You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a trapezoid \( MNPQ \) with bases \( MQ \) and \( NP \). A line parallel to the bases intersects the lateral side \( MN \) at point \( A \), and the lateral side \( PQ \) at point \( B \). The ratio of the areas of the trapezoids \( ANPB \) and \( MABQ \) is \( \frac{2}{7} \). Find \( AB \) if \( NP = 4 \) and \( MQ = 6 \).

Answer: $\frac{2\sqrt{46}}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.

Answer: $306$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a certain country, there are 200 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that it should be possible to travel from any city to any other city using the flights of each airline (possibly with layovers). What is the maximum number of airlines for which this is possible?

Answer: $100$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible?

Answer: $42$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given \(3 \sin^{2} \alpha + 2 \sin^{2} \beta = 1\) and \(3 (\sin \alpha + \cos \alpha)^{2} - 2 (\sin \beta + \cos \beta)^{2} = 1\), find \(\cos 2 (\alpha + \beta) = \quad \) .

Answer: $-\frac{1}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, but Carlson requires 5 minutes to do the same. In total, they had 10 jars of jam and 10 pots of honey. For how many minutes did they eat everything, given that they started and finished eating at the same time? (Each eats a pot of honey or a jar of jam completely.)

Answer: $48$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always two that are connected by a bridge. What is the largest possible value of \( N \)?

Answer: $36$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( A \) and \( B \) be the endpoints of a semicircular arc of radius \( 3 \). The arc is divided into five congruent arcs by four equally spaced points \( C_1, C_2, C_3, C_4 \). All chords of the form \( \overline{AC_i} \) or \( \overline{BC_i} \) are drawn. Find the product of the lengths of these eight chords.

Answer: $32805$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the largest natural number \( n \) such that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.

Answer: $2004$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye is worth 315 points. How many points is hitting the blue (second-to-last) zone worth?

Answer: $45$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\{1, 2, \cdots, 60\}$, there must be three numbers that are pairwise coprime.

Answer: $41$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:

(i) \( f(1)=0 \),

(ii) \( f(p)=1 \) for all prime numbers \( p \),

(iii) \( f(xy)=y f(x)+x f(y) \) for all \( x, y \in \mathbb{Z}_{>0} \).

Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n)=n \).

(Gerhard J. Woeginger)

Answer: $3125$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), the area of the square is:

Answer: $36$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

What is the maximum value that can be taken by the sum

$$
\left|x_{1}-1\right|+\left|x_{2}-2\right|+\ldots+\left|x_{63}-63\right|
$$

if $x_{1}, x_{2}, \ldots, x_{63}$ are some permutation of the numbers $1, 2, 3, \ldots, 63$?

Answer: $1984$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a function $f: \{1, 2, 3\} \rightarrow \{1, 2, 3\}$ that satisfies $f(f(x)) = f(x)$, determine the total number of such functions.

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the sequence ${a_n}$ satisfying $a_1=1$, $a_2=2$, $a_3=3$, $a_{n+3}=a_n$ ($n\in\mathbb{N}^*$). If $a_n=A\sin(\omega n+\varphi)+c$ $(ω>0,|\varphi|<\frac{\pi}{2})$, find the real number $A$.

Answer: $-\frac{2\sqrt{3}}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone.

Answer: $\frac{1152}{125} \pi$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A string of 33 pearls has its middle pearl as the largest and most valuable. The values of the remaining pearls decrease by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the string is 25 times the value of the fourth pearl from the middle on the more expensive side?

Answer: $90000$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \).

Answer: $15$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The base of a prism is an equilateral triangle $ABC$. The lateral edges of the prism $AA_1$, $BB_1$, and $CC_1$ are perpendicular to the base. A sphere, whose radius is equal to the edge of the base of the prism, touches the plane $A_1B_1C_1$ and the extensions of the segments $AB_1$, $BC_1$, and $CA_1$ beyond the points $B_1$, $C_1$, and $A_1$, respectively. Find the sides of the base of the prism, given that the lateral edges are each equal to 1.

Answer: $\sqrt{44} - 6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A square board with three rows and three columns contains nine cells. In how many different ways can we write the three letters A, B, and C in three different cells, so that exactly one of these three letters is written in each row?

Answer: $162$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first round. How many people were sitting at the table?

