the segments Q0Q1 and Q1Q2 the points R0 and R1 are drawn such that Q0R0/Q0Q1=Q1R1/Q1Q2=t for the same value of t.
On the segment R0R1 the point B is drawn such that R0B/R0R1=t for the same value of t.
The Bézier curve defined by the points P0, P1, P2, P3 is the locus of B as Q0 takes all possible positions on the segment P0P1. (Please note that for all points the value of t is the same.)
In the applet to the right you can drag the points P0, P1, P2 and P3 to see what the Bézier curve (green curve) defined by those points looks like. You can also drag the point Q0 along the segment P0P1.
From the construction it is clear that the Bézier curve will be tangent to the segments P0P1 in P0 and P2P3 in P3.
A cubic Bézier curve with P0=(1,0), P1=(1,v), P2=(v,1) and P3=(0,1) is used to approximate a quarter circle.
The value v>0 is chosen such that the area enclosed by the lines OP0, OP3 and the curve is equal to π/4 (the area of the quarter circle).
By how many percent does the length of the curve differ from the length of the quarter circle?
That is, if L is the length of the curve, calculate 100*(L-π/2)/(π/2).
Give your answer rounded to 10 digits behind the decimal point.
Two players, Anton and Bernhard, are playing the following game.
There is one pile of n stones.
The first player may remove any positive number of stones, but not the whole pile.
Thereafter, each player may remove at most twice the number of stones his opponent took on the previous move.
The player who removes the last stone wins.
E.g. n=5
If the first player takes anything more than one stone the next player will be able to take all remaining stones.
If the first player takes one stone, leaving four, his opponent will take also one stone, leaving three stones.
The first player cannot take all three because he may take at most 2x1=2 stones. So let's say he takes also one stone, leaving 2. The second player can take the two remaining stones and wins.
So 5 is a losing position for the first player.
For some winning positions there is more than one possible move for the first player.
E.g. when n=17 the first player can remove one or four stones.
Let M(n) be the maximum number of stones the first player can take from a winning position at his first turn and M(n)=0 for any other position.
∑M(n) for n≤100 is 728.
Find ∑M(n) for n≤1018. Give your answer modulo 108.
Bozo sort, not to be confused with the slightly less efficient bogo sort, consists out of checking if the input sequence is sorted and if not swapping randomly two elements. This is repeated until eventually the sequence is sorted.
If we consider all permutations of the first 4 natural numbers as input the expectation value of the number of swaps, averaged over all 4! input sequences is 24.75.
The already sorted sequence takes 0 steps.
In this problem we consider the following variant on bozo sort.
If the sequence is not in order we pick three elements at random and shuffle these three elements randomly.
All 3!=6 permutations of those three elements are equally likely.
The already sorted sequence will take 0 steps.
If we consider all permutations of the first 4 natural numbers as input the expectation value of the number of shuffles, averaged over all 4! input sequences is 27.5.
Consider as input sequences the permutations of the first 11 natural numbers.
Averaged over all 11! input sequences, what is the expected number of shuffles this sorting algorithm will perform?
Give your answer rounded to the nearest integer.
An integer partition of a number n is a way of writing n as a sum of positive integers.
Partitions that differ only in the order of their summands are considered the same. A partition of n into distinct parts is a partition of n in which every part occurs at most once.
The partitions of 5 into distinct parts are:
5, 4+1 and 3+2.
Let f(n) be the maximum product of the parts of any such partition of n into distinct parts and let m(n) be the number of elements of any such partition of n with that product.
So f(5)=6 and m(5)=2.
For n=10 the partition with the largest product is 10=2+3+5, which gives f(10)=30 and m(10)=3.
And their product, f(10)·m(10) = 30·3 = 90
It can be verified that
∑f(n)·m(n) for 1 ≤ n ≤ 100 = 1683550844462.
Find ∑f(n)·m(n) for 1 ≤ n ≤ 1014.
Give your answer modulo 982451653, the 50 millionth prime.
Let Sn be an integer sequence produced with the following pseudo-random number generator:
Let A(i, j) be the minimum of the numbers Si, Si+1, ... , Sj for i ≤ j.
Let M(N) = ΣA(i, j) for 1 ≤ i ≤ j ≤ N.
We can verify that M(10) = 432256955 and M(10 000) = 3264567774119.
Find M(2 000 000 000).
Consider the following set of dice with nonstandard pips:
Die A: 1 4 4 4 4 4
Die B: 2 2 2 5 5 5
Die C: 3 3 3 3 3 6
A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
If the first player picks die A and the second player picks die B we get
P(second player wins) = 7/12 > 1/2
If the first player picks die B and the second player picks die C we get
If the first player picks die C and the second player picks die A we get
P(second player wins) = 25/36 > 1/2
So whatever die the first player picks, the second player can pick another die and have a larger than 50% chance of winning.
A set of dice having this property is called a nontransitive set of dice.
We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:
There are three six-sided dice with each side having between 1 and N pips, inclusive.
Dice with the same set of pips are equal, regardless of which side on the die the pips are located.
The same pip value may appear on multiple dice; if both players roll the same value neither player wins.
The sets of dice {A,B,C}, {B,C,A} and {C,A,B} are the same set.
For N = 7 we find there are 9780 such sets.
How many are there for N = 30 ?
An m×n maze is an m×n rectangular grid with walls placed between grid cells such that there is exactly one path from the top-left square to any other square.
The following are examples of a 9×12 maze and a 15×20 maze:
Let C(m,n) be the number of distinct m×n mazes. Mazes which can be formed by rotation and reflection from another maze are considered distinct.
It can be verified that C(1,1) = 1, C(2,2) = 4, C(3,4) = 2415, and C(9,12) = 2.5720e46 (in scientific notation rounded to 5 significant digits).
Find C(100,500) and write your answer in scientific notation rounded to 5 significant digits.
When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is 1234567891011 then the answer format would be 1.2346e12.
A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit. A polygon consists of at least three sides and does not self-intersect.
A set S of positive numbers is said to generate a polygon P if:
no two sides of P are the same length,
the length of every side of P is in S, and
S contains no other value.
The set {3, 4, 5} generates a polygon with sides 3, 4, and 5 (a triangle).
The set {6, 9, 11, 24} generates a polygon with sides 6, 9, 11, and 24 (a quadrilateral).
The sets {1, 2, 3} and {2, 3, 4, 9} do not generate any polygon at all.
Consider the sequence s, defined as follows:
s1 = 1, s2 = 2, s3 = 3
sn = sn-1 + sn-3 for n > 3.
Let Un be the set {s1, s2, ..., sn}. For example, U10 = {1, 2, 3, 4, 6, 9, 13, 19, 28, 41}.
Let f(n) be the number of subsets of Un which generate at least one polygon.
For example, f(5) = 7, f(10) = 501 and f(25) = 18635853.
Find the last 9 digits of f(1018).
Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).
E.g.: a(5) = a(1012) = 0, a(6) = a(1102) = 1, a(7) = a(1112) = 2
Define the sequence b(n) = (-1)a(n).
This sequence is called the Rudin-Shapiro sequence.
Also consider the summatory sequence of b(n): .
