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

Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=10$. A beam of light is emitted from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 3 units from $\overline{BG}$ and 4 units from $\overline{BC}$. The beam continues its path, reflecting off the faces of the cube. The length of the light path from when it leaves point $A$ until it reaches another vertex of the cube for the first time is expressed as $r\sqrt{s}$, where $r$ and $s$ are integers and $s$ is not divisible by the square of any prime. Determine $r+s$.

Answer: $55$


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

Rhombus $ABCD$ is inscribed in rectangle $WXYZ$ such that vertices $A$, $B$, $C$, and $D$ are on sides $\overline{WX}$, $\overline{XY}$, $\overline{YZ}$, and $\overline{ZW}$, respectively. It is given that $WA=12$, $XB=9$, $BD=15$, and the diagonal $AC$ of rhombus equals side $XY$ of the rectangle. Calculate the perimeter of rectangle $WXYZ$.

Answer: $66$


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

If two circles $(x-m)^2+y^2=4$ and $(x+1)^2+(y-2m)^2=9$ are tangent internally, then the real number $m=$ ______ .

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


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

Fifteen square tiles with side 10 units long are arranged as shown. An ant walks along the edges of the tiles, always keeping a black tile on its left. Find the shortest distance that the ant would walk in going from point \( P \) to point \( Q \).

Answer: $80$


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>.

Let \( x \) and \( y \) be real numbers such that
\[
  1 < \frac{x - y}{x + y} < 3.
\]
If \( \frac{x}{y} \) is an integer, what is its value?

Answer: $-2$


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

Arrange the 9 numbers 12, 13, ..., 20 in a row such that the sum of every three consecutive numbers is a multiple of 3. How many such arrangements are there?

Answer: $216$


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

Determine the integer $x$ that satisfies the following set of congruences:
\begin{align*}
4+x &\equiv 3^2 \pmod{2^3} \\
6+x &\equiv 4^2 \pmod{3^3} \\
8+x &\equiv 6^2 \pmod{5^3}
\end{align*}
Find the remainder when $x$ is divided by $120$.

Answer: $37$


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

Let $g$ be defined by  \[g(x) = \left\{
\begin{array}{cl}
x+3 & \text{ if } x \leq 2, \\
x^2 - 4x + 5 & \text{ if } x > 2.
\end{array}
\right.\]Calculate $g^{-1}(1)+g^{-1}(6)+g^{-1}(11)$.

Answer: $2 + \sqrt{5} + \sqrt{10}$


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

Given an angle $α$ whose terminal side passes through the point $(-1, \sqrt{2})$, find the values of $\tan α$ and $\cos 2α$.

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


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

The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \), and \( A \neq B \). When \( (A, B) \) and \( (B, A) \) are considered as different pairs, how many such pairs \( (A, B) \) are there?

Answer: $27$


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>.

In the Cartesian coordinate system $xOy$, the parametric equation of line $l_{1}$ is $\begin{cases} x=t- \sqrt {3} \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_{2}$ is $\begin{cases} x= \sqrt {3}-m \\ y= \frac {m}{3k}\end{cases}$ (where $m$ is the parameter). Let $p$ be the intersection point of $l_{1}$ and $l_{2}$. When $k$ varies, the trajectory of $p$ is curve $C_{1}$
(Ⅰ) Write the general equation and parametric equation of $C_{1}$;
(Ⅱ) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. Suppose the polar equation of curve $C_{2}$ is $p\sin (\theta+ \frac {\pi}{4})=4 \sqrt {2}$. Let $Q$ be a moving point on curve $C_{1}$, find the minimum distance from point $Q$ to $C_{2}$.

Answer: $3 \sqrt {2}$


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>.

Suppose \(a\), \(b\), and \(c\) are real numbers such that:
\[
\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -12
\]
and
\[
\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 15.
\]
Compute the value of:
\[
\frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}.
\]

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 the following expression:
\[ 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4)))))))) \]

Answer: $1398100$


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

The axis cross-section $SAB$ of a cone with an equal base triangle side length of 2, $O$ as the center of the base, and $M$ as the midpoint of $SO$. A moving point $P$ is on the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ is ( ).