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the prism \( A B C - A_{1} B_{1} C_{1} \), \( A B \) is perpendicular to the lateral face \( B B_{1} C_{1} C \). Point \( E \) lies on edge \( C C_{1} \) such that \( E \ne C \) and \( E \ne C_{1} \). Given that \( E A \perp E B_1 \), \( A B = \sqrt{2} \), \( B B_1 = 2 \), \( B C = 1 \), and \( \angle B C C_1 = \frac{\pi}{3} \), find the tangent of the dihedral angle between the planes \( A-E B_1 \) and \( A_1 \).

Answer: $\frac{\sqrt{2}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed?

Answer: $3600$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find all odd natural numbers greater than 500 but less than 1000, each of which has the property that the sum of the last digits of all its divisors (including 1 and the number itself) is equal to 33.

Answer: $729$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89.

Answer: $1383$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

For the four-digit number \(\overline{abcd}\) where \(1 \leqslant a \leqslant 9\) and \(0 \leqslant b, c, d \leqslant 9\), if \(a > b, b < c, c > d\), then \(\overline{abcd}\) is called a \(P\)-type number. If \(a < b, b > c, c < d\), then \(\overline{abcd}\) is called a \(Q\)-type number. Let \(N(P)\) and \(N(Q)\) represent the number of \(P\)-type and \(Q\)-type numbers respectively. Find the value of \(N(P) - N(Q)\).

Answer: $285$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Three positive integers are each greater than $1$, have a product of $1728$, and are pairwise relatively prime. What is their sum?

Answer: $43$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.

Answer: $1/8$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region.

Answer: $9 \sqrt{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base.

Answer: $\frac{1}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the eccentricity is \( \frac{\sqrt{17}}{3} \). Let \( F \) be the right focus, and points \( A \) and \( B \) lie on the right branch of the hyperbola. Let \( D \) be the point symmetric to \( A \) with respect to the origin \( O \), with \( D F \perp A B \). If \( \overrightarrow{A F} = \lambda \overrightarrow{F B} \), find \( \lambda \).

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a building with 10 mailboxes, a distributor places a flyer in 5 of the mailboxes. Later, another distributor also places a flyer in 5 of the mailboxes. What is the probability that at least 8 mailboxes receive a flyer?

Answer: $1/2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( T \) be a right triangle with sides having lengths 3, 4, and 5. A point \( P \) is called awesome if \( P \) is the center of a parallelogram whose vertices all lie on the boundary of \( T \). What is the area of the set of awesome points?

Answer: $3/2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a connected simple graph \( G \) with \( e \) edges and pieces placed on each vertex of \( G \) (where each piece can only be placed on a single vertex of \( G \)), you are allowed to perform the following operation: if the number of pieces on a vertex \( v \) is at least the number of vertices adjacent to \( v \) (denoted as \( d \)), you can choose \( d \) pieces from \( v \) and distribute them to the adjacent vertices, giving each one piece. If the number of pieces on each vertex is less than the number of adjacent vertices, you cannot perform any operation.

Find the minimum value of \( m \) (the total number of pieces) such that there is an initial placement of pieces which allows you to perform an infinite number of operations.

Answer: $e$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).

Majka created one three-digit funny number and one three-digit cheerful number, using six different digits without including 0. The sum of these two numbers was 1617. The product of these two numbers ended with the digits 40.

Determine Majka's numbers and calculate their product.

Answer: $635040$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given \( 5 \sin 2 \alpha = \sin 2^\circ \), find the value of \( \frac{\tan (\alpha + 1^\circ)}{\tan (\alpha - 1^\circ)} \).

Answer: $-\frac{3}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In tetrahedron \(ABCD\), \(AB = 1\), \(BC = 5\), \(CD = 7\), \(DA = 5\), \(AC = 5\), \(BD = 2\sqrt{6}\). Find the distance between skew lines \(AC\) and \(BD\).

Answer: $\frac{3\sqrt{11}}{10}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Evaluate $(128)^{\frac{1}{3}}(729)^{\frac{1}{2}}$.

Answer: $108 \cdot 2^{\frac{1}{3}}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A seven-digit phone number \(d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7}\) is called "memorable" if the initial three digits \(d_{1} d_{2} d_{3}\) match either the middle three digits \(d_{4} d_{5} d_{6}\) or the last three digits \(d_{5} d_{6} d_{7}\) (it is possible for all three groups to be the same). Each digit can be one of the digits \(0,1,2,3,\cdots,8,9\). How many different "memorable" phone numbers are there?

Answer: $19990$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers \( n \) satisfy \( f(n) = f(2005) \) ?

Answer: $225$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \(4: 7: 14\). On the twelfth day, more people participated in the survey, which changed the ratio of the response frequencies to \(6: 9: 16\). What is the minimum number of people who must have responded to the survey on the twelfth day?