The first couple of values of these sequences are:
n 0 1 2 3 4 5 6 7
a(n) 0 0 0 1 0 0 1 2
b(n) 1 1 1 -1 1 1 -1 1
s(n) 1 2 3 2 3 4 3 4
The sequence s(n) has the remarkable property that all elements are positive and every positive integer k occurs exactly k times.
Define g(t,c), with 1 ≤ c ≤ t, as the index in s(n) for which t occurs for the c'th time in s(n).
E.g.: g(3,3) = 6, g(4,2) = 7 and g(54321,12345) = 1220847710.
Let F(n) be the fibonacci sequence defined by:
F(0)=F(1)=1 and
F(n)=F(n-1)+F(n-2) for n>1.
Define GF(t)=g(F(t),F(t-1)).
Find ΣGF(t) for 2≤t≤45.
For any triangle T in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside T.
For a given n, consider triangles T such that:
- the vertices of T have integer coordinates with absolute value ≤ n, and
- the foci1 of the largest-area ellipse inside T are (√13,0) and (-√13,0).
Let A(n) be the sum of the areas of all such triangles.
For example, if n = 8, there are two such triangles. Their vertices are (-4,-3),(-4,3),(8,0) and (4,3),(4,-3),(-8,0), and the area of each triangle is 36. Thus A(8) = 36 + 36 = 72.
It can be verified that A(10) = 252, A(100) = 34632 and A(1000) = 3529008.
Find A(1 000 000 000).
1The foci (plural of focus) of an ellipse are two points A and B such that for every point P on the boundary of the ellipse, AP + PB is constant.
Let n be an integer and S(n) be the set of factors of n.
A subset A of S(n) is called an antichain of S(n) if A contains only one element or if none of the elements of A divides any of the other elements of A.
For example: S(30) = {1, 2, 3, 5, 6, 10, 15, 30}
{2, 5, 6} is not an antichain of S(30).
{2, 3, 5} is an antichain of S(30).
Let N(n) be the maximum length of an antichain of S(n).
Find ΣN(n) for 1 ≤ n ≤ 108
A Harshad or Niven number is a number that is divisible by the sum of its digits.
201 is a Harshad number because it is divisible by 3 (the sum of its digits.)
When we truncate the last digit from 201, we get 20, which is a Harshad number.
When we truncate the last digit from 20, we get 2, which is also a Harshad number.
Let's call a Harshad number that, while recursively truncating the last digit, always results in a Harshad number a right truncatable Harshad number.
201/3=67 which is prime.
Let's call a Harshad number that, when divided by the sum of its digits, results in a prime a strong Harshad number.
Now take the number 2011 which is prime.
When we truncate the last digit from it we get 201, a strong Harshad number that is also right truncatable.
Let's call such primes strong, right truncatable Harshad primes.
You are given that the sum of the strong, right truncatable Harshad primes less than 10000 is 90619.
Find the sum of the strong, right truncatable Harshad primes less than 1014.
Consider all lattice points (a,b,c) with 0 ≤ a,b,c ≤ N.
From the origin O(0,0,0) all lines are drawn to the other lattice points.
Let D(N) be the number of distinct such lines.
You are given that D(1 000 000) = 831909254469114121.
Find D(1010). Give as your answer the first nine digits followed by the last nine digits.
An unbiased single 4-sided die is thrown and its value, T, is noted.
T unbiased 6-sided dice are thrown and their scores are added together. The sum, C, is noted.
C unbiased 8-sided dice are thrown and their scores are added together. The sum, O, is noted.
O unbiased 12-sided dice are thrown and their scores are added together. The sum, D, is noted.
D unbiased 20-sided dice are thrown and their scores are added together. The sum, I, is noted.
Find the variance of I, and give your answer rounded to 4 decimal places.
Let sk be the number of 1's when writing the numbers from 0 to k in binary.
For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1's, so s5 = 7.
The sequence S = {sk : k ≥ 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.
A game is played by two players. Before the game starts, a number n is chosen. A counter c starts at 0. At each turn, the player chooses a number from 1 to n (inclusive) and increases c by that number. The resulting value of c must be a member of S. If there are no more valid moves, the player loses.
Let n = 5. c starts at 0.
Player 1 chooses 4, so c becomes 0 + 4 = 4.
Player 1 chooses 3, so c becomes 9 + 3 = 12.
Note that c must always belong to S, and each player can increase c by at most n.
Let M(n) be the highest number the first player can choose at her first turn to force a win, and M(n) = 0 if there is no such move. For example, M(2) = 2, M(7) = 1 and M(20) = 4.
Given Σ(M(n))3 = 8150 for 1 ≤ n ≤ 20.
Find Σ(M(n))3 for 1 ≤ n ≤ 1000.
A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
An example of such grid is logarithmic graph paper.
Consider rectilinear grids in the Cartesian coordinate system with the following properties:
The gridlines are parallel to the axes of the Cartesian coordinate system.
There are N+2 vertical and N+2 horizontal gridlines. Hence there are (N+1) x (N+1) rectangular cells.
The equations of the two outer vertical gridlines are x = -1 and x = 1.
The equations of the two outer horizontal gridlines are y = -1 and y = 1.
The grid cells are colored red if they overlap with the unit circle, black otherwise.
For this problem we would like you to find the postions of the remaining N inner horizontal and N inner vertical gridlines so that the area occupied by the red cells is minimized.
E.g. here is a picture of the solution for N = 10:
The area occupied by the red cells for N = 10 rounded to 10 digits behind the decimal point is 3.3469640797.
Find the positions for N = 400.
Give as your answer the area occupied by the red cells rounded to 10 digits behind the decimal point.
Jeff eats a pie in an unusual way.
The pie is circular. He starts with slicing an initial cut in the pie along a radius.
While there is at least a given fraction F of pie left, he performs the following procedure:
- He makes two slices from the pie centre to any point of what is remaining of the pie border, any point on the remaining pie border equally likely. This will divide the remaining pie into three pieces.
- Going counterclockwise from the initial cut, he takes the first two pie pieces and eats them.
When less than a fraction F of pie remains, he does not repeat this procedure. Instead, he eats all of the remaining pie.
For x ≥ 1, let E(x) be the expected number of times Jeff repeats the procedure above with F = 1/x.
It can be verified that E(1) = 1, E(2) ≈ 1.2676536759, and E(7.5) ≈ 2.1215732071.
Find E(40) rounded to 10 decimal places behind the decimal point.
For any positive integer n, the nth weak Goodstein sequence {g1, g2, g3, ...} is defined as:
g1 = n
for k > 1, gk is obtained by writing gk-1 in base k, interpreting it as a base k + 1 number, and subtracting 1.
The sequence terminates when gk becomes 0.
For example, the 6th weak Goodstein sequence is {6, 11, 17, 25, ...}:
g1 = 6.
g2 = 11 since 6 = 1102, 1103 = 12, and 12 - 1 = 11.
g3 = 17 since 11 = 1023, 1024 = 18, and 18 - 1 = 17.
It can be shown that every weak Goodstein sequence terminates.
Let G(n) be the number of nonzero elements in the nth weak Goodstein sequence.
It can be verified that G(2) = 3, G(4) = 21 and G(6) = 381.
