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 value of $\cos(-\pi - \alpha)$ given a point $P(-3, 4)$ on the terminal side of angle $\alpha$.

Answer: $-\dfrac{3}{5}$


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>.

In space, there are four spheres with radii 2, 2, 3, and 3. Each sphere is externally tangent to the other three spheres. Additionally, there is a smaller sphere that is externally tangent to all four of these spheres. Find the radius of this smaller sphere.

Answer: $\frac{6}{2 - \sqrt{26}}$


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>.

Given that the focus of the parabola $y^{2}=ax$ coincides with the left focus of the ellipse $\frac{x^{2}}{6}+ \frac{y^{2}}{2}=1$, find the value of $a$.

Answer: $-16$


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

Let \(a, b\), and \(c\) be the roots of the cubic polynomial \(3x^3 - 4x^2 + 100x - 3\). Compute \[(a+b+2)^2 + (b+c+2)^2 + (c+a+2)^2.\]

Answer: $119.888...$


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>.

Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {0,1,2,3,4}. Determine the number of such quadruples that make the expression $a \cdot d - b \cdot c + 1$ even.

Answer: $136$


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 store sold an air conditioner for 2000 yuan and a color TV for 2000 yuan. The air conditioner made a 30% profit, while the color TV incurred a 20% loss. Could you help the store owner calculate whether the store made a profit or a loss on this transaction, and by how much?

Answer: $38.5$


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>.

George is trying to find the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,2)$, and $C$ is at $(5,4)$. He guesses that the point is at $P = (3,1)$. Compute the sum of the distances from $P$ to the vertices of $\triangle ABC$. If he obtains $x + y\sqrt{z}$, where $x$, $y$, and $z$ are integers, what is $x + y + z$?

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 a \(10 \times 10\) grid (where the sides of the cells have a unit length), \(n\) cells are selected, and a diagonal is drawn in each of them with an arrow pointing in one of two directions. It turns out that for any two arrows, either the end of one coincides with the beginning of the other, or the distance between their ends is at least 2. What is the largest possible \(n\)?

Answer: $48$


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

If $a$ and $b$ are positive integers such that $ab - 7a + 6b = 559$, what is the minimal possible value of $|a - b|$?

Answer: $587$


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>.

Seven frogs are sitting in a row. They come in four colors: two green, two red, two yellow, and one blue. Green frogs refuse to sit next to red frogs, and yellow frogs refuse to sit next to blue frogs. In how many ways can the frogs be positioned respecting these restrictions?

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>.

During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream?

Answer: $10$


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

In the coordinate plane, a point  is called a  $\text{lattice point}$  if both of its coordinates are integers. Let  $A$   be the point  $(12,84)$ . Find the number of right  angled triangles  $ABC$   in the coordinate plane  $B$  and  $C$  are lattice points, having a right angle at vertex  $A$   and whose incenter is at the origin  $(0,0)$ .

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>.

In isosceles $\triangle ABC$ with $AB=AC$, let $D$ be the midpoint of $AC$ and $BD=1$. Find the maximum area of $\triangle ABC$.

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


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 minimum value of $\frac{b^{2}+1}{\sqrt{3}a}$ is what occurs when the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is 2.

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


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

Compute $\tan(-405^\circ)$.

Answer: $-1$


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>.

If the surface area of a cone is $3\pi$, and its lateral surface unfolds into a semicircle, then the diameter of the base of the cone is ___.

Answer: $\sqrt{6}$


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

The net change in the population over these four years is a 20% increase, then a 30% decrease, then a 20% increase, and finally a 30% decrease. Calculate the net change in the population over these four years.

Answer: $-29$


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

Let a  $9$ -digit number be balanced if it has all numerals  $1$  to  $9$ . Let  $S$  be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of  $k$  such that any two subsequences of  $S$  which has consecutive  $k$  numerals are different from each other.

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>.

Find the largest prime divisor of \( 16^2 + 81^2 \).

Answer: $53$


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>.

Regular decagon `ABCDEFGHIJ` has its center at `K`. Each of the vertices and the center are to be associated with one of the digits `1` through `10`, with each digit used once, in such a way that the sums of the numbers on the lines `AKE`, `BKF`, `CKG`, `DLH` and `EJI` are all equal. In how many ways can this be done?