Answer: $75$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find a positive integer \( n \) with 1000 digits, none of which are 0, such that we can group the digits into 500 pairs so that the sum of the products of the numbers in each pair divides \( n \).


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Two teams of 20 people each participated in a relay race from Moscow to Petushki. Each team divided the distance into 20 segments, not necessarily of equal length, and assigned them to participants such that each member runs exactly one segment (each participant maintains a constant speed, but the speeds of different participants may vary). The first participants of both teams started simultaneously, and the baton exchange occurs instantaneously. What is the maximum number of overtakes that could occur in such a relay race? Note that an overtaking at the boundary of the stages is not considered as an overtake.

(2023-96, E. Neustroeva)

Answer: $38$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:

- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adventurer has rubies, they either have emeralds or diamonds (but not both simultaneously).

What is the minimum number of adventurers that could be in this group?

Answer: $17$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the function \( f: \mathbf{R} \rightarrow \mathbf{R} \), for any real numbers \( x, y, z \), the inequality \(\frac{1}{3} f(x y) + \frac{1}{3} f(x z) - f(x) f(y z) \geq \frac{1}{9} \) always holds. Find the value of \(\sum_{i=1}^{100} [i f(i)]\), where \([x]\) represents the greatest integer less than or equal to \( x \).

Answer: $1650$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R}_{+} \), if the function has real roots, determine the maximum value of \( \min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\} \).

Answer: $5/4$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The base of the pyramid is a parallelogram with adjacent sides of 9 cm and 10 cm, and one of the diagonals measuring 11 cm. The opposite lateral edges are equal, and each of the longer edges is 10.5 cm. Calculate the volume of the pyramid.

Answer: $200$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[2\mathbf{a} + 3\mathbf{b} + 4\mathbf{c} = \mathbf{0}.\]Compute $2(\mathbf{a} \cdot \mathbf{b}) + 3(\mathbf{a} \cdot \mathbf{c}) + 4(\mathbf{b} \cdot \mathbf{c}).$

Answer: $-673/4$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 contain the digit one, 19 contain the digit two, and 30 contain neither one nor two. How many numbers did Misha erase?

Answer: $33$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.

How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calculated to the nearest $\frac{1}{10} \mathrm{~mm}$. Air resistance, etc., are not to be considered.)

Answer: $34.6$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $\sum_{k=1}^{36}\sin 4k=\tan \frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\frac{p}{q}<90,$ find $p+q.$

Answer: $73$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a city, there are 10,000 bicycles with all possible numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered does not contain the digit 8?

Answer: $0.6561$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Fill in the appropriate numbers in the parentheses:

(1) 7÷9= $$\frac {(    )}{(    )}$$

(2) $$\frac {12}{7}$$=\_\_\_\_÷\_\_\_\_\_

(3) 3 $$\frac {5}{8}$$= $$\frac {(    )}{(    )}$$

(4) 6= $$\frac {()}{11}$$

Answer: $\frac {66}{11}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the coefficient of \(x^{29}\) in the expansion of \(\left(1 + x^{5} + x^{7} + x^{9}\right)^{16}\).

Answer: $65520$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $\overline{2 a 1 b 9}$ represents a five-digit number, how many ordered digit pairs $(a, b)$ are there such that
$$
\overline{2 a 1 b 9}^{2019} \equiv 1 \pmod{13}?
$$

Answer: $23$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability  $\frac{1}{2}$ . What is the expected value of the number of games they will play?

Answer: $14$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a regular triangular prism \(ABC-A_1B_1C_1\), all 9 edges are equal in length. Point \(P\) is the midpoint of \(CC_1\). The dihedral angle \(B-A_1P-B_1 = \alpha\). Find \(\sin \alpha\).

Answer: $\frac{\sqrt{10}}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \).

Answer: $999$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7.

Answer: $16687$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Which integers from 1 to 60,000 (inclusive) are more numerous and by how much: those containing only even digits in their representation, or those containing only odd digits in their representation?

Answer: $780$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a board game played with dice, our piece is four spaces away from the finish line. If we roll at least a four, we reach the finish line. If we roll a three, we are guaranteed to finish in the next roll.

What is the probability that we will reach the finish line in more than two rolls?

Answer: $1/12$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection.

Answer: $40$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The greatest common divisor (GCD) of 17 and 51 is     , and the least common multiple (LCM) is     . The GCD of 6 and 8 is     , and the LCM of 8 and 9 is     .