It can also be verified that ΣG(n) = 2517 for 1 ≤ n < 8.
Find the last 9 digits of ΣG(n) for 1 ≤ n < 16.
The first 15 fibonacci numbers are:
1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.
It can be seen that 8 and 144 are not squarefree: 8 is divisible by 4 and 144 is divisible by 4 and by 9.
So the first 13 squarefree fibonacci numbers are:
1,1,2,3,5,13,21,34,55,89,233,377 and 610.
The 200th squarefree fibonacci number is: 971183874599339129547649988289594072811608739584170445.
The last sixteen digits of this number are: 1608739584170445 and in scientific notation this number can be written as 9.7e53.
Find the 100 000 000th squarefree fibonacci number.
Give as your answer its last sixteen digits followed by a comma followed by the number in scientific notation (rounded to one digit after the decimal point).
For the 200th squarefree number the answer would have been: 1608739584170445,9.7e53
For this problem, assume that for every prime p, the first fibonacci number divisible by p is not divisible by p2 (this is part of Wall's conjecture). This has been verified for primes ≤ 3 × 1015, but has not been proven in general.
If it happens that the conjecture is false, then the accepted answer to this problem isn't guaranteed to be the 100 000 000th squarefree fibonacci number, rather it represents only a lower bound for that number.
We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions. Each number (question) we ask, we get one of three possible answers:
"Your guess is lower than the hidden number" (and you incur a cost of a), or
"Your guess is higher than the hidden number" (and you incur a cost of b), or
"Yes, that's it!" (and the game ends).
Given the value of n, a, and b, an optimal strategy minimizes the total cost for the worst possible case.
For example, if n = 5, a = 2, and b = 3, then we may begin by asking "2" as our first question.
If we are told that 2 is higher than the hidden number (for a cost of b=3), then we are sure that "1" is the hidden number (for a total cost of 3).
If we are told that 2 is lower than the hidden number (for a cost of a=2), then our next question will be "4".
If we are told that 4 is higher than the hidden number (for a cost of b=3), then we are sure that "3" is the hidden number (for a total cost of 2+3=5).
If we are told that 4 is lower than the hidden number (for a cost of a=2), then we are sure that "5" is the hidden number (for a total cost of 2+2=4).
Thus, the worst-case cost achieved by this strategy is 5. It can also be shown that this is the lowest worst-case cost that can be achieved. So, in fact, we have just described an optimal strategy for the given values of n, a, and b.
Let C(n, a, b) be the worst-case cost achieved by an optimal strategy for the given values of n, a, and b.
C(5, 2, 3) = 5
C(500, √2, √3) = 13.22073197...
C(20000, 5, 7) = 82
C(2000000, √5, √7) = 49.63755955...
Let Fk be the Fibonacci numbers: Fk = Fk-1 + Fk-2 with base cases F1 = F2 = 1.
Find ∑1≤k≤30 C(1012, √k, √Fk), and give your answer rounded to 8 decimal places behind the decimal point.
Let n be a positive integer. Consider nim positions where:
There are n non-empty piles.
Each pile has size less than 2n.
No two piles have the same size.
Let W(n) be the number of winning nim positions satisfying the above conditions (a position is winning if the first player has a winning strategy). For example, W(1) = 1, W(2) = 6, W(3) = 168, W(5) = 19764360 and W(100) mod 1 000 000 007 = 384777056.
Find W(10 000 000) mod 1 000 000 007.
Let n be a positive integer. Suppose there are stations at the coordinates (x, y) = (2i mod n, 3i mod n) for 0 ≤ i ≤ 2n. We will consider stations with the same coordinates as the same station.
We wish to form a path from (0, 0) to (n, n) such that the x and y coordinates never decrease.
Let S(n) be the maximum number of stations such a path can pass through.
For example, if n = 22, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, S(22) = 5. The case is illustrated below, with an example of an optimal path:
It can also be verified that S(123) = 14 and S(10000) = 48.
Find ∑ S(k5) for 1 ≤ k ≤ 30.
6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits in decreasing order, we get 7641-1467=6174.
Even more remarkable is that if we start from any 4 digit number and repeat this process of sorting and subtracting, we'll eventually end up with 6174 or immediately with 0 if all digits are equal.
This also works with numbers that have less than 4 digits if we pad the number with leading zeroes until we have 4 digits.
E.g. let's start with the number 0837:
8730-0378=8352
6174 is called the Kaprekar constant. The process of sorting and subtracting and repeating this until either 0 or the Kaprekar constant is reached is called the Kaprekar routine.
We can consider the Kaprekar routine for other bases and number of digits.
Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers.
However, it can be shown that for 5 digits and a base b = 6t+3≠9, a Kaprekar constant exists.
E.g. base 15: (10,4,14,9,5)15
base 21: (14,6,20,13,7)21
Define Cb to be the Kaprekar constant in base b for 5 digits. Define the function sb(i) to be
0 if i = Cb or if i written in base b consists of 5 identical digits
the number of iterations it takes the Kaprekar routine in base b to arrive at Cb, otherwise
Note that we can define sb(i) for all integers i < b5. If i written in base b takes less than 5 digits, the number is padded with leading zero digits until we have 5 digits before applying the Kaprekar routine.
Define S(b) as the sum of sb(i) for 0 < i < b5.
E.g. S(15) = 5274369
S(111) = 400668930299
Find the sum of S(6k+3) for 2 ≤ k ≤ 300.
Give the last 18 digits as your answer.
A set of lattice points S is called a titanic set if there exists a line passing through exactly two points in S.
An example of a titanic set is S = {(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)}, where the line passing through (0, 1) and (2, 0) does not pass through any other point in S.
On the other hand, the set {(0, 0), (1, 1), (2, 2), (4, 4)} is not a titanic set since the line passing through any two points in the set also passes through the other two.
For any positive integer N, let T(N) be the number of titanic sets S whose every point (x, y) satisfies 0 ≤ x, y ≤ N. It can be verified that T(1) = 11, T(2) = 494, T(4) = 33554178, T(111) mod 108 = 13500401 and T(105) mod 108 = 63259062.
Find T(1011) mod 108.
A row of n squares contains a frog in the leftmost square. By successive jumps the frog goes to the rightmost square and then back to the leftmost square. On the outward trip he jumps one, two or three squares to the right, and on the homeward trip he jumps to the left in a similar manner. He cannot jump outside the squares. He repeats the round-trip travel m times.
Let F(m, n) be the number of the ways the frog can travel so that at most one square remains unvisited.
For example, F(1, 3) = 4, F(1, 4) = 15, F(1, 5) = 46, F(2, 3) = 16 and F(2, 100) mod 109 = 429619151.
Find the last 9 digits of F(10, 1012).
A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:
1/2 = 0.5
1/3 = 0.(3)
1/4 = 0.25
1/6 = 0.1(6)
1/7 = 0.(142857)
1/8 = 0.125
1/10 = 0.1
Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.
Unit fractions whose denominator has no other prime factors than 2 and/or 5 are not considered to have a recurring cycle.
We define the length of the recurring cycle of those unit fractions as 0.
Let L(n) denote the length of the recurring cycle of 1/n. You are given that ∑L(n) for 3 ≤ n ≤ 1 000 000 equals 55535191115.
