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 \(10 \times 1\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at least one green and one yellow tile should be used, and adjacent tiles should have different colors?

Answer: $1022$


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>.

Petya and Vasya calculated that if they walk at a speed of 4 km per hour to the neighboring village, which is 4 kilometers away, they will be 10 minutes late for the football match held there for the district championship. How should they proceed to arrive at the match on time and achieve the greatest time gain, having at their disposal a bicycle that can only be ridden by one person but goes three times faster than walking? How many minutes before the start of the match will they arrive?

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>.

Let \( P \) be a regular polygon with 2006 sides. A diagonal of \( P \) is called good if its endpoints divide the perimeter of \( P \) into two parts, each having an odd number of sides of \( P \). The sides of \( P \) are also called good. Suppose \( P \) has been subdivided into triangles by 2003 diagonals that do not intersect at an interior point of \( P \). Find the maximum number of isosceles triangles that can appear in such a subdivision, where two sides of the isosceles triangles are good.

Answer: $1003$


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 line \( l \) passes through the focus \( F \) of the parabola \( y^2 = 4x \) and intersects the parabola at points \( A \) and \( B \). Point \( M \) is given as \( (4,0) \). Extending \( AM \) and \( BM \) intersects the parabola again at points \( C \) and \( D \), respectively. Find the value of \(\frac{S_{\triangle CDM}}{S_{\triangle ABM}}\).

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>.

In triangle \( \triangle ABC \), \( AB = 8 \), \( BC = 11 \), and \( AC = 6 \). The points \( P \) and \( Q \) are on \( BC \) such that \( \triangle PBA \) and \( \triangle QAC \) are each similar to \( \triangle ABC \). What is the length of \( PQ \)?

Answer: $\frac{21}{11}$


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>.

For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\).

Answer: $312$


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 sequence \(\{a_n\}\) has consecutive terms \(a_n\) and \(a_{n+1}\) as the roots of the equation \(x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0\), with initial term \(a_1 = 2\). Find the sum of the infinite series \(c_1, c_2, \cdots, c_n, \cdots\).

Answer: $\frac{9}{2}$


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>.

For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?

Answer: $256$


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 isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vertices fill the leg. Determine the sum of the areas of these equilateral triangles.

Answer: $324$


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 $1,2, \ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible?

Answer: $1017072$


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 each positive integer \( n \), define \( A_{n} = \frac{20^{n} + 11^{n}}{n!} \), where \( n! = 1 \times 2 \times \cdots \times n \). Find the value of \( n \) that maximizes \( A_{n} \).

Answer: $19$


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 notebook, there is a grid rectangle of size $3 \times 7$. Igor's robot was asked to trace all the lines with a marker, and it took him 26 minutes (the robot draws lines at a constant speed). How many minutes will it take him to trace all the lines in a $5 \times 5$ grid square with the marker?

Answer: $30$


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>.

Point \( M \) divides the side \( BC \) of parallelogram \( ABCD \) in the ratio \( BM: MC = 1: 3 \). Line \( AM \) intersects diagonal \( BD \) at point \( K \). Find the area of quadrilateral \( CMKD \) if the area of parallelogram \( ABCD \) is 1.

Answer: $\frac{19}{40}$


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>.

Dad says he is exactly 35 years old, not counting weekends. How old is he really?

Answer: $49$


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 an isosceles trapezoid, one base is \(40 \text{ cm}\) and the other is \(24 \text{ cm}\). The diagonals of this trapezoid are mutually perpendicular. Find its area.

Answer: $1024$


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 9-digit numbers that are divisible by 5 can be formed by permuting the digits of the number 377353752?

Answer: $1120$


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 \(X_{0}\) be the interior of a triangle with side lengths 3, 4, and 5. For all positive integers \(n\), define \(X_{n}\) to be the set of points within 1 unit of some point in \(X_{n-1}\). The area of the region outside \(X_{20}\) but inside \(X_{21}\) can be written as \(a\pi + b\), for integers \(a\) and \(b\). Compute \(100a + b\).

Answer: $4112$


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 pyramid with a triangular base has edges of unit length, and the angles between its edges are \(60^{\circ}, 90^{\circ},\) and \(120^{\circ}\). What is the volume of the pyramid?

Answer: $\frac{\sqrt{2}}{12}$


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>.

Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "There are no more than 17 of us in the hut," and then noted, "Some of us are from the Ah tribe." A third person said, "There are five of us in the hut," and looking around, added, "There are at least three Uh tribe members among us." How many Ah tribe members are in the hut?

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>.

Let point \( P \) lie on the face \( ABC \) of a tetrahedron \( ABCD \) with edge length 2. The distances from \( P \) to the planes \( DAB \), \( DBC \), and \( DCA \) form an arithmetic sequence. Find the distance from \( P \) to the plane \( DBC \).

Answer: $\frac{2\sqrt{6}}{9}$


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>.

There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.

Answer: $153$


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>.

Person A and Person B start walking towards each other from points $A$ and $B$ respectively, which are 10 kilometers apart. If they start at the same time, they will meet at a point 1 kilometer away from the midpoint of $A$ and $B$. If Person A starts 5 minutes later than Person B, they will meet exactly at the midpoint of $A$ and $B$. Determine how long Person A has walked in minutes in this scenario.

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 fixed points \( A(3,0) \), \( B(0,4) \), and point \( P \) on the incircle of triangle \( \triangle AOB \) (where \( O \) is the origin), find the maximum value of \( |PA|^2 + |PB|^2 + |PO|^2 \).

Answer: $22$


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 $ABCD$, $AD > AB$, $AD \perp AB$, $AD \perp AC$, $\angle BAC = \frac{\pi}{3}$. Let the areas of triangles $ADB$, $ADC$, $ABC$, and $BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively. It is known that $S_{1} + S_{2} = S_{3} + S_{4}$. Find the value of $\frac{S_{3}}{S_{1}} + \frac{S_{3}}{S_{2}}$.

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>.

Find the maximum value of the expression

$$
\frac{a}{x} + \frac{a+b}{x+y} + \frac{a+b+c}{x+y+z}
$$

where \( a, b, c \in [2,3] \), and the triplet of numbers \( x, y, z \) is some permutation of the triplet \( a, b, c \).

Answer: $15/4$


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 \(2x^2 + 3xy + 2y^2 = 1\), find the minimum value of \(f(x, y) = x + y + xy\).

Answer: $-\frac{9}{8}$


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>.

Let \( T \) be the set of all positive divisors of \( 60^{100} \). \( S \) is a subset of \( T \) such that no number in \( S \) is a multiple of another number in \( S \). Find the maximum value of \( |S| \).

Answer: $10201$


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>.

Point \( M \) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \( M \) to the lines containing the other sides of the hexagon.

Answer: $30\sqrt{3}$


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>.

Given real numbers \(a\) and \(b\) that satisfy \(0 \leqslant a, b \leqslant 8\) and \(b^2 = 16 + a^2\), find the sum of the maximum and minimum values of \(b - a\).

Answer: $12 - 4\sqrt{3}$


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>.

Let \( \triangle DEF \) be a triangle and \( H \) the foot of the altitude from \( D \) to \( EF \). If \( DE = 60 \), \( DF = 35 \), and \( DH = 21 \), what is the difference between the minimum and the maximum possible values for the area of \( \triangle DEF \)?

Answer: $588$


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 start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to purchase gifts from all three stores and still have money left?

Answer: $17/27$


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 largest natural number whose all digits in its decimal representation are different and which decreases 5 times if you cross out the first digit.

Answer: $3750$


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>.

Divide each natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number). Is there a smallest quotient among the obtained quotients, and if so, which one is it?

Answer: $1/9$


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>.

Suppose \(A, B\) are the foci of a hyperbola and \(C\) is a point on the hyperbola. Given that the three sides of \(\triangle ABC\) form an arithmetic sequence, and \(\angle ACB = 120^\circ\), determine the eccentricity of the hyperbola.

Answer: $7/2$


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>.

For the pair of positive integers \((x, y)\) such that \(\frac{x^{2}+y^{2}}{11}\) is an integer and \(\frac{x^{2}+y^{2}}{11} \leqslant 1991\), find the number of such pairs \((x, y)\) (where \((a, b)\) and \((b, a)\) are considered different pairs if \(a \neq b\)).