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


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

Find the maximum value of the expression \( x + y \) if \( (2 \sin x - 1)(2 \cos y - \sqrt{3}) = 0 \), \( x \in [0, \frac{3\pi}{2}] \), \( y \in [\pi, 2\pi] \).

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


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>.

A finite set  $\{a_1, a_2, ... a_k\}$  of positive integers with  $a_1 < a_2 < a_3 < ... < a_k$  is named  *alternating* if  $i+a$  for  $i = 1, 2, 3, ..., k$  is even. The empty set is also considered to be alternating. The number of alternating subsets of  $\{1, 2, 3,..., n\}$  is denoted by  $A(n)$ .
Develop a method to determine  $A(n)$  for every  $n \in N$  and calculate hence  $A(33)$ .

Answer: $5702887$


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

Let $M$ denote the number of $9$-digit positive integers in which the digits are in increasing order, given that repeated digits are allowed and the digit ‘0’ is permissible. Determine the remainder when $M$ is divided by $1000$.

Answer: $620$


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

Suppose $X$ and $Y$ are digits in base $d > 8$ such that $\overline{XY}_d + \overline{XX}_d = 234_d$. Find $X_d - Y_d$ in base $d$.

Answer: $-2$


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

A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element?

Answer: $87$


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

In a cube \(A B C D-A_{1} B_{1} C_{1} D_{1}\) with edge length 1, \(P, Q, R\) are the midpoints of edges \(A B, A D,\) and \(A A_1\) respectively. A right triangular prism is constructed with \(\triangle PQR\) as its base, such that the vertices of the other base also lie on the surface of the cube. Find the volume of this right triangular prism.

Answer: $\frac{3}{16}$


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>.

A certain middle school assigns numbers to each student, where the last digit indicates the gender of the student: 1 for male and 2 for female. If 028432 represents "a female student who is number 43 in class 8 and enrolled in the year 2002," then the number for a male student who is number 23 in class 6 and enrolled in the year 2008 is.

Answer: $086231$


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

Square the numbers \(a=1001\) and \(b=1001001\). Extract the square root of the number \(c=1002003004005004003002001\).

Answer: $1001001001001$


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

In triangle $XYZ$ where $XY=60$ and $XZ=15$, the area of the triangle is given as $225$. Let $W$ be the midpoint of $\overline{XY}$, and $V$ be the midpoint of $\overline{XZ}$. The angle bisector of $\angle YXZ$ intersects $\overline{WV}$ and $\overline{YZ}$ at points $P$ and $Q$, respectively. Determine the area of quadrilateral $PQYW$.
A) $110$
B) $120$
C) $123.75$
D) $130$

Answer: $123.75$


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>.

In the diagram, \(O\) is the center of a circle with radii \(OA=OB=7\). A quarter circle arc from \(A\) to \(B\) is removed, creating a shaded region. What is the perimeter of the shaded region?

Answer: $14 + 10.5\pi$


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>.

Two thirds of a pitcher is filled with orange juice and the remaining part is filled with apple juice. The pitcher is emptied by pouring an equal amount of the mixture into each of 6 cups. Calculate the percentage of the total capacity of the pitcher that each cup receives.

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 $\triangle ABC$, we have $AC = BC = 10$, and $AB = 8$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 12$. What is $BD$?

Answer: $2\sqrt{15}$


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>.