Find ∑L(n) for 3 ≤ n ≤ 100 000 000
Let n be a positive integer. An integer triple (a, b, c) is called a factorisation triple of n if:
1 ≤ a ≤ b ≤ c
a·b·c = n.
Define f(n) to be a + b + c for the factorisation triple (a, b, c) of n which minimises c / a. One can show that this triple is unique.
For example, f(165) = 19, f(100100) = 142 and f(20!) = 4034872.
Find f(43!).
The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
The sequence starts with 1 and all other members are obtained by describing the previous member in terms of consecutive digits.
It helps to do this out loud:
1 is 'one one' → 11
11 is 'two ones' → 21
21 is 'one two and one one' → 1211
1211 is 'one one, one two and two ones' → 111221
111221 is 'three ones, two twos and one one' → 312211
Define A(n), B(n) and C(n) as the number of ones, twos and threes in the n'th element of the sequence respectively.
One can verify that A(40) = 31254, B(40) = 20259 and C(40) = 11625.
Find A(n), B(n) and C(n) for n = 1012.
Give your answer modulo 230 and separate your values for A, B and C by a comma.
E.g. for n = 40 the answer would be 31254,20259,11625
A positive integer matrix is a matrix whose elements are all positive integers.
Some positive integer matrices can be expressed as a square of a positive integer matrix in two different ways. Here is an example:
We define F(N) as the number of the 2x2 positive integer matrices which have a trace less than N and which can be expressed as a square of a positive integer matrix in two different ways.
We can verify that F(50) = 7 and F(1000) = 1019.
Find F(107).
Numbers of the form n15+1 are composite for every integer n > 1.
For positive integers n and m let s(n,m) be defined as the sum of the distinct prime factors of n15+1 not exceeding m.
E.g. 215+1 = 3×3×11×331.
So s(2,10) = 3 and s(2,1000) = 3+11+331 = 345.
Also 1015+1 = 7×11×13×211×241×2161×9091.
So s(10,100) = 31 and s(10,1000) = 483.
Find ∑ s(n,108) for 1 ≤ n ≤ 1011.
Let H be the hyperbola defined by the equation 12x2 + 7xy - 12y2 = 625.
Next, define X as the point (7, 1). It can be seen that X is in H.
Now we define a sequence of points in H, {Pi : i ≥ 1}, as:
P1 = (13, 61/4).
P2 = (-43/6, -4).
For i > 2, Pi is the unique point in H that is different from Pi-1 and such that line PiPi-1 is parallel to line Pi-2X. It can be shown that Pi is well-defined, and that its coordinates are always rational.
You are given that P3 = (-19/2, -229/24), P4 = (1267/144, -37/12) and P7 = (17194218091/143327232, 274748766781/1719926784).
Find Pn for n = 1114 in the following format:
If Pn = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.
For n = 7, the answer would have been: 806236837.
Let n be a positive integer.
A 6-sided die is thrown n times. Let c be the number of pairs of consecutive throws that give the same value.
For example, if n = 7 and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:
(1,1,5,6,6,6,3)
Therefore, c = 3 for (1,1,5,6,6,6,3).
Define C(n) as the number of outcomes of throwing a 6-sided die n times such that c does not exceed π(n).1
For example, C(3) = 216, C(4) = 1290, C(11) = 361912500 and C(24) = 4727547363281250000.
Define S(L) as ∑ C(n) for 1 ≤ n ≤ L.
For example, S(50) mod 1 000 000 007 = 832833871.
Find S(50 000 000) mod 1 000 000 007.
1 π denotes the prime-counting function, i.e. π(n) is the number of primes ≤ n.
The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The common rules of kakuro puzzles can be found easily on numerous internet sites. Other related information can also be currently found at krazydad.com whose author has provided the puzzle data for this challenge.)
The downloadable text file (kakuro200.txt) contains the description of 200 such puzzles, a mix of 5x5 and 6x6 types. The first puzzle in the file is the above example which is coded as follows:
6,X,X,(vCC),(vI),X,X,X,(hH),B,O,(vCA),(vJE),X,(hFE,vD),O,O,O,O,(hA),O,I,(hJC,vB),O,O,(hJC),H,O,O,O,X,X,X,(hJE),O,O,X
The first character is a numerical digit indicating the size of the information grid. It would be either a 6 (for a 5x5 kakuro puzzle) or a 7 (for a 6x6 puzzle) followed by a comma (,). The extra top line and left column are needed to insert information.
The content of each cell is then described and followed by a comma, going left to right and starting with the top line.
X = Gray cell, not required to be filled by a digit.
O (upper case letter)= White empty cell to be filled by a digit.
A = Or any one of the upper case letters from A to J to be replaced by its equivalent digit in the solved puzzle.
( ) = Location of the encrypted sums. Horizontal sums are preceded by a lower case "h" and vertical sums are preceded by a lower case "v". Those are followed by one or two upper case letters depending if the sum is a single digit or double digit one. For double digit sums, the first letter would be for the "tens" and the second one for the "units". When the cell must contain information for both a horizontal and a vertical sum, the first one is always for the horizontal sum and the two are separated by a comma within the same set of brackets, ex.: (hFE,vD). Each set of brackets is also immediately followed by a comma.
The description of the last cell is followed by a Carriage Return/Line Feed (CRLF) instead of a comma.
The required answer to each puzzle is based on the value of each letter necessary to arrive at the solution and according to the alphabetical order. As indicated under the example puzzle, its answer would be 8426039571. At least 9 out of the 10 encrypting letters are always part of the problem description. When only 9 are given, the missing one must be assigned the remaining digit.
You are given that the sum of the answers for the first 10 puzzles in the file is 64414157580.
Find the sum of the answers for the 200 puzzles.
Two positive numbers A and B are said to be connected (denoted by "A ↔ B") if one of these conditions holds:
(1) A and B have the same length and differ in exactly one digit; for example, 123 ↔ 173.
(2) Adding one digit to the left of A (or B) makes B (or A); for example, 23 ↔ 223 and 123 ↔ 23.
We call a prime P a 2's relative if there exists a chain of connected primes between 2 and P and no prime in the chain exceeds P.
For example, 127 is a 2's relative. One of the possible chains is shown below:
2 ↔ 3 ↔ 13 ↔ 113 ↔ 103 ↔ 107 ↔ 127
However, 11 and 103 are not 2's relatives.
Let F(N) be the sum of the primes ≤ N which are not 2's relatives.
We can verify that F(103) = 431 and F(104) = 78728.
Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty boxes, 2 occupied boxes, 1 empty box, and 2 occupied boxes can be denoted by the sequence (2, 2, 2, 1, 2), in which the number of consecutive occupied and empty boxes appear alternately.
A turn consists of moving each ball exactly once according to the following rule: Transfer the leftmost ball which has not been moved to the nearest empty box to its right.
After one turn the sequence (2, 2, 2, 1, 2) becomes (2, 2, 1, 2, 3) as can be seen below; note that we begin the new sequence starting at the first occupied box.
A system like this is called a Box-Ball System or BBS for short.
It can be shown that after a sufficient number of turns, the system evolves to a state where the consecutive numbers of occupied boxes is invariant. In the example below, the consecutive numbers of occupied boxes evolves to [1, 2, 3]; we shall call this the final state.