Answer: $72$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A polynomial with integer coefficients is of the form
\[12x^3 - 4x^2 + a_1x + 18 = 0.\]
Determine the number of different possible rational roots of this polynomial.

Answer: $20$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?

Answer: $\frac{\sqrt{2}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins. If Nathaniel goes first, determine the probability that he ends up winning.

Answer: $\frac{5}{11}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Draw a tangent line MN to the circle $(x-2)^2+(y-2)^2=1$ at point N, where N is the point of tangency. If $|MN|=|MO|$ (where O is the origin), then the minimum value of $|MN|$ is \_\_\_\_\_\_.

Answer: $\frac{7\sqrt{2}}{8}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\). Point \(P\) is a moving point on side \(DC\) (including \(D\) and \(C\)), and point \(Q\) is a moving point on the extension of side \(CB\) (including point \(B\)), such that \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). Find the minimum value of the dot product \(\overrightarrow{PA} \cdot \overrightarrow{PQ}\).

Answer: $3/4$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Today our cat gave birth to kittens! It is known that the two lightest kittens together weigh 80 g, the four heaviest kittens together weigh 200 g, and the total weight of all the kittens is 500 g. How many kittens did the cat give birth to?

Answer: $11$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a rectangular coordinate system with origin point $O$, vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfy $|\overrightarrow {a}|=|\overrightarrow {b}|=1$ and $\overrightarrow {a}\cdot \overrightarrow {b}=\frac {1}{2}$. Let $\overrightarrow {c} = (m, 1-m)$ and $\overrightarrow {d} = (n, 1-n)$. For any real numbers $m$ and $n$, the inequality $|\overrightarrow {a} - \overrightarrow {c}| + |\overrightarrow {b} - \overrightarrow {d}| \geq T$ holds. Determine the range of possible values for the real number $T$.

Answer: $\frac{\sqrt{6} - \sqrt{2}}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

What is the smallest positive integer that is divisible by 111 and has the last four digits as 2004?

Answer: $662004$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \mathrm{~km/h}$. If they run at full speed either towards $A$ or $B$ station, they can just catch the train. What is the maximum speed at which they can run?

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In every acyclic graph with 2022 vertices we can choose  $k$  of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of  $k$ .

Answer: $1517$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the largest integer $x$ for which $4^{27} + 4^{1010} + 4^{x}$ is a perfect square.

Answer: $1992$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given.

Find the side \(AC\), if it is known that the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\).

Answer: $\sqrt{13}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.

Answer: $288$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $\cos A= \frac {3}{4}$, $\cos C= \frac {1}{8}$,
(I) find the ratio $a:b:c$;
(II) if $| \overrightarrow{AC}+ \overrightarrow{BC}|= \sqrt {46}$, find the area of $\triangle ABC$.

Answer: $\frac {15 \sqrt {7}}{4}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A positive integer is *happy* if:

1. All its digits are different and not  $0$ ,
2. One of its digits is equal to the sum of the other digits.

For example, 253 is a *happy* number. How many *happy* numbers are there?

Answer: $32$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

By expanding the expression \((1+\sqrt{11})^{214}\) using the binomial theorem, we obtain terms of the form \(C_{214}^{k}(\sqrt{11})^{k}\). Find the value of \( k \) for which this term has the greatest value.

Answer: $165$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Inside a cube \( C \) with side length 1, an inscribed sphere \( O_{1} \) is placed. Another sphere \( O_{2} \) is also placed inside \( C \) such that it is tangent to the inscribed sphere \( O_{1} \) and also tangent to three faces of the cube. What is the surface area of sphere \( O_{2} \)?

Answer: $(7 - 4\sqrt{3})\pi$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A regular hexagon \( A B C D E K \) is inscribed in a circle of radius \( 3 + 2\sqrt{3} \). Find the radius of the circle inscribed in the triangle \( B C D \).

Answer: $\frac{3}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

My friend Ana likes numbers that are divisible by 8. How many different pairs of last two digits are possible in numbers that Ana likes?

Answer: $13$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a corridor that is 100 meters long, there are 20 rugs with a total length of 1 kilometer. Each rug is as wide as the corridor. What is the maximum possible total length of the sections of the corridor that are not covered by the rugs?

Answer: $50$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Acute angles \( A \) and \( B \) of a triangle satisfy the equation \( \tan A - \frac{1}{\sin 2A} = \tan B \) and \( \cos^2 \frac{B}{2} = \frac{\sqrt{6}}{3} \). Determine the value of \( \sin 2A \).