Answer: $131$


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 \( M_{n} = \left\{ 0 . \overline{a_{1} a_{2} \cdots a_{n}} \mid a_{i} \ \text{is either 0 or 1 for} \ i=1,2, \cdots, n-1, \ a_{n}=1 \right\} \). \( T_{n} \) is the number of elements in \( M_{n} \) and \( S_{n} \) is the sum of all elements in \( M_{n} \). Find \( \lim_{n \rightarrow \infty} \frac{S_{n}}{T_{n}} \).

Answer: $1/18$


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>.

When measuring a part, random errors occur that follow a normal distribution with a parameter $\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15$ mm.

Answer: $0.8664$


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 write on the board the equation

$$
(x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016),
$$

where there are 2016 linear factors on each side. What is the smallest positive value of $k$ such that we can omit exactly $k$ of these 4032 linear factors in such a way that there is at least one linear factor on each side, and the resulting equation has no real roots?

Answer: $2016$


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 kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such divisions into circles are considered the same if one can be obtained from the other by swapping the trees (along with their respective circles) and rotating each circle around its tree. In how many ways can the children be divided into the circles?

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>.

Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), then \( \{a, b\} \) and \( \{c, d\} \) are called a balanced pair of the set \( M \). Find the number of balanced pairs in the set \( M \).

Answer: $476$


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>.

There is a simple pendulum with a period of $T=1$ second in summer. In winter, the length of the pendulum shortens by 0.01 centimeters. How many seconds faster is this pendulum in winter over a 24-hour period compared to summer? (Round to the nearest second).

Note: The formula for the period of a simple pendulum is $T=2 \pi \sqrt{\frac{l}{g}}$, where the period $T$ is in seconds, the length $l$ is in centimeters, and the acceleration due to gravity $g$ is $980$ cm/s².

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>.

On a highway, there are checkpoints D, A, C, and B arranged in sequence. A motorcyclist and a cyclist started simultaneously from A and B heading towards C and D, respectively. After meeting at point E, they exchanged vehicles and each continued to their destinations. As a result, the first person spent 6 hours traveling from A to C, and the second person spent 12 hours traveling from B to D. Determine the distance of path AB, given that the speed of anyone riding a motorcycle is 60 km/h, and the speed on a bicycle is 25 km/h. Additionally, the average speed of the first person on the path AC equals the average speed of the second person on the path BD.

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>.

The country Omega grows and consumes only vegetables and fruits. It is known that in 2014, 1200 tons of vegetables and 750 tons of fruits were grown in Omega. In 2015, 900 tons of vegetables and 900 tons of fruits were grown. During the year, the price of one ton of vegetables increased from 90,000 to 100,000 rubles, and the price of one ton of fruits decreased from 75,000 to 70,000 rubles. By what percentage (%) did the real GDP of this country change in 2015, if the base year in Omega is 2014? Round your answer to two decimal places. If the real GDP of the country decreased, put a minus sign in the answer, and if it increased, put a plus sign.

Answer: $-9.59$


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 identical cylindrical vessels are connected by a small tube with a valve at the bottom. Initially, the valve is closed, and water is poured into the first vessel while oil is poured into the second vessel, such that the liquid levels are equal and are $h=40$ cm. At what level will the water be in the first vessel if the valve is opened? The density of water is 1000 kg/m³, and the density of oil is 700 kg/m³. Neglect the volume of the connecting tube. Provide the answer in centimeters.

Answer: $32.94$


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 triangle with sides 6 cm, 10 cm, and 12 cm, an inscribed circle is tangent to the two longer sides. Find the perimeter of the resulting triangle formed by the tangent line and the two longer sides.

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>.

At 30 palm trees on different parts of an uninhabited island, a sign is attached.

- On 15 of them it says: "Exactly under 15 signs a treasure is buried."
- On 8 of them it says: "Exactly under 8 signs a treasure is buried."
- On 4 of them it says: "Exactly under 4 signs a treasure is buried."
- On 3 of them it says: "Exactly under 3 signs a treasure is buried."

It is known that only those signs under which there is no treasure are truthful.

Under the smallest number of signs can a treasure be buried?

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>.

Five monkeys share a pile of peanuts. The first monkey divides the peanuts into five piles, leaving one peanut which it eats, and takes away one pile. The second monkey then divides the remaining peanuts into five piles, leaving exactly one peanut, eats it, and takes away one pile. This process continues in the same manner until the fifth monkey, who also evenly divides the remaining peanuts into five piles and has one peanut left over. What is the minimum number of peanuts in the pile originally?