We define the sequence {ti}:
sk+1 = sk2 mod 50515093
tk = (sk mod 64) + 1
Starting from the initial configuration (t0, t1, …, t10), the final state becomes [1, 3, 10, 24, 51, 75].
Starting from the initial configuration (t0, t1, …, t10 000 000), find the final state.
Give as your answer the sum of the squares of the elements of the final state. For example, if the final state is [1, 2, 3] then 14 ( = 12 + 22 + 32) is your answer.
A sequence of integers S = {si} is called an n-sequence if it has n elements and each element si satisfies 1 ≤ si ≤ n. Thus there are nn distinct n-sequences in total. For example, the sequence S = {1, 5, 5, 10, 7, 7, 7, 2, 3, 7} is a 10-sequence.
For any sequence S, let L(S) be the length of the longest contiguous subsequence of S with the same value. For example, for the given sequence S above, L(S) = 3, because of the three consecutive 7's.
Let f(n) = ∑ L(S) for all n-sequences S.
For example, f(3) = 45, f(7) = 1403689 and f(11) = 481496895121.
Find f(7 500 000) mod 1 000 000 009.
Let a, b and c be positive numbers.
Let W, X, Y, Z be four collinear points where |WX| = a, |XY| = b, |YZ| = c and |WZ| = a + b + c.
Let Cin be the circle having the diameter XY.
Let Cout be the circle having the diameter WZ.
The triplet (a, b, c) is called a necklace triplet if you can place k ≥ 3 distinct circles C1, C2, ..., Ck such that:
Ci has no common interior points with any Cj for 1 ≤ i, j ≤ k and i ≠ j,
Ci is tangent to both Cin and Cout for 1 ≤ i ≤ k,
Ci is tangent to Ci+1 for 1 ≤ i < k, and
Ck is tangent to C1.
For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not.
Let T(n) be the number of necklace triplets (a, b, c) such that a, b and c are positive integers, and b ≤ n. For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.
Find T(1 000 000 000).
A unitary divisor d of a number n is a divisor of n that has the property gcd(d, n/d) = 1.
The unitary divisors of 4! = 24 are 1, 3, 8 and 24.
The sum of their squares is 12 + 32 + 82 + 242 = 650.
Let S(n) represent the sum of the squares of the unitary divisors of n. Thus S(4!)=650.
Find S(100 000 000!) modulo 1 000 000 009.
N disks are placed in a row, indexed 1 to N from left to right.
Each disk has a black side and white side. Initially all disks show their white side.
At each turn, two, not necessarily distinct, integers A and B between 1 and N (inclusive) are chosen uniformly at random.
All disks with an index from A to B (inclusive) are flipped.
The following example shows the case N = 8. At the first turn A = 5 and B = 2, and at the second turn A = 4 and B = 6.
Let E(N, M) be the expected number of disks that show their white side after M turns.
We can verify that E(3, 1) = 10/9, E(3, 2) = 5/3, E(10, 4) ≈ 5.157 and E(100, 10) ≈ 51.893.
Find E(1010, 4000).
Give your answer rounded to 2 decimal places behind the decimal point.
Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when he discovers that it is circular. Quentin, the representative from the company that installed the silo, explains that they only manufacture cylindrical silos, but he points out that it is resting on a square base. Fred is not amused and insists that it is removed from his property.
Quick thinking Quentin explains that when granular materials are delivered from above a conical slope is formed and the natural angle made with the horizontal is called the angle of repose. For example if the angle of repose, $\alpha = 30$ degrees, and grain is delivered at the centre of the silo then a perfect cone will form towards the top of the cylinder. In the case of this silo, which has a diameter of 6m, the amount of space wasted would be approximately 32.648388556 m3. However, if grain is delivered at a point on the top which has a horizontal distance of $x$ metres from the centre then a cone with a strangely curved and sloping base is formed. He shows Fred a picture.
We shall let the amount of space wasted in cubic metres be given by $V(x)$. If $x = 1.114785284$, which happens to have three squared decimal places, then the amount of space wasted, $V(1.114785284) \approx 36$. Given the range of possible solutions to this problem there is exactly one other option: $V(2.511167869) \approx 49$. It would be like knowing that the square is king of the silo, sitting in splendid glory on top of your grain.
Fred's eyes light up with delight at this elegant resolution, but on closer inspection of Quentin's drawings and calculations his happiness turns to despondency once more. Fred points out to Quentin that it's the radius of the silo that is 6 metres, not the diameter, and the angle of repose for his grain is 40 degrees. However, if Quentin can find a set of solutions for this particular silo then he will be more than happy to keep it.
If Quick thinking Quentin is to satisfy frustratingly fussy Fred the farmer's appetite for all things square then determine the values of $x$ for all possible square space wastage options and calculate $\sum x$ correct to 9 decimal places.
Let S(n,m) = ∑φ(n × i) for 1 ≤ i ≤ m. (φ is Euler's totient function)
You are given that S(510510,106 )= 45480596821125120.
Find S(510510,1011).
Give the last 9 digits of your answer.
Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algorithm. More formally:
x1 = y0, y1 = x0 mod y0
xn = yn-1, yn = xn-1 mod yn-1
E(x0, y0) is the smallest n such that yn = 0.
We have E(1,1) = 1, E(10,6) = 3 and E(6,10) = 4.
Define S(N) as the sum of E(x,y) for 1 ≤ x,y ≤ N.
We have S(1) = 1, S(10) = 221 and S(100) = 39826.
Find S(5·106).
Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent.
Graphs can be embedded in Euclidean space by associating each vertex with a point in the Euclidean space.
A flexible graph is an embedding of a graph where it is possible to move one or more vertices continuously so that the distance between at least two nonadjacent vertices is altered while the distances between each pair of adjacent vertices is kept constant.
A rigid graph is an embedding of a graph which is not flexible.
Informally, a graph is rigid if by replacing the vertices with fully rotating hinges and the edges with rods that are unbending and inelastic, no parts of the graph can be moved independently from the rest of the graph.
The grid graphs embedded in the Euclidean plane are not rigid, as the following animation demonstrates:
However, one can make them rigid by adding diagonal edges to the cells. For example, for the 2x3 grid graph, there are 19 ways to make the graph rigid:
Note that for the purposes of this problem, we do not consider changing the orientation of a diagonal edge or adding both diagonal edges to a cell as a different way of making a grid graph rigid.
Let R(m,n) be the number of ways to make the m × n grid graph rigid.
E.g. R(2,3) = 19 and R(5,5) = 23679901
Define S(N) as ∑R(i,j) for 1 ≤ i, j ≤ N.
E.g. S(5) = 25021721.
Find S(100), give your answer modulo 1000000033
The Fibonacci numbers {fn, n ≥ 0} are defined recursively as fn = fn-1 + fn-2 with base cases f0 = 0 and f1 = 1.
Define the polynomials {Fn, n ≥ 0} as Fn(x) = ∑fixi for 0 ≤ i ≤ n.
For example, F7(x) = x + x2 + 2x3 + 3x4 + 5x5 + 8x6 + 13x7, and F7(11) = 268357683.