Answer: $\frac{2\sqrt{6} - 3}{3}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?

Answer: $26$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( ABC \) be a triangle with \(\angle A = 60^\circ\). Line \(\ell\) intersects segments \( AB \) and \( AC \) and splits triangle \( ABC \) into an equilateral triangle and a quadrilateral. Let \( X \) and \( Y \) be on \(\ell\) such that lines \( BX \) and \( CY \) are perpendicular to \(\ell\). Given that \( AB = 20 \) and \( AC = 22 \), compute \( XY \).

Answer: $21$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find all values of \( x \) for which the smaller of the numbers \( \frac{1}{x} \) and \( \sin x \) is greater than \( \frac{1}{2} \). In the answer, provide the total length of the resulting intervals on the number line, rounding to the nearest hundredth if necessary.

Answer: $2.09$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years.

Answer: $0.8$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

For what is the largest natural number \( m \) such that \( m! \cdot 2022! \) is a factorial of a natural number?

Answer: $2022! - 1$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a quadrilateral pyramid \(S A B C D\):

- The lateral faces \(S A B, S B C, S C D, S D A\) have areas of 9, 9, 27, and 27 respectively;
- The dihedral angles at the edges \(A B, B C, C D, D A\) are equal;
- The quadrilateral \(A B C D\) is inscribed in a circle, and its area is 36.

Find the volume of the pyramid \(S A B C D\).

Answer: $54$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

How many times must a die be rolled so that the probability of the inequality

\[ \left| \frac{m}{n} - \frac{1}{6} \right| \leq 0.01 \]

is at least as great as the probability of the opposite inequality, where \( m \) is the number of times a specific face appears in \( n \) rolls of the die?

Answer: $632$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

1. Given non-negative real numbers \( x, y, z \) satisfying \( x^{2} + y^{2} + z^{2} + x + 2y + 3z = \frac{13}{4} \), determine the maximum value of \( x + y + z \).

2. Given \( f(x) \) is an odd function defined on \( \mathbb{R} \) with a period of 3, and when \( x \in \left(0, \frac{3}{2} \right) \), \( f(x) = \ln \left(x^{2} - x + 1\right) \). Find the number of zeros of the function \( f(x) \) in the interval \([0,6]\).

Answer: $\frac{3}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pants, a shirt, and shoes, and wore a jacket for some lessons. On any two lessons, at least one element of his attire or shoes was different. It is known that if he had taken one more shirt, he could have conducted 36 more lessons; if he had taken one more pair of pants, he could have conducted 72 more lessons; if he had taken one more pair of shoes, he could have conducted 54 more lessons. What is the maximum number of lessons he could have conducted under these conditions?

Answer: $216$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\), satisfying
1. \(x_2 - x_1 = \lambda\);
2. \(x_3 > \frac{1}{2}(x_1 + x_2)\).

Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).

Answer: $\frac{3 \sqrt{3}}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given vectors $\overrightarrow{a}=(-1,0)$ and $\overrightarrow{b}=\left( \frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: $\frac{5\pi}{6}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). If the reflex angle at \( G \) is \( x^\circ \), then \( x \) equals:

Answer: $340$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers $n$, $k$, and $l$, where $n! = 1 \cdot 2 \cdots n$. In your answer, indicate 0 if there are no solutions, specify the value of $n$ if there is only one solution, or provide the sum of all $n$ values if there are multiple solutions. Recall that a solution is a triple $(n, k, l)$; if they differ in at least one component, they are considered different solutions.

Answer: $10$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Fill the first eight positive integers in a $2 \times 4$ table, one number per cell, such that each row's four numbers increase from left to right, and each column's two numbers increase from bottom to top. How many different ways can this be done?

Answer: $14$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A graph shows the number of books read in June by the top readers in a school library. The data points given are:
- 4 readers read 3 books each
- 5 readers read 5 books each
- 2 readers read 7 books each
- 1 reader read 10 books
Determine the mean (average) number of books read by these readers.

Answer: $5.0833$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In $\triangle ABC$, $AB = 10$ and the height from $A$ to $BC$ is 3. When the product $AC \cdot BC$ is minimized, what is $AC + BC$?

Answer: $4\sqrt{10}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \, \text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both boys spent 10 minutes at the respective relative's house. On their way back, they met again $400 \, \text{m}$ away from Károly's house. How far apart do the two families live? What assumptions can we make to answer this question?