Given that $m$ and $n$ are two non-coincident lines, and $\alpha$, $\beta$, $\gamma$ are three pairwise non-coincident planes, consider the following four propositions:
$(1)$ If $m \perp \alpha$ and $m \perp \beta$, then $\alpha \parallel \beta$;
$(2)$ If $\alpha \perp \gamma$ and $\beta \perp \gamma$, then $\alpha \parallel \beta$;
$(3)$ If $m \subset \alpha$ and $n \subset \beta$, then $\alpha \parallel \beta$;
$(4)$ If $m \nparallel \beta$ and $\beta \nparallel \gamma$, then $m \nparallel \gamma$.
Among these propositions, the incorrect ones are __________. (Fill in all the correct proposition numbers)

Answer: $(2)(3)(4)$


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

Jia and his four friends each have a private car. The last digit of Jia's license plate is 0, and the last digits of his four friends' license plates are 0, 2, 1, 5, respectively. To comply with the local traffic restrictions from April 1st to 5th (cars with odd-numbered last digits are allowed on odd days, and cars with even-numbered last digits are allowed on even days), the five people discussed carpooling, choosing any car that meets the requirements each day. However, Jia's car can only be used for one day at most. The total number of different car use plans is \_\_\_\_\_\_.

Answer: $64$


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

Let \(x_{1}, x_{2}\) be the roots of the quadratic equation \(ax^{2} + bx + c = 0\) with real coefficients. If \(x_{1}\) is an imaginary number and \(\frac{x_{1}^{2}}{x_{2}}\) is a real number, determine the value of \(S = 1 + \frac{x_{1}}{x_{2}} + \left(\frac{x_{1}}{x_{2}}\right)^{2} + \left(\frac{x_{1}}{x_{2}}\right)^{4} + \cdots + \left(\frac{x_{1}}{x_{2}}\right)^{1999}\).

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>.

Determine the maximum number of elements in the set \( S \) that satisfy the following conditions:
1. Each element in \( S \) is a positive integer not exceeding 100;
2. For any two different elements \( a \) and \( b \) in \( S \), there exists another element \( c \) in \( S \) such that the greatest common divisor of \( a + b \) and \( c \) is 1;
3. For any two different elements \( a \) and \( b \) in \( S \), there exists another element \( c \) in \( S \) such that the greatest common divisor of \( a + b \) and \( c \) is greater than 1.

Answer: $50$


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

Let  $S = {1, 2, \cdots, 100}.$   $X$  is a subset of  $S$  such that no two distinct elements in  $X$  multiply to an element in  $X.$  Find the maximum number of elements of  $X$ .

*2022 CCA Math Bonanza Individual Round #3*

Answer: $91$


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

Xiao Zhang and Xiao Zhao can only take on the first two roles, while the other three can take on any of the four roles, calculate the total number of different selection schemes.

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>.

There are 10 cards, labeled from 1 to 10. Three cards denoted by  $ a,\ b,\ c\ (a > b > c)$  are drawn from the cards at the same time.
Find the probability such that  $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$ .

Answer: $1/30$


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

In △ABC, the sides opposite to angles A, B, C are a, b, c, respectively. Given cosB= $\frac {2}{5}$, and sinAcosB - (2c - cosA)•sinB = 0.
(1) Find the value of b;
(2) Find the maximum value of the perimeter of △ABC.

Answer: $\frac { \sqrt {30}}{6} + \frac {1}{2}$


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

What is the smallest positive value of $x$ such that $x + 3956$ results in a palindrome?

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>.

Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \(d\), \(e\), and \(f\) are integers and \(d > 0\). What are the values of \(d + e + f\)?

Answer: $174$


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

On a grid, there are three locations $A, B, C$, with each small square having a side length of 100 meters. If roads are to be built along the grid lines to connect the three locations, what is the minimum total length of the roads in meters?

Answer: $1000$


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

Given that children enter at a discounted rate, half that of an adult ticket, and the total cost for $6$ adult tickets and $5$ child tickets amounts to $32.50$, calculate the total cost for $10$ adult tickets and $8$ child tickets.

Answer: $53.50$


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

In the equation $\frac{1}{(\;\;\;)} + \frac{4}{(\;\;\;)} + \frac{9}{(\;\;\;\;)} = 1$, fill in the three brackets in the denominators with a positive integer, respectively, such that the equation holds true. The minimum value of the sum of these three positive integers is $\_\_\_\_\_\_$.