Answer: $3121$


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 skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from the turning point. Find the speed of the second skier.

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>.

In parallelogram \(ABCD\), \(AB = 1\), \(BC = 4\), and \(\angle ABC = 60^\circ\). Suppose that \(AC\) is extended from \(A\) to a point \(E\) beyond \(C\) so that triangle \(ADE\) has the same area as the parallelogram. Find the length of \(DE\).

Answer: $2\sqrt{3}$


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>.

The teacher asked the students to calculate \(\overline{AB} . C + D . E\). Xiao Hu accidentally missed the decimal point in \(D . E\), getting an incorrect result of 39.6; while Da Hu mistakenly saw the addition sign as a multiplication sign, getting an incorrect result of 36.9. What should the correct calculation result be?

Answer: $26.1$


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>.

There is a house at the center of a circular field. From it, 6 straight roads radiate, dividing the field into 6 equal sectors. Two geologists start their journey from the house, each choosing a road at random and traveling at a speed of 4 km/h. Determine the probability that the distance between them after an hour is at least 6 km.

Answer: $0.5$


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>.

Find all positive integers \( n \) such that \( n \) is equal to 100 times the number of positive divisors of \( n \).

Answer: $2000$


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 scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley?

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>.

The sequence $\left\{a_{n}\right\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Calculate $S_{2016}$.

Answer: $7066$


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 plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________.

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>.

A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles?

Answer: $1002$


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 an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(3:1\) starting from vertex \(B\), and point \(E\) is the midpoint of segment \(AD\). It is known that \(BE = \sqrt{7}\) and \(CE = 3\). Find the radius of the circumcircle of triangle \(ABC\).

Answer: $\frac{8}{3}$


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 maximum constant $k$ such that $\frac{k a b c}{a+b+c} \leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$.

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>.

At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the speed of the current if, at the moment the boats met, the raft had drifted 7.5 km from the pier?

Answer: $2.5$


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>.

Ten points are given in the plane, and no three points are collinear. Four distinct segments connecting pairs of these points are chosen at random, all with the same probability. What is the probability that three of the chosen segments will form a triangle?

Answer: $16/473$


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 \( M_n \) be the set of \( n \)-digit pure decimals in decimal notation \( 0.\overline{a_1a_2\cdots a_n} \), where \( a_i \) can only take the values 0 or 1 for \( i=1,2,\cdots,n-1 \), and \( a_n = 1 \). Let \( T_n \) be the number of elements in \( M_n \), and \( S_n \) be the sum of all elements in \( M_n \). Find \( \lim_{n \rightarrow \infty} \frac{S_n}{T_n} \).

Answer: $\frac{1}{18}$


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 sum of all numbers in the following square matrix?
$$
\begin{array}{l}
1, 2, 3, \ldots, 98, 99, 100 \\
2, 3, 4, \ldots, 99, 100, 101 \\
3, 4, 5, \ldots, 100, 101, 102 \\
\ldots \\
100, 101, 102, \ldots, 197, 198, 199
\end{array}
$$

Answer: $1000000$


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>.

$ABCD$ is a square of side length 1. $X$ and $Y$ are points on $BC$ and $CD$ respectively such that $CX = CY = m$. When extended, $AB$ meets $DX$ at $P$, $AD$ meets $BY$ at $Q$, $AX$ meets $DC$ at $R$, and $AY$ meets $BC$ at $S$. If $P, Q, R, S$ are collinear, find $m$.

Answer: $\frac{3 - \sqrt{5}}{2}$


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>.

A triangle \(ABC\) is to be constructed so that \(A\) is at \((3,2)\), \(B\) is on the line \(y=x\), and \(C\) is on the \(x\)-axis. Find the minimum possible perimeter of \(\triangle ABC\).

Answer: $\sqrt{26}$


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>.

Karlsson eats three jars of jam and one jar of honey in 25 minutes, while Little Brother does it in 55 minutes. Karlsson eats one jar of jam and three jars of honey in 35 minutes, while Little Brother does it in 1 hour 25 minutes. How long will it take them to eat six jars of jam together?

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>.

Let the set \( S = \{1,2,3, \cdots, 10\} \). Given that subset \( A \) of \( S \) satisfies \( A \cap \{1,2,3\} \neq \varnothing \) and \( A \cup \{4,5,6\} \neq S \), determine the number of such subsets \( A \).