Answer: $3840$


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

Let $a,$ $b,$ and $c$ be positive real numbers such that $ab^2c^3 = 256$. Find the minimum value of
\[a^2 + 8ab + 16b^2 + 2c^5.\]

Answer: $768$


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 convex pentagon $FGHIJ$ where $\angle F = \angle G = 100^\circ$. Let $FI = IJ = JG = 3$ and $GH = HF = 5$. Calculate the area of pentagon $FGHIJ$.

Answer: $\frac{9\sqrt{3}}{4} + 24.62$


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>.

Let $\triangle PQR$ have side lengths $PQ=13$, $PR=15$, and $QR=14$. Inside $\angle QPR$ are two circles: one is tangent to rays $\overline{PQ}$, $\overline{PR}$, and segment $\overline{QR}$; the other is tangent to the extensions of $\overline{PQ}$ and $\overline{PR}$ beyond $Q$ and $R$, and also tangent to $\overline{QR}$. Compute the distance between the centers of these two circles.

Answer: $5\sqrt{13}$


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>.

The equation of one of the axes of symmetry for the graph of the function $f(x)=\sin \left(x- \frac {\pi}{4}\right)$ $(x\in\mathbb{R})$ can be found.

Answer: $-\frac{\pi}{4}$


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>.

In acute triangle \( ABC \), \( M \) and \( N \) are the midpoints of sides \( AB \) and \( BC \), respectively. The tangents to the circumcircle of triangle \( BMN \) at \( M \) and \( N \) meet at \( P \). Suppose that \( AP \) is parallel to \( BC \), \( AP = 9 \), and \( PN = 15 \). Find \( AC \).

Answer: $20\sqrt{2}$


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>.

Quadrilateral $ABCD$ is a parallelogram with an area of $50$ square units. Points $P$ and $Q$ are located on sides $AB$ and $CD$ respectively, such that $AP = \frac{1}{3}AB$ and $CQ = \frac{2}{3}CD$. What is the area of triangle $APD$?

Answer: $16.67$


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 decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the number \( M \) have? List all possible answers.

Answer: $29$


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 hyperbola has its center at the origin O, with its foci on the x-axis and two asymptotes denoted as l₁ and l₂. A line perpendicular to l₁ passes through the right focus F intersecting l₁ and l₂ at points A and B, respectively. It is known that the magnitudes of vectors |OA|, |AB|, and |OB| form an arithmetic sequence, and the vectors BF and FA are in the same direction. Determine the eccentricity of the hyperbola.

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


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>.

Given \( a_{n} = \mathrm{C}_{200}^{n} \cdot (\sqrt[3]{6})^{200-n} \cdot \left( \frac{1}{\sqrt{2}} \right)^{n} \) for \( n = 1, 2, \ldots, 95 \), find the number of integer terms in the sequence \(\{a_{n}\}\).

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>.

A truncated right circular cone has a large base radius of 10 cm and a small base radius of 5 cm. The height of the truncated cone is 10 cm. Calculate the volume of this solid.

Answer: $583.33\pi$


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

Let \(x\), \(y\), and \(z\) be positive real numbers such that \(x + y + z = 3.\) Find the maximum value of \(x^3 y^3 z^2.\)

Answer: $\frac{4782969}{390625}$


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 integers between 1 and 3015 are either multiples of 5 or 7 but not multiples of 35?

Answer: $948$


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

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


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>.

In triangle $ABC,$ $\angle A = 45^\circ,$ $\angle B = 75^\circ,$ and $AC = 6.$ Find $BC$.

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


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

Given $a = 1 + 2\binom{20}{1} + 2^2\binom{20}{2} + \ldots + 2^{20}\binom{20}{20}$, and $a \equiv b \pmod{10}$, determine the possible value(s) for $b$.

Answer: $2011$


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 \(2, 3, 4, 5, 6, 7, 8\) are to be placed, one per square, in the diagram shown so that the sum of the four numbers in the horizontal row equals 21 and the sum of the four numbers in the vertical column also equals 21. In how many different ways can this be done?

Answer: $12$


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>.

In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{c}{2} = b - a\cos C$,
(1) Determine the measure of angle $A$.
(2) If $a=\sqrt{15}$ and $b=4$, find the length of side $c$.

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>.