Let n = 1015. Find the sum [∑0≤x≤100 Fn(x)] mod 1307674368000 (= 15!).
Julie proposes the following wager to her sister Louise.
She suggests they play a game of chance to determine who will wash the dishes.
For this game, they shall use a generator of independent random numbers uniformly distributed between 0 and 1.
The game starts with S = 0.
The first player, Louise, adds to S different random numbers from the generator until S > 1 and records her last random number 'x'.
The second player, Julie, continues adding to S different random numbers from the generator until S > 2 and records her last random number 'y'.
The player with the highest number wins and the loser washes the dishes, i.e. if y > x the second player wins.
For example, if the first player draws 0.62 and 0.44, the first player turn ends since 0.62+0.44 > 1 and x = 0.44.
If the second players draws 0.1, 0.27 and 0.91, the second player turn ends since 0.62+0.44+0.1+0.27+0.91 > 2 and y = 0.91. Since y > x, the second player wins.
Louise thinks about it for a second, and objects: "That's not fair".
What is the probability that the second player wins?
Give your answer rounded to 10 places behind the decimal point in the form 0.abcdefghij
When we calculate 8n modulo 11 for n=0 to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2, 5, 7.
As we see all possible values from 1 to 10 occur. So 8 is a primitive root of 11.
But there is more:
If we take a closer look we see:
8+9=17≡6 mod 11
4+10=14≡3 mod 11
10+3=13≡2 mod 11
5+7=12≡1 mod 11.
So the powers of 8 mod 11 are cyclic with period 10, and 8n + 8n+1 ≡ 8n+2 (mod 11).
8 is called a Fibonacci primitive root of 11.
Not every prime has a Fibonacci primitive root.
There are 323 primes less than 10000 with one or more Fibonacci primitive roots and the sum of these primes is 1480491.
Find the sum of the primes less than 100,000,000 with at least one Fibonacci primitive root.
For an n-tuple of integers t = (a1, ..., an), let (x1, ..., xn) be the solutions of the polynomial equation xn + a1xn-1 + a2xn-2 + ... + an-1x + an = 0.
Consider the following two conditions:
x1, ..., xn are all real.
If x1, ..., xn are sorted, ⌊xi⌋ = i for 1 ≤ i ≤ n. (⌊·⌋: floor function.)
In the case of n = 4, there are 12 n-tuples of integers which satisfy both conditions.
We define S(t) as the sum of the absolute values of the integers in t.
For n = 4 we can verify that ∑S(t) = 2087 for all n-tuples t which satify both conditions.
Find ∑S(t) for n = 7.
Let d(k) be the sum of all divisors of k.
We define the function S(N) = ∑1≤i≤N ∑1≤j≤N d(i·j).
For example, S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59.
You are given that S(103) = 563576517282 and S(105) mod 109 = 215766508.
Find S(1011) mod 109.
We want to tile a board of length n and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:
For example, here are some of the ways to tile a board of length n = 8:
Let T(n) be the number of ways to tile a board of length n as described above.
For example, T(1) = 10 and T(2) = 101.
Let S(L) be the triple sum ∑a,b,c gcd(T(ca), T(cb)) for 1 ≤ a, b, c ≤ L.
S(2) = 10444
S(3) = 1292115238446807016106539989
S(4) mod 987 898 789 = 670616280.
Find S(2000) mod 987 898 789.
For an integer M, we define R(M) as the sum of 1/(p·q) for all the integer pairs p and q which satisfy all of these conditions:
1 ≤ p < q ≤ M
p + q ≥ M
p and q are coprime.
We also define S(N) as the sum of R(i) for 2 ≤ i ≤ N.
We can verify that S(2) = R(2) = 1/2, S(10) ≈ 6.9147 and S(100) ≈ 58.2962.
Find S(107). Give your answer rounded to four decimal places.
An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of 11 except 1.
For example, 2404 and 13431 are eleven-free, while 911 and 4121331 are not.
Let E(n) be the nth positive eleven-free integer. For example, E(3) = 3, E(200) = 213 and E(500 000) = 531563.
Find E(1018).
Let g(n) be a sequence defined as follows:
g(4) = 13,
g(n) = g(n-1) + gcd(n, g(n-1)) for n > 4.
The first few values are:
n 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
g(n) 13 14 16 17 18 27 28 29 30 31 32 33 34 51 54 55 60 ...
You are given that g(1 000) = 2524 and g(1 000 000) = 2624152.
Find g(1015).
A group of p people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £p, with no two tickets alike. The goal of the game is for each person to maximize his ticket winnings upon leaving the game.
An arbitrary person is chosen to be the first player. Going around the table, each player has only one of two options:
1. The player can scratch his ticket and reveal its worth to everyone at the table.
2. The player can trade his unscratched ticket for a previous player's scratched ticket, and then leave the game with that ticket. The previous player then scratches his newly-acquired ticket and reveals its worth to everyone at the table.
The game ends once all tickets have been scratched. All players still remaining at the table must leave with their currently-held tickets.
Assume that each player uses the optimal strategy for maximizing the expected value of his ticket winnings.
Let E(p) represent the expected number of players left at the table when the game ends in a game consisting of p players (e.g. E(111) = 5.2912 when rounded to 5 significant digits).
Let S1(N) = E(p)
Let Sk(N) = Sk-1(p) for k > 1
Find S20(1014) and write the answer in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent (e.g. S3(100) = 5.983679014e5).
For every integer n>1, the family of functions fn,a,b is defined by fn,a,b(x)≡ax+b mod n for a,b,x integer and 0<a<n, 0≤b<n, 0≤x<n.
We will call fn,a,b a retraction if fn,a,b(fn,a,b(x))≡fn,a,b(x) mod n for every 0≤x<n.
Let R(n) be the number of retractions for n.
You are given that
∑ R(c) for c=C(100 000,k), and 1 ≤ k ≤99 999 ≡628701600 (mod 1 000 000 007).
(C(n,k) is the binomial coefficient).
Find ∑ R(c) for c=C(10 000 000,k), and 1 ≤k≤ 9 999 999.
Give your answer modulo 1 000 000 007.
F(N)=∑R(n4+4) for 1≤n≤N.
F(1024)=77532377300600.
Find F(107) (mod 1 000 000 007)
F(N)=∑R(n) for 2≤n≤N.
F(107)≡638042271 (mod 1 000 000 007).
Find F(1014) (mod 1 000 000 007).
The function lcm(a,b) denotes the least common multiple of a and b.
Let A(n) be the average of the values of lcm(n,i) for 1≤i≤n.
E.g: A(2)=(2+2)/2=2 and A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32.
Let S(n)=∑A(k) for 1≤k≤n.
S(100)=122726.
Find S(99999999019) mod 999999017.
Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: b2x2 + b2y2 + a2z2 = a2b2.
Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick.
If a=1 mm and b=1 mm, the amount of chocolate required is
π mm3
If a=2 mm and b=1 mm, the amount of chocolate required is approximately 60.35475635 mm3.
Find the amount of chocolate in mm3 required if a=3 mm and b=1 mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.
A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
$x(t) = (R - r) \cos(t) + r \cos(\frac {R - r} r t)$
$y(t) = (R - r) \sin(t) - r \sin(\frac {R - r} r t)$
Where R is the radius of the large circle and r the radius of the small circle.