Answer: $1760$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

On the board, there are three two-digit numbers. One starts with 5, another starts with 6, and the third one starts with 7. The teacher asked three students to each choose any two of these numbers and add them together. The first student got 147, and the results of the second and third students were different three-digit numbers starting with 12. What could be the number starting with 7? If there are multiple answers, list them all.

Answer: $78$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \( n \) be an integer between 1 and 1990, and let \( x^2 + x - 3n \) be factored into the product of two linear factors with integer coefficients. Determine the number of integers \( n \) that satisfy this condition.

Answer: $50$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that $11^{-1} \equiv 3 \pmod{31}$, find $20^{-1} \pmod{31}$. Provide the result as a residue modulo 31 (a number between 0 and 30, inclusive).

Answer: $28$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Consider a $4 \times 4$ matrix filled with each number from $1$ to $16$. Each number must be placed such that the entries in each row and in each column are in increasing order. Find the number of such matrices that exist.

Answer: $24$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The product of several distinct positive integers is divisible by  ${2006}^{2}$ . Determine the minimum value the sum of such numbers can take.

Answer: $228$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given set \( A = \{0, 1, 2, 3, 4, 5, 9\} \), and \( a, b \in A \) where \( a \neq b \). The number of functions of the form \( y = -a x^2 + (4 - b)x \) whose vertex lies in the first quadrant is ___.

Answer: $21$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

The segments of two lines, enclosed between two parallel planes, are in the ratio of \( 5:9 \), and the acute angles between these lines and one of the planes are in the ratio of \( 2:1 \), respectively. Find the cosine of the smaller angle.

Answer: $0.9$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the last two digits of \(\left[(\sqrt{29}+\sqrt{21})^{1984}\right]\).

Answer: $71$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In how many ways can one arrange the natural numbers from 1 to 9 in a $3 \times 3$ square table so that the sum of the numbers in each row and each column is odd? (Numbers can repeat)

Answer: $6 * 4^6 * 5^3 + 9 * 4^4 * 5^5 + 5^9$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the number of four-digit numbers, composed of the digits 1, 2, 3, 4, 5, 6, 7 (each digit can be used no more than once), that are divisible by 15.

Answer: $36$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit?

Answer: $70$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the isosceles triangle \(ABC\) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\). In what ratio does the line \(AD\) divide the altitude \(BE\) of triangle \(ABC\), counting from vertex \(B\)?

Answer: $1:2$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given are  $100$  positive integers whose sum equals their product. Determine the minimum number of  $1$ s that may occur among the  $100$  numbers.

Answer: $95$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the definite integral:

$$
\int_{\pi / 4}^{\arcsin \sqrt{2 / 3}} \frac{8 \tan x \, dx}{3 \cos ^{2} x+8 \sin 2 x-7}
$$

Answer: $\frac{4}{21} \ln \left| \frac{7\sqrt{2} - 2}{5} \right| - \frac{4}{3} \ln |2 - \sqrt{2}|$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let \(ABC\) be a triangle such that the altitude from \(A\), the median from \(B\), and the internal angle bisector from \(C\) meet at a single point. If \(BC = 10\) and \(CA = 15\), find \(AB^2\).

Answer: $205$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

A $7 \times 7$ board is either empty or contains an invisible $2 \times 2$ ship placed "on the cells." Detectors can be placed on certain cells of the board and then activated simultaneously. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detectors needed to guarantee determining whether the ship is on the board and, if so, identifying the cells it occupies?

Answer: $16$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Let  $X=\{2^m3^n|0 \le m, \ n \le 9 \}$ . How many quadratics are there of the form  $ax^2+2bx+c$ , with equal roots, and such that  $a,b,c$  are distinct elements of  $X$ ?

Answer: $9900$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\).

Answer: $2020032$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the integer root of the polynomial \[2x^3 + ax^2 + bx + c = 0,\] where $a, b$, and $c$ are rational numbers. The equation has $4-2\sqrt{3}$ as a root and another root whose sum with $4-2\sqrt{3}$ is 8.

Answer: $-8$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given a parabola \(C\) with the center of ellipse \(E\) as its focus, the parabola \(C\) passes through the two foci of the ellipse \(E\), and intersects the ellipse \(E\) at exactly three points. Find the eccentricity of the ellipse \(E\).

Answer: $\frac{2 \sqrt{5}}{5}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Then $b$ is:

Answer: $\frac{1}{2}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Through two vertices of an equilateral triangle \(ABC\) with an area of \(21 \sqrt{3} \ \text{cm}^2\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle.

Answer: $2\sqrt{7}$