Given the sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $a_{n}$ represents the $n$-th digit after the decimal point of $\sqrt{2}=1.41421356237⋯$ (for example, $a_{1}=4$, $a_{6}=3)$, and $b_{1}=a_{1}$, with ${b_{n+1}}={a_{{b_n}}}$ for all $n\in N^{*}$, find all values of $n$ that satisfy $b_{n}=n-2022$.

Answer: $675$


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 Samuel has a $3 \times 7$ index card and shortening the length of one side by $1$ inch results in an area of $15$ square inches, determine the area of the card in square inches if instead he shortens the length of the other side by $1$ inch.

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 that point $P$ is inside isosceles triangle $ABC$, with $AB = BC$ and $\angle BPC = 108^{\circ}$. Let $D$ be the midpoint of side $AC$, and let $BD$ intersect $PC$ at point $E$. If $P$ is the incenter of $\triangle ABE$, find $\angle PAC$.

Answer: $48$


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

Let $y=f(x)$ be a quadratic function, and the equation $f(x)=0$ has two equal real roots. Also, $f'(x)=2x+2$.
1. Find the expression for $y=f(x)$.
2. Find the area of the shape enclosed by the graph of $y=f(x)$ and the two coordinate axes.
3. If the line $x=-t$ ($0<t<1$) divides the area enclosed by the graph of $y=f(x)$ and the two coordinate axes into two equal parts, find the value of $t$.

Answer: $1-\frac{1}{32}$


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

If the digits of a natural number can be divided into two groups such that the sum of the digits in each group is equal, the number is called a "balanced number". For example, 25254 is a "balanced number" because $5+2+2=4+5$. If two adjacent natural numbers are both "balanced numbers", they are called a pair of "twin balanced numbers". What is the sum of the smallest pair of "twin balanced numbers"?

Answer: $1099$


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

By multiplying a natural number by the number that is one greater than it, the product takes the form $ABCD$, where $A, B, C, D$ are different digits. Starting with the number that is 3 less, the product takes the form $CABD$. Starting with the number that is 30 less, the product takes the form $BCAD$. Determine these numbers.

Answer: $8372$


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>.

Note that there are exactly three ways to write the integer  $4$  as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let  $f(n)$  be the number of ways to write a natural number  $n$  as a sum of positive odd integers where the order of the summands matters.  Find the remainder when  $f(2008)$  is divided by  $100$ .

Answer: $71$


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

Given the two-digit integer $MM$, where both digits are equal, when multiplied by the one-digit integer $K$ (different from $M$ and only $1\leq K \leq 9$), it results in a three-digit number $NPK$. Identify the digit pairs $(M, K)$ that yield the highest value of $NPK$.

Answer: $891$


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

Define the operations:
\( a \bigcirc b = a^{\log _{7} b} \),
\( a \otimes b = a^{\frac{1}{\log ^{6} b}} \), where \( a, b \in \mathbb{R} \).

A sequence \(\{a_{n}\} (n \geqslant 4)\) is given such that:
\[ a_{3} = 3 \otimes 2, \quad a_{n} = (n \otimes (n-1)) \bigcirc a_{n-1}. \]

Then, the integer closest to \(\log _{7} a_{2019}\) is:

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>.

If
\[
\begin{pmatrix}
1 & 3 & b \\
0 & 1 & 5 \\
0 & 0 & 1
\end{pmatrix}^m = \begin{pmatrix}
1 & 33 & 6006 \\
0 & 1 & 55 \\
0 & 0 & 1
\end{pmatrix},
\]
then find $b + m.$

Answer: $432$


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 the broad money supply $\left(M2\right)$ balance was 2912000 billion yuan, express this number in scientific notation.

Answer: $2.912 \times 10^{6}$


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

In a triangle, two angles measure 45 degrees and 60 degrees. The side opposite the 45-degree angle measures 8 units. Calculate the sum of the lengths of the other two sides.

Answer: $19.3$


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>.

8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For example, if there was a fragment ...044566667..., then it becomes ...04567...

a) Find the probability that the sequence will shorten by exactly one digit.

b) Find the expected length of the new sequence.

Answer: $1813.6$


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

Determine the value of $\sin {15}^{{}^\circ }+\cos {15}^{{}^\circ }$.