Let $C(R, r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that $\sin(t)$ and $\cos(t)$ are rational numbers.
Let $S(R, r) = \sum_{(x,y) \in C(R, r)} |x| + |y|$ be the sum of the absolute values of the x and y coordinates of the points in $C(R, r)$.
Let $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\lfloor \frac {R - 1} 2 \rfloor} S(R, r)$ be the sum of $S(R, r)$ for R and r positive integers, $R\leq N$ and $2r < R$.
You are given:
C(3, 1) = {(3, 0), (-1, 2), (-1,0), (-1,-2)}
C(2500, 1000) =
{(2500, 0), (772, 2376), (772, -2376), (516, 1792), (516, -1792), (500, 0), (68, 504), (68, -504),
(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)}
Note: (-625, 0) is not an element of C(2500, 1000) because $\sin(t)$ is not a rational number for the corresponding values of t.
S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10
T(3) = 10; T(10) = 524 ;T(100) = 580442; T(103) = 583108600.
Find T(106).
There are eight positive numbers less than 15 which are coprime to 15: 1, 2, 4, 7, 8, 11, 13, 14.
The modular inverses of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14
1*1 mod 15=1
2*8=16 mod 15=1
7*13=91 mod 15=1
11*11=121 mod 15=1
Let I(n) be the largest positive number m smaller than n-1 such that the modular inverse of m modulo n equals m itself.
So I(15)=11.
Also I(100)=51 and I(7)=1.
Find ∑I(n) for 3≤n≤2·107
Define F(m,n) as the number of n-tuples of positive integers for which the product of the elements doesn't exceed m.
F(10, 10) = 571.
F(106, 106) mod 1 234 567 891 = 252903833.
Find F(109, 109) mod 1 234 567 891.
A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.
Let Q(m, n) be the number of simple quadrilaterals whose vertices are lattice points with coordinates (x,y) satisfying 0 ≤ x ≤ m and 0 ≤ y ≤ n.
For example, Q(2, 2) = 94 as can be seen below:
It can also be verified that Q(3, 7) = 39590, Q(12, 3) = 309000 and Q(123, 45) = 70542215894646.
Find Q(12345, 6789) mod 135707531.
In the following equation x, y, and n are positive integers.
For a limit L we define F(L) as the number of solutions which satisfy x < y ≤ L.
Find F(1012).
Let f(n) be the largest positive integer x less than 109 such that the last 9 digits of nx form the number x (including leading zeros), or zero if no such integer exists.
f(4) = 411728896 (4411728896 = ...490411728896)
f(10) = 0
f(157) = 743757 (157743757 = ...567000743757)
Σf(n), 2 ≤ n ≤ 103 = 442530011399
Find Σf(n), 2 ≤ n ≤ 106.
xn = (1248n mod 32323) - 16161
yn = (8421n mod 30103) - 15051
Pn = {(x1, y1), (x2, y2), ..., (xn, yn)}
For example, P8 = {(-14913, -6630), (-10161, 5625), (5226, 11896), (8340, -10778), (15852, -5203), (-15165, 11295), (-1427, -14495), (12407, 1060)}.
Let C(n) be the number of triangles whose vertices are in Pn which contain the origin in the interior.
C(8) = 20
C(600) = 8950634
C(40 000) = 2666610948988
Find C(2 000 000).
Let f(n) = n2 - 3n - 1.
Let p be a prime.
Let R(p) be the smallest positive integer n such that f(n) mod p2 = 0 if such an integer n exists, otherwise R(p) = 0.
Let SR(L) be ∑R(p) for all primes not exceeding L.
Find SR(107).
Consider the alphabet A made out of the letters of the word "project": A={c,e,j,o,p,r,t}.
Let T(n) be the number of strings of length n consisting of letters from A that do not have a substring that is one of the 5040 permutations of "project".
T(7)=77-7!=818503.
Find T(1012). Give the last 9 digits of your answer.
The flipping game is a two player game played on a N by N square board.
Each square contains a disk with one side white and one side black.
The game starts with all disks showing their white side.
A turn consists of flipping all disks in a rectangle with the following properties:
the upper right corner of the rectangle contains a white disk
the rectangle width is a perfect square (1, 4, 9, 16, ...)
the rectangle height is a triangular number (1, 3, 6, 10, ...)
Players alternate turns. A player wins by turning the grid all black.
Let W(N) be the number of winning moves for the first player on a N by N board with all disks white, assuming perfect play.
W(1) = 1, W(2) = 0, W(5) = 8 and W(102) = 31395.
For N=5, the first player's eight winning first moves are:
Find W(106).
On the Euclidean plane, an ant travels from point A(0, 1) to point B(d, 1) for an integer d.
In each step, the ant at point (x0, y0) chooses one of the lattice points (x1, y1) which satisfy x1 ≥ 0 and y1 ≥ 1 and goes straight to (x1, y1) at a constant velocity v. The value of v depends on y0 and y1 as follows:
If y0 = y1, the value of v equals y0.
If y0 ≠ y1, the value of v equals (y1 - y0) / (ln(y1) - ln(y0)).
The left image is one of the possible paths for d = 4. First the ant goes from A(0, 1) to P1(1, 3) at velocity (3 - 1) / (ln(3) - ln(1)) ≈ 1.8205. Then the required time is sqrt(5) / 1.8205 ≈ 1.2283.
From P1(1, 3) to P2(3, 3) the ant travels at velocity 3 so the required time is 2 / 3 ≈ 0.6667. From P2(3, 3) to B(4, 1) the ant travels at velocity (1 - 3) / (ln(1) - ln(3)) ≈ 1.8205 so the required time is sqrt(5) / 1.8205 ≈ 1.2283.
Thus the total required time is 1.2283 + 0.6667 + 1.2283 = 3.1233.
The right image is another path. The total required time is calculated as 0.98026 + 1 + 0.98026 = 2.96052. It can be shown that this is the quickest path for d = 4.
Let F(d) be the total required time if the ant chooses the quickest path. For example, F(4) ≈ 2.960516287.
We can verify that F(10) ≈ 4.668187834 and F(100) ≈ 9.217221972.
Find F(10000). Give your answer rounded to nine decimal places.
Let fn(k) = ek/n - 1, for all non-negative integers k.
Remarkably, f200(6) + f200(75) + f200(89) + f200(226) = 3.141592644529… ≈ π.
In fact, it is the best approximation of π of the form fn(a) + fn(b) + fn(c) + fn(d) for n = 200.
Let g(n) = a2 + b2 + c2 + d 2 for a, b, c, d that minimize the error: | fn(a) + fn(b) + fn(c) + fn(d) - π|
(where |x| denotes the absolute value of x).
You are given g(200) = 62 + 752 + 892 + 2262 = 64658.
Find g(10000).
A 3-smooth number is an integer which has no prime factor larger than 3. For an integer N, we define S(N) as the set of 3-smooth numbers less than or equal to N . For example, S(20) = { 1, 2, 3, 4, 6, 8, 9, 12, 16, 18 }.
We define F(N) as the number of permutations of S(N) in which each element comes after all of its proper divisors.