Answer: $36$


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

The sequence $\{a_n\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\{a_n\}$.

Answer: $1830$


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

In a plane Cartesian coordinate system, a point whose x and y coordinates are both integers is called a "lattice point." How many lattice points are there inside and on the boundaries of the triangle formed by the line $7x + 11y = 77$ and the coordinate axes?

Answer: $49$


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

In the Cartesian coordinate system, A and B are points moving on the x-axis and y-axis, respectively. If the circle C with AB as its diameter is tangent to the line $3x+y-4=0$, then the minimum area of circle C is \_\_\_\_\_\_.

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


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 smallest positive integer  $n$  for which  $315^2-n^2$  evenly divides  $315^3-n^3$ .

*Proposed by Kyle Lee*

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>.

In the polar coordinate system, the curve $\rho=4\sin \left( \theta- \frac{\pi}{3} \right)$ is symmetric about what axis?

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


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

If
\[1 \cdot 1992 + 2 \cdot 1991 + 3 \cdot 1990 + \dots + 1991 \cdot 2 + 1992 \cdot 1 = 1992 \cdot 996 \cdot y,\]
compute the integer \(y\).

Answer: $664$


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

Let  $k$  be a real number such that the product of real roots of the equation   $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$   is  $-2013$ . Find the sum of the squares of these real roots.

Answer: $4027$


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

Given functions \(f(x)\) and \(g(x)\) are defined on \(\mathbb{R}\) with \(g(x)\neq 0\), and \(f(x)g'(x) > f'(x)g(x)\). It is also given that \(f(x)=a^{x}\cdot g(x)\) where \(a > 0, a\neq 1\), and \(\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}\). Considering the finite sequence \(\left\{\frac{f(n)}{g(n)}\right\}_{n=1,2,\ldots,10}\), find the probability that the sum of the first \(k\) terms is greater than \(\frac{15}{16}\) for any positive integer \(k\) with \(1 \leq k \leq 10\).

Answer: $\frac{3}{5}$


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>.

Given that $k$ is a composite number, and $1 < k < 100$, when the sum of the digits of $k$ is a prime number, this prime number is called the "derived prime" of $k$.
(1) If the "derived prime" of $k$ is 2, then $k=$
(2) Let set $A=\{P(k) | P(k)$ is the "derived prime" of $k\}$, $B=\{k | P(k)$ is the "derived prime" of $k\}$, then the number of elements in the set $A \cup B$ is

Answer: $30$


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

The following diagram shows equilateral triangle  $\vartriangle ABC$  and three other triangles congruent to it. The other three triangles are obtained by sliding copies of  $\vartriangle  ABC$  a distance  $\frac18 AB$  along a side of  $\vartriangle ABC$  in the directions from  $A$  to  $B$ , from  $B$  to  $C$ , and from  $C$  to  $A$ . The shaded region inside all four of the triangles has area  $300$ . Find the area of  $\vartriangle ABC$ .
![Image](https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png)

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>.

How many three-eighths are there in $8\frac{5}{3} - 3$?

Answer: $17\frac{7}{9}$


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>.

Given vectors $\overrightarrow{a}=(2\cos\omega x,-2)$ and $\overrightarrow{b}=(\sqrt{3}\sin\omega x+\cos\omega x,1)$, where $\omega\ \ \gt 0$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$. The distance between two adjacent symmetric centers of the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find $\omega$;
$(2)$ Given $a$, $b$, $c$ are the opposite sides of the three internal angles $A$, $B$, $C$ of scalene triangle $\triangle ABC$, and $f(A)=f(B)=\sqrt{3}$, $a=\sqrt{2}$, find the area of $\triangle ABC$.

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


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

The product of the digits of 4321 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 18?

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>.