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


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 cao has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns.
An estimate of $E>0$ earns $\left\lfloor 20 \min (N / E, E / N)^{4}\right\rfloor$ points.

Answer: $1416528$


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  $ABCD$  be a parallelogram such that  $\angle BAD = 60^{\circ}.$  Let  $K$  and  $L$  be the midpoints of  $BC$  and  $CD,$  respectively. Assuming that  $ABKL$  is a cyclic quadrilateral, find  $\angle ABD.$

Answer: $75$


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

Given $\tan\alpha= \frac {1}{2}$ and $\tan(\alpha-\beta)=- \frac {2}{5}$, calculate the value of $\tan(2\alpha-\beta)$.

Answer: $-\frac{1}{12}$


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

Let \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) be two matrices such that \(\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}\). Assuming \(4b \neq c\), find \(\frac{a - 2d}{c - 4b}\).

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


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 same amount of steel used to create six solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball?

Answer: $4\sqrt[3]{2}$


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

Let $WXYZ$ be a rhombus with diagonals $WY = 20$ and $XZ = 24$. Let $M$ be a point on $\overline{WX}$, such that $WM = MX$. Let $R$ and $S$ be the feet of the perpendiculars from $M$ to $\overline{WY}$ and $\overline{XZ}$, respectively. Find the minimum possible value of $RS$.

Answer: $\sqrt{244}$


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 four points \( K, L, M, N \) that are not coplanar. A sphere touches the planes \( K L M \) and \( K L N \) at points \( M \) and \( N \) respectively. Find the surface area of the sphere, knowing that \( M L = 1 \), \( K M = 2 \), \( \angle M N L = 60^\circ \), and \( \angle K M L = 90^\circ \).

Answer: $\frac{64\pi}{11}$


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>.

The function $g : \mathbb{R} \to \mathbb{R}$ satisfies
\[g(x) + 3g(1 - x) = 2x^2 + 1\]for all $x.$  Find $g(5).$

Answer: $-9$


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>.

A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Kira considers a license plate lucky if the second letter is a vowel, the second digit is odd, and the third digit is even (other symbols have no restrictions). How many license plates does Kira consider lucky?

Answer: $359999$


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 Ravi shortens the length of one side of a $5 \times 7$ index card by $1$ inch, the card would have an area of $24$ square inches. What is the area of the card in square inches if instead he shortens the length of the other side by $1$ inch?

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>.

Tina is trying to solve the equation by completing the square: $$25x^2+30x-55 = 0.$$ She needs to rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)?

Answer: $-38$


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

Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\overline{PQ}$ is tangent to segments $\overline{PR}$ and $\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what?

Answer: $2850$


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

Given the function $f(x)= \frac {1}{3}x^{3}+x^{2}+ax+1$, and the slope of the tangent line to the curve $y=f(x)$ at the point $(0,1)$ is $-3$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Find the extrema of $f(x)$.

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


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>.

Consider a modified sequence whose $n$th term is defined by $a_n = (-1)^n \cdot \lfloor \frac{3n+1}{2} \rfloor$. What is the average of the first $150$ terms of this sequence?

Answer: $-\frac{1}{6}$


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

Let $a,$ $b,$ $c,$ and $d$ be real numbers such that $ab = 2$ and $cd = 18.$ Find the minimum value of
\[(ac)^2 + (bd)^2.\]

Answer: $12$


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>.

The sides of  $\triangle ABC$  measure 11,20, and 21 units. We fold it along  $PQ,QR,RP$  where  $P,Q,R$  are the midpoints of its sides until  $A,B,C$  coincide. What is the volume of the resulting tetrahedron?

Answer: $45$


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

The sum of 9 standard dice rolled has the same probability of occurring as the sum of 20. Find the value of the sum of 9 standard dice that shares this same probability.

Answer: $43$


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

Given a right triangle \( ABC \) with legs \( AC = 3 \) and \( BC = 4 \). Construct triangle \( A_1 B_1 C_1 \) by successively translating point \( A \) a certain distance parallel to segment \( BC \) to get point \( A_1 \), then translating point \( B \) parallel to segment \( A_1 C \) to get point \( B_1 \), and finally translating point \( C \) parallel to segment \( A_1 B_1 \) to get point \( C_1 \). If it turns out that angle \( A_1 B_1 C_1 \) is a right angle and \( A_1 B_1 = 1 \), what is the length of segment \( B_1 C_1 \)?