(Note: The original problem includes determining the count of subset \( A \) that meets the given conditions.)

Answer: $888$


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>.

Is the following number rational or irrational?

$$
\sqrt[3]{2016^{2} + 2016 \cdot 2017 + 2017^{2} + 2016^{3}} ?
$$

Answer: $2017$


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>.

Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \( A \) such that, in any such set of numbers, each number does not exceed \( A \).

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>.

Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, d \) are constants, and given that \( p(1) = 1993 \), \( p(2) = 3986 \), and \( p(3) = 5979 \). Calculate \( \frac{1}{4} [p(11) + p(-7)] \).

Answer: $5233$


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 rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy?

Answer: $90$


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 square-based pyramid (with base vertices $A, B, C, D$) with equal side edges, and let the apex be $E$. Let $P$ be the point that divides the side edge $A E$ in a ratio of 3:1, such that $E P : P A = 3$, and let $Q$ be the midpoint of the side edge $C E$. In what ratio does the plane passing through points $D, P$, and $Q$ divide the side edge $B E$?

Answer: $4/3$


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 fraction with the smallest possible denominator that lies between \(\frac{7}{25}\) and \(\frac{8}{25}\). Provide all possible solutions.

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


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 parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \). On edge \( A_{1} D_{1} \), point \( X \) is selected, and on edge \( B C \), point \( Y \) is selected. It is known that \( A_{1} X = 5 \), \( B Y = 3 \), and \( B_{1} C_{1} = 14 \). The plane \( C_{1} X Y \) intersects the ray \( D A \) at point \( Z \). Find \( D Z \).

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>.

There are five distinct nonzero natural numbers; the smallest one is 7. If one of them is decreased by 20, and the other four numbers are each increased by 5, the resulting set of numbers remains the same. What is the sum of these five numbers?

Answer: $85$


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 convex quadrilateral $ABCD$, $M$ and $N$ are the midpoints of sides $AD$ and $BC$, respectively. Given that $|\overrightarrow{AB}|=2$, $|\overrightarrow{MN}|=\frac{3}{2}$, and $\overrightarrow{MN} \cdot (\overrightarrow{AD} - \overrightarrow{BC}) = \frac{3}{2}$, find $\overrightarrow{AB} \cdot \overrightarrow{CD}$.

Answer: $-2$


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>.

There are 4 people in the family. If Masha's scholarship is doubled, the total family income will increase by 5%. If the mother's salary is doubled instead, the total family income will increase by 15%. If the father's salary is doubled, the total family income will increase by 25%. By how many percent will the family's income increase if the grandfather's pension is doubled?

Answer: $55$


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 cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of this ellipse is ___.

Answer: $25$


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>.

Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, the number of fish caught in the net was no greater than the number caught on the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three specific days: Monday, Wednesday, and Friday?

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>.

The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 155 and 13 respectively, and its lateral sides are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AC}\) and \(\overrightarrow{BD}\).

Answer: $-2015$


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 \(\{a_n\}\) with the first term 2, and it satisfies
\[ 6 S_n = 3 a_{n+1} + 4^n - 1. \]

Find the maximum value of \(S_n\).

Answer: $35$


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 minimum number of vertices in a graph that contains no cycle of length less than 6 and where every vertex has a degree of 3?

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>.

Cyclic quadrilateral \(ABCD\) has side lengths \(AB = 1\), \(BC = 2\), \(CD = 3\), and \(AD = 4\). Determine \(\frac{AC}{BD}\).

Answer: $5/7$


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 the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is the center $O$ of a sphere with radius 10. The sphere intersects the face $A A_{1} D_{1} D$ along a circle with radius 1, the face $A_{1} B_{1} C_{1} D_{1}$ along a circle with radius 1, and the face $C D D_{1} C_{1}$ along a circle with radius 3. Find the length of the segment $O D_{1}$.

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>.

A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \frac{\text{km}}{\text{hour}}$ on a horizontal track with a friction coefficient of $\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $e=0.030$?

(Note: In this problem, $\rho$ is the friction coefficient and $e$ is the sine of the inclination angle.)

Answer: $10.3$


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 quadrilateral \(P Q R S\) is inscribed in a circle. Diagonals \(P R\) and \(Q S\) are perpendicular and intersect at point \(M\). It is known that \(P S = 13\), \(Q M = 10\), and \(Q R = 26\). Find the area of the quadrilateral \(P Q R S\).