This is one of the possible permutations for N = 20.
- 1, 2, 4, 3, 9, 8, 16, 6, 18, 12.
This is not a valid permutation because 12 comes before its divisor 6.
- 1, 2, 4, 3, 9, 8, 12, 16, 6, 18.
We can verify that F(6) = 5, F(8) = 9, F(20) = 450 and F(1000) ≈ 8.8521816557e21.
Find F(1018). Give as your answer its scientific notation rounded to ten digits after the decimal point.
When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is 112,233,445,566,778,899 then the answer format would be 1.1223344557e17.
The function $f$ is defined for all positive integers as follows:
$f(1)=1$
$f(2n)=f(n)$
$f(4n + 1)=2f(2n + 1) - f(n)$
$f(4n + 3)=3f(2n + 1) - 2f(n)$
The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$.
$S(8)=22$ and $S(100)=3604$.
Find $S(3^{37})$. Give the last 9 digits of your answer.
The Möbius function, denoted μ(n), is defined as:
μ(n) = (-1)ω(n) if n is squarefree (where ω(n) is the number of distinct prime factors of n)
μ(n) = 0 if n is not squarefree.
Let P(a,b) be the number of integers n in the interval [a,b] such that μ(n) = 1.
Let N(a,b) be the number of integers n in the interval [a,b] such that μ(n) = -1.
For example, P(2,10) = 2 and N(2,10) = 4.
Let C(n) be the number of integer pairs (a,b) such that:
1 ≤ a ≤ b ≤ n,
99·N(a,b) ≤ 100·P(a,b), and
99·P(a,b) ≤ 100·N(a,b).
For example, C(10) = 13, C(500) = 16676 and C(10 000) = 20155319.
Find C(20 000 000).
The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible. We define a polar polygon as a polygon for which the origin is strictly contained inside its kernel.
For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have self-intersection and cannot have zero area.
For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):
Notice that the first polygon has three consecutive collinear vertices.
Let P(n) be the number of polar polygons such that the vertices (x, y) have integer coordinates whose absolute values are not greater than n.
Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0),(0,3),(1,1),(3,0)] is distinct from the polygon with vertices [(0,0),(0,3),(1,1),(3,0),(1,0)].
For example, P(1) = 131, P(2) = 1648531, P(3) = 1099461296175 and P(343) mod 1 000 000 007 = 937293740.
Find P(713) mod 1 000 000 007.
Let P(m,n) be the number of distinct terms in an m×n multiplication table.
For example, a 3×4 multiplication table looks like this:
There are 8 distinct terms {1,2,3,4,6,8,9,12}, therefore P(3,4) = 8.
You are given that:
P(64,64) = 1263,
P(12,345) = 1998, and
P(32,1015) = 13826382602124302.
Find P(64,1016).
An integer s is called a superinteger of another integer n if the digits of n form a subsequence of the digits of s.
For example, 2718281828 is a superinteger of 18828, while 314159 is not a superinteger of 151.
Let p(n) be the nth prime number, and let c(n) be the nth composite number. For example, p(1) = 2, p(10) = 29, c(1) = 4 and c(10) = 18.
{p(i) : i ≥ 1} = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...}
{c(i) : i ≥ 1} = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...}
Let PD the sequence of the digital roots of {p(i)} (CD is defined similarly for {c(i)}):
PD = {2, 3, 5, 7, 2, 4, 8, 1, 5, 2, ...}
CD = {4, 6, 8, 9, 1, 3, 5, 6, 7, 9, ...}
Let Pn be the integer formed by concatenating the first n elements of PD (Cn is defined similarly for CD).
P10 = 2357248152
C10 = 4689135679
Let f(n) be the smallest positive integer that is a common superinteger of Pn and Cn.
For example, f(10) = 2357246891352679, and f(100) mod 1 000 000 007 = 771661825.
Find f(10 000) mod 1 000 000 007.
An integer is called B-smooth if none of its prime factors is greater than B.
Let SB(n) be the largest B-smooth divisor of n.
S1(10) = 1
S4(2100) = 12
S17(2496144) = 5712
Define F(n) = ∑1≤B≤n ∑0≤r≤n SB(C(n,r)). Here, C(n,r) denotes the binomial coefficient.
F(11) = 3132
F(1 111) mod 1 000 000 993 = 706036312
F(111 111) mod 1 000 000 993 = 22156169
Find F(11 111 111) mod 1 000 000 993.
In a room N chairs are placed around a round table.
Knights enter the room one by one and choose at random an available empty chair.
To have enough elbow room the knights always leave at least one empty chair between each other.
When there aren't any suitable chairs left, the fraction C of empty chairs is determined.
We also define E(N) as the expected value of C.
We can verify that E(4) = 1/2 and E(6) = 5/9.
Find E(1018). Give your answer rounded to fourteen decimal places in the form 0.abcdefghijklmn.
Consider a single game of Ramvok:
Let t represent the maximum number of turns the game lasts. If t = 0, then the game ends immediately. Otherwise, on each turn i, the player rolls a die. After rolling, if i < t the player can either stop the game and receive a prize equal to the value of the current roll, or discard the roll and try again next turn. If i = t, then the roll cannot be discarded and the prize must be accepted. Before the game begins, t is chosen by the player, who must then pay an up-front cost ct for some constant c. For c = 0, t can be chosen to be infinite (with an up-front cost of 0). Let R(d, c) be the expected profit (i.e. net gain) that the player receives from a single game of optimally-played Ramvok, given a fair d-sided die and cost constant c. For example, R(4, 0.2) = 2.65. Assume that the player has sufficient funds for paying any/all up-front costs.
Now consider a game of Super Ramvok:
In Super Ramvok, the game of Ramvok is played repeatedly, but with a slight modification. After each game, the die is altered. The alteration process is as follows: The die is rolled once, and if the resulting face has its pips visible, then that face is altered to be blank instead. If the face is already blank, then it is changed back to its original value. After the alteration is made, another game of Ramvok can begin (and during such a game, at each turn, the die is rolled until a face with a value on it appears). The player knows which faces are blank and which are not at all times. The game of Super Ramvok ends once all faces of the die are blank.
Let S(d, c) be the expected profit that the player receives from an optimally-played game of Super Ramvok, given a fair d-sided die to start (with all sides visible), and cost constant c. For example, S(6, 1) = 208.3.
Let F(n) = ∑4≤d≤n ∑0≤c≤n S(d, c).
Calculate F(20), rounded to the nearest integer.
The triangle ΔABC is inscribed in an ellipse with equation $\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$, 0 < 2b < a, a and b integers.
Let r(a,b) be the radius of the incircle of ΔABC when the incircle has center (2b, 0) and A has coordinates $\left( \frac a 2, \frac {\sqrt 3} 2 b\right)$.
For example, r(3,1) = ½, r(6,2) = 1, r(12,3) = 2.
Let $G(n) = \sum_{a=3}^n \sum_{b=1}^{\lfloor \frac {a - 1} 2 \rfloor} r(a, b)$
You are given G(10) = 20.59722222, G(100) = 19223.60980 (rounded to 10 significant digits).
Give your answer in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent.
For G(10) the answer would have been 2.059722222e1.