Answer: $12$


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>.

Compute $\arccos(\cos 9).$ All functions are in radians.

Answer: $9 - 2\pi$


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

Find the number of triples of natural numbers \( m, n, k \) that are solutions to the equation \( m + \sqrt{n+\sqrt{k}} = 2023 \).

Answer: $27575680773$


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 nearest integer to $(3 + 2)^6$?

Answer: $9794$


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 triangle  $ABC$ . Let  $A_1B_1$ ,  $A_2B_2$ , $ ...$ ,  $A_{2008}B_{2008}$  be  $2008$  lines parallel to  $AB$  which divide triangle  $ABC$  into  $2009$  equal areas. Calculate the value of   $$  \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor $$

Answer: $29985$


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 $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[17]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?

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>.

**polyhedral**

we call a  $12$ -gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points  $(0,1)$ , $(1,0)$ , $(-1,0)$  and  $(0,-1)$ .
find the faces of the <u>massivest</u> polyhedral that it's image on every three plane  $xy$ , $yz$  and  $zx$  is a good  $12$ -gon.
(it's obvios that centers of these three  $12$ -gons are the origin of coordinates for three dimensions.)

time allowed for this question is 1 hour.

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 $\triangle ABC$, $AB= 400$, $BC=480$, and $AC=560$. An interior point $P$ is identified, and segments are drawn through $P$ parallel to the sides of the triangle. These three segments are of equal length $d$. Determine $d$.

Answer: $218\frac{2}{9}$


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>.

In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $a$, $b$, $c$ form an arithmetic sequence in that order.
$(1)$ If the vectors $\overrightarrow{m}=(3,\sin B)$ and $\overrightarrow{n}=(2,\sin C)$ are collinear, find the value of $\cos A$;
$(2)$ If $ac=8$, find the maximum value of the area $S$ of $\triangle ABC$.

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>.

Given that the students are numbered from 01 to 70, determine the 7th individual selected by reading rightward starting from the number in the 9th row and the 9th column of the random number table.

Answer: $44$


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>.

Answer: $\sqrt{5}$


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>.

Let $Q(x) = 0$ be the polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[4]{13} + \sqrt[4]{169}$ as a root. Compute the product of all of the roots of $Q(x) = 0.$

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>.

Compute
\[\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\]

Answer: $962$


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 largest $2$-digit prime factor of the integer $n = {300 \choose 150}$?

Answer: $89$


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 distinct points are arranged on a plane such that the segments connecting them form lengths $a$, $a$, $a$, $a$, $a$, $b$, $b$, $2a$, and $c$. The shape formed by these points is no longer restricted to a simple polygon but could include one bend (not perfectly planar). What is the ratio of $c$ to $a$?
**A)** $\sqrt{3}$
**B)** $2$
**C)** $2\sqrt{3}$
**D)** $3$
**E)** $\pi$

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>.

Given that both $m$ and $n$ are non-negative integers, find the number of "simple" ordered pairs $(m, n)$ with a value of 2019.

Answer: $60$


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 $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$.

Answer: $306$


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

Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\).

Answer: $1/2$


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

Let \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \).

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>.

A, B, and C start from the same point on a circular track with a circumference of 360 meters: A starts first and runs in the counterclockwise direction; before A completes a lap, B and C start simultaneously and run in the clockwise direction; when A and B meet for the first time, C is exactly half a lap behind them; after some time, when A and C meet for the first time, B is also exactly half a lap behind them. If B’s speed is 4 times A’s speed, then how many meters has A run when B and C start?

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>.

The polynomial \( x^{2n} + 1 + (x+1)^{2n} \) cannot be divided by \( x^2 + x + 1 \) under the condition that \( n \) is equal to:

Answer: $21$


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

Determine the area of the Crescent Gemini.

Answer: $\frac{17\pi}{4}$


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>.

In an acute-angled $\triangle ABC$, find the minimum value of $3 \tan B \tan C + 2 \tan A \tan C + \tan A \tan B$.

Answer: $6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6}$


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

In a bookshelf, there are four volumes of Astrid Lindgren's collected works in order, each containing 200 pages. A little worm living in these volumes burrowed a path from the first page of the first volume to the last page of the fourth volume. How many pages did the worm burrow through?