Answer: $319$


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 maximum value of the expression \((\sin 3x + \sin 2y + \sin z)(\cos 3x + \cos 2y + \cos z)\).

(Possible points: 15)

Answer: $4.5$


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>.

We use \( S_{k} \) to represent an arithmetic sequence with the first term \( k \) and common difference \( k^{2} \). For example, \( S_{3} \) is \( 3, 12, 21, \cdots \). If 306 is a term in \( S_{k} \), the sum of all possible \( k \) that satisfy this condition is ____.

Answer: $326$


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 an isosceles trapezoid with bases \(a = 21\), \(b = 9\) and height \(h = 8\), find the radius of the circumscribed circle.

Answer: $\frac{85}{8}$


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>.

The route not passing through the Zoo is 11 times shorter.

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>.

A four-digit number $\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification).

Answer: $1979$


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 parallelepiped \(A B C D A_{1} B_{1} C_{1} D_{1}\). Point \(X\) is chosen on edge \(A_{1} D_{1}\) and point \(Y\) is chosen on edge \(B C\). It is known that \(A_{1} X = 5\), \(B Y = 3\), and \(B_{1} C_{1} = 14\). The plane \(C_{1} X Y\) intersects the ray \(D A\) at point \(Z\). Find \(D Z\).

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>.

In \(\triangle ABC\), \(AB : AC = 4 : 3\) and \(M\) is the midpoint of \(BC\). \(E\) is a point on \(AB\) and \(F\) is a point on \(AC\) such that \(AE : AF = 2 : 1\). It is also given that \(EF\) and \(AM\) intersect at \(G\) with \(GF = 72 \mathrm{~cm}\) and \(GE = x \mathrm{~cm}\). Find the value of \(x\).

Answer: $108$


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 2017 lines separated into three sets such that lines in the same set are parallel to each other, what is the largest possible number of triangles that can be formed with vertices on these lines?

Answer: $673 * 672^2$


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 regular dodecagon $A B C D E F G H I J K L$ inscribed in a circle with a radius of $6 \mathrm{~cm}$, determine the perimeter of the pentagon $A C F H K$.

Answer: $18 + 12\sqrt{2}$


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>.

Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} x \, dx}{(1+\cos x+\sin x)^{2}}
$$

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


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>.

At a certain meeting, a total of \(12k\) people attended. Each person greeted exactly \(3k+6\) other people. For any two individuals, the number of people who greeted both of them is the same. How many people attended the meeting?

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>.

In a $4 \times 5$ grid, place 5 crosses such that each row and each column contains at least one cross. How many ways can this be done?

Answer: $240$


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 black and white chocolate bar consists of individual pieces arranged in $n$ horizontal rows and $m$ vertical columns, painted in a checkerboard pattern. Ian ate all the black pieces, and Max ate all the white pieces. What is the sum of $m + n$ if it is known that Ian ate $8 \frac{1}{3} \%$ more pieces than Max?

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>.

The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) grid such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?

Answer: $25$


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 square $A B C D$ is enlarged from vertex $A$ resulting in the square $A B^{\prime} C^{\prime} D^{\prime}$. The intersection point of the diagonals of the enlarged square is $M$. It is given that $M C = B B^{\prime}$. What is the scale factor of the enlargement?

Answer: $\sqrt{2}$


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>.

An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01?

Answer: $11475$


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>.

100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Problem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of Correct Answers & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}

Criteria: To pass, one must answer at least 6 questions correctly. Based on the table, calculate the minimum number of people who passed.

Answer: $62$


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 natural numbers \( m \) and \( n \) where \( n > m > 1 \), the last three digits of the decimal representation of \( 1978^m \) and \( 1978^n \) are the same. Find \( m \) and \( n \) such that \( m+n \) is minimized.

Answer: $106$


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 Three-Body Problem," the operation involves intercepting the "Judgment Day" ship with a "nano-blade" material at the Panama Canal locks. As long as the "Judgment Day" ship passes through the "nano-blade" material completely undetected, the operation is a success. If the entire length of the "Judgment Day" ship is 400 meters, and it takes 50 seconds to pass through a 100-meter long tunnel at a constant speed, how many seconds will it take for the ship to pass through the "nano-blade" material at the same speed?