Answer: $400$


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 $DEF$, $\angle E = 45^\circ$, $DE = 100$, and $DF = 100 \sqrt{2}$. Find the sum of all possible values of $EF$.

Answer: $\sqrt{30000 + 5000(\sqrt{6} - \sqrt{2})}$


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>.

In right triangle $\triangle ABC$ with $\angle BAC = 90^\circ$, medians $\overline{AD}$ and $\overline{BE}$ are given such that $AD = 18$ and $BE = 24$. If $\overline{AD}$ is the altitude from $A$, find the area of $\triangle ABC$.

Answer: $432$


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 real numbers x and y satisfy x + y = 5 and x * y = -3, find the value of x + x^4 / y^3 + y^4 / x^3 + y.

Answer: $5 + \frac{2829}{27}$


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>.

Let $x,$ $y,$ $z$ be real numbers such that $0 \leq x, y, z \leq 1$. Find the maximum value of
\[
\frac{1}{(2 - x)(2 - y)(2 - z)} + \frac{1}{(2 + x)(2 + y)(2 + z)}.
\]

Answer: $\frac{12}{27}$


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>.

Given that the probability of bus No. 3 arriving at the bus stop within 5 minutes is 0.20 and the probability of bus No. 6 arriving within 5 minutes is 0.60, calculate the probability that the passenger can catch the bus he needs within 5 minutes.

Answer: $0.80$


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 expression $\frac{\sqrt{3}\tan 12^{\circ} - 3}{(4\cos^2 12^{\circ} - 2)\sin 12^{\circ}}$ equals \_\_\_\_\_\_.

Answer: $-4\sqrt{3}$


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>.

Determine constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2 - 96x + 2210}{x^2 + 65x - 3510}$. What is $\alpha + \beta$?

Answer: $112$


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 $\triangle ABC$ satisfies that $\cot A, \cot B, \cot C$ form an arithmetic sequence, then the maximum value of $\angle B$ is ______.

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


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>.

Give an example of a function that, when \( x \) is equal to a known number, takes the form \( \frac{0}{0} \), but as \( x \) approaches this number, tends to a certain limit.

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


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>.

The diagram below shows part of a city map. The small rectangles represent houses, and the spaces between them represent streets. A student walks daily from point $A$ to point $B$ on the streets shown in the diagram, and can only walk east or south. At each intersection, the student has an equal probability ($\frac{1}{2}$) of choosing to walk east or south (each choice is independent of others). What is the probability that the student will walk through point $C$?

Answer: $\frac{21}{32}$


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

Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers?

Answer: $128$


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 an arithmetic sequence $\{a_n\}$ with a non-zero common difference, some terms $a_{k_1}$, $a_{k_2}$, $a_{k_3}$, ... form a geometric sequence $\{a_{k_n}\}$, and it is given that $k_1 \neq 1$, $k_2 \neq 2$, $k_3 \neq 6$. Find the value of $k_4$.

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>.

Each twin from the first 4 sets shakes hands with all twins except his/her sibling and with one-third of the triplets; the remaining 8 sets of twins shake hands with all twins except his/her sibling but does not shake hands with any triplet; and each triplet shakes hands with all triplets except his/her siblings and with one-fourth of all twins from the first 4 sets only.

Answer: $394$


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

We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way?

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>.

A right square pyramid with base edges of length $12$ units each and slant edges of length $15$ units each is cut by a plane that is parallel to its base and $4$ units above its base. What is the volume, in cubic units, of the top pyramid section that is cut off by this plane?

Answer: $\frac{1}{3} \times \left(\frac{(144 \cdot (153 - 8\sqrt{153}))}{153}\right) \times (\sqrt{153} - 4)$


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

A wooden cube with edges of length $3$ meters has square holes, of side one meter, centered in each face, cut through to the opposite face. Find the entire surface area, including the inside, of this cube in square meters.

Answer: $72$


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

Find the number of permutations of \( n \) distinct elements \( a_1, a_2, \cdots, a_n \) (where \( n \geqslant 2 \)) such that \( a_1 \) is not in the first position and \( a_2 \) is not in the second position.

Answer: $32,527,596$


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 triangular box is to be cut from an equilateral triangle of length 30 cm. Find the largest possible volume of the box (in cm³).

Answer: $500$


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 two distinguishable flagpoles, and there are $17$ flags, of which $11$ are identical red flags, and $6$ are identical white flags. Determine the number of distinguishable arrangements using all of the flags such that each flagpole has at least one flag and no two white flags on either pole are adjacent. Compute the remainder when this number is divided by $1000$.

Answer: $164$


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 the center of circle $M$ lies on the $y$-axis, the radius is $1$, and the chord intercepted by line $l: y = 2x + 2$ on circle $M$ has a length of $\frac{4\sqrt{5}}{5}$. Additionally, the circle center $M$ is located below line $l$.

(1) Find the equation of circle $M$;

(2) Let $A(t, 0), B(t + 5, 0) \, (-4 \leqslant t \leqslant -1)$, if $AC, BC$ are tangent lines to circle $M$, find the minimum value of the area of $\triangle ABC$.

Answer: $\frac{125}{21}$


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

Find all real numbers \( p \) such that the cubic equation \( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \) has three roots, all of which are positive integers.

Answer: $76$


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>.

Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of  $5$  random keystrokes; at each stroke, each key is equally likely to be pressed.  The calculator then evaluates the entire expression, yielding a result of  $E$ .  Find the expected value of  $E$ .
(Note: Negative numbers are permitted, so 13-22 gives  $E = -9$ . Any excess operators are parsed as signs, so -2-+3 gives  $E=-5$  and -+-31 gives  $E = 31$ .  Trailing operators are discarded, so 2++-+ gives  $E=2$ .  A string consisting only of operators, such as -++-+, gives  $E=0$ .)

*Proposed by Lewis Chen*

Answer: $1866$


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 the largest four-digit number $M$ has digits with a product of $72$, find the sum of its digits.

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>.

Simplify $$\frac{13!}{10! + 3\cdot 9!}$$

Answer: $1320$


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 of the four endpoints of the axes of an ellipse are, in some order, \[(10, -3), \; (15, 7), \; (25, -3).\] Find the distance between the foci of the ellipse.

Answer: $11.18$


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>.

The maximum number that can be formed by the digits 0, 3, 4, 5, 6, 7, 8, 9, given that there are only thirty 2's and twenty-five 1's.

Answer: $199$


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

Find a whole number, $M$, such that $\frac{M}{5}$ is strictly between 9.5 and 10.5.

Answer: $51$


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 the difference between each number in a row and the number immediately to its left in the given diagram is the same, and the quotient of each number in a column divided by the number immediately above it is the same, then $a + b \times c =\quad$

Answer: $540$


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>.

9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the winner could have obtained?

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>.

Let  $p = 2017,$  a prime number. Let  $N$  be the number of ordered triples  $(a,b,c)$  of integers such that  $1 \le a,b \le p(p-1)$  and  $a^b-b^a=p \cdot c$ . Find the remainder when  $N$  is divided by  $1000000.$ *Proposed by Evan Chen and Ashwin Sah*

*Remark:* The problem was initially proposed for  $p = 3,$  and  $1 \le a, b \le 30.$

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>.

Let  $\mathcal{P}$   be a convex polygon with  $50$  vertices. A set  $\mathcal{F}$   of diagonals of  $\mathcal{P}$  is said  to be *$minimally friendly$* if any diagonal  $d \in \mathcal{F}$  intersects at most one other diagonal  in  $\mathcal{F}$   at a point interior to  $\mathcal{P}.$  Find the largest  possible number of  elements  in a  $\text{minimally friendly}$  set  $\mathcal{F}$ .

Answer: $72$


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

What is the minimum number of cells required to mark on a chessboard so that each cell of the board (marked or unmarked) is adjacent by side to at least one marked cell?

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>.

The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$.

Answer: $-6$


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

Cindy wants to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 16 possible values for $Y$ given all of the restrictions, what is the smallest number of coins she could have?

Answer: $131072$


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 value of \(\log _{2}\left[2^{3} 4^{4} 8^{5} \cdots\left(2^{20}\right)^{22}\right]\).
Choose one of the following options:
(a) 3290
(b) 3500
(c) 3710
(d) 4172

Answer: $5950$


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 cells of an  $8\times 8$  board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.

Answer: $36$