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>.

How many different integral solutions \((x, y)\) does \(3|x| + 5|y| = 100\) have?

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>.

\(\triangle ABC\) is equilateral with side length 4. \(D\) is a point on \(BC\) such that \(BD = 1\). If \(r\) and \(s\) are the radii of the inscribed circles of \(\triangle ADB\) and \(\triangle ADC\) respectively, find \(rs\).

Answer: $4 - \sqrt{13}$


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>.

Given a parallelepiped \(A B C D A_{1} B_{1} C_{1} D_{1}\), a point \(X\) is chosen on edge \(A_{1} D_{1}\), and a point \(Y\) is chosen on edge \(B C\). It is known that \(A_{1} X = 5\), \(B Y = 3\), and \(B_{1} C_{1} = 14\). The plane \(C_{1} X Y\) intersects the ray \(D A\) at point \(Z\). Find \(D Z\).

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>.

Three balls are lying on a table, each pair touching each other. The radii of the balls form a geometric progression with a common ratio \( q \neq 1 \). The radius of the middle ball is 2012. Find the ratio of the sum of the squares of the sides of the triangle formed by the points of contact of the balls with the table to the sum of the sides of the triangle formed by the centers of the balls.

Answer: $4024$


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 does the digit 0 appear in the integer equal to \( 20^{10} \)?

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>.

In the process of making a steel cable, it was determined that the length of the cable is the same as the curve given by the system of equations:

$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$

Find the length of the cable.

Answer: $4 \pi \sqrt{\frac{23}{3}}$


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>.

Given \(a\) and \(b\) are real numbers, satisfying:
\[
\sqrt[3]{a} - \sqrt[3]{b} = 12, \quad ab = \left( \frac{a + b + 8}{6} \right)^3.
\]

Find \(a - b\).

Answer: $468$


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>.

Masha looked at the drawing and said: "There are seven rectangles here: one big one and six small ones." "There are also various middle-sized rectangles here," said her mother. How many rectangles are there in total in this drawing? Explain your answer.

Answer: $18$


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 minimum value of the function

$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-102)^{2}
$$

If you obtain a non-integer number, round the result to the nearest whole number.

Answer: $46852$


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 right triangle with legs of 5 and 12, a segment is drawn connecting the shorter leg and the hypotenuse, touching the inscribed circle and parallel to the longer leg. Find its length.

Answer: $2.4$


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>.

Sasha wrote the numbers $7, 8, 9, \ldots, 17$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the maximum value that the sum of the remaining numbers on the board can have?

Answer: $121$


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>.

It is known that 999973 has exactly three distinct prime factors. Find the sum of these prime factors.

Answer: $171$


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 area enclosed by the curves \( y = \sin x \) and \( y = \left(\frac{4}{\pi}\right)^{2} \sin \left(\frac{\pi}{4}\right) x^{2} \) (the latter is a quadratic function).

Answer: $1 - \frac{\sqrt{2}}{2}\left(1 + \frac{\pi}{12}\right)$


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>.

If \( AC = 1 \ \text{cm} \) and \( AD = 4 \ \text{cm} \), what is the relationship between the areas of triangles \( \triangle ABC \) and \( \triangle CBD \)?

Answer: $1/3$


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>.

\( AB \) and \( AC \) are tangents to a circle with center \( O \), \( M \) is the point of intersection of the line \( AO \) with the circle; \( DE \) is the segment of the tangent drawn through the point \( M \) between \( AB \) and \( AC \). Find \( DE \), if the radius of the circle is 15 and \( AO = 39 \).

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>.

A two-digit number, when multiplied by 109, yields a four-digit number. It is divisible by 23, and the quotient is a one-digit number. The maximum value of this two-digit number is $\qquad$.

Answer: $69$


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 \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if both parts of \( P \)'s boundary divided by this diagonal contain an odd number of edges. Each side of \( P \) is also defined as a "good edge."

Given that 2003 non-intersecting diagonals inside \( P \) divide it into several triangles, determine the maximum number of isosceles triangles with two "good edges" in this division.

Answer: $1003$


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 \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\) and \(\sin \beta = 2 \cos (\alpha + \beta) \cdot \sin \alpha \left(\alpha + \beta \neq \frac{\pi}{2}\right)\), find the maximum value of \(\tan \beta\).

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