Given a set $A_n = \{1, 2, 3, \ldots, n\}$, define a mapping $f: A_n \rightarrow A_n$ that satisfies the following conditions:
① For any $i, j \in A_n$ with $i \neq j$, $f(i) \neq f(j)$;
② For any $x \in A_n$, if the equation $x + f(x) = 7$ has $K$ pairs of solutions, then the mapping $f: A_n \rightarrow A_n$ is said to contain $K$ pairs of "good numbers." Determine the number of such mappings for $f: A_6 \rightarrow A_6$ that contain 3 pairs of good numbers.

Answer: $40$


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

Given the ten digits 0, 1, 2, 3, …, 9 and the imaginary unit i, determine the total number of distinct imaginary numbers that can be formed.

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>.

What is the smallest positive integer with exactly 12 positive integer divisors?

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>.

A certain product in a shopping mall sells an average of 30 items per day, with a profit of 50 yuan per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every 1 yuan reduction in price, the mall can sell an additional 2 items per day. Let $x$ represent the price reduction per item. Based on this rule, please answer:
$(1)$ The daily sales volume of the mall increases by ______ items, and the profit per item is ______ yuan (expressed in algebraic expressions containing $x$);
$(2)$ Under the same conditions as above, in normal sales situations, how much should each item be reduced in price so that the mall's daily profit reaches 2100 yuan?

Answer: $20$


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

Let  $ a,b,c,d>0$  for which the following conditions:: $a)$   $(a-c)(b-d)=-4$  $b)$   $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression  $a+c$

Answer: $4\sqrt{2}$


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>.

There are 2 dimes of Chinese currency, how many ways can they be exchanged into coins (1 cent, 2 cents, and 5 cents)?

Answer: $28$


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

Solve the equation: $2\left(x-1\right)^{2}=x-1$.

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


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

What is the area of the portion of the circle defined by the equation $x^2 + 6x + y^2 = 50$ that lies below the $x$-axis and to the left of the line $y = x - 3$?

Answer: $\frac{59\pi}{4}$


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>.

Let \( a_1, a_2, \ldots, a_{2020} \) be the roots of the polynomial
\[ x^{2020} + x^{2019} + \cdots + x^2 + x - 2022 = 0. \]
Compute
\[ \sum_{n = 1}^{2020} \frac{1}{1 - a_n}. \]

Answer: $2041210$


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

Amelia and Blaine are playing a modified game where they toss their respective coins. Amelia's coin lands on heads with a probability of $\frac{3}{7}$, and Blaine's lands on heads with a probability of $\frac{1}{3}$. They begin their game only after observing at least one head in a simultaneous toss of both coins. Once the game starts, they toss coins alternately with Amelia starting first, and the player to first toss heads twice consecutively wins the game. What is the probability that Amelia wins the game?
A) $\frac{1}{2}$
B) $\frac{9}{49}$
C) $\frac{2401}{6875}$
D) $\frac{21609}{64328}$

Answer: $\frac{21609}{64328}$


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

Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order.

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 $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$.

Answer: $28$


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

How many whole numbers between 1 and 2000 do not contain the digits 1 or 2?

Answer: $511$


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

Given the function $f(x)=ax^{3}+2bx^{2}+3cx+4d$, where $a,b,c,d$ are real numbers, $a < 0$, and $c > 0$, is an odd function, and when $x\in[0,1]$, the range of $f(x)$ is $[0,1]$. Find the maximum value of $c$.

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


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>.

In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange?

Answer: $19$


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

The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 8 feet, and its height is 64 feet. The water in the tank is $40\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers, and $b$ is not divisible by a perfect cube greater than 1. What is $a+b$?

Answer: $66$


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

Ten circles of diameter 1 are arranged in the first quadrant of a coordinate plane. Five circles are in the base row with centers at $(0.5, 0.5)$, $(1.5, 0.5)$, $(2.5, 0.5)$, $(3.5, 0.5)$, $(4.5, 0.5)$, and the remaining five directly above the first row with centers at $(0.5, 1.5)$, $(1.5, 1.5)$, $(2.5, 1.5)$, $(3.5, 1.5)$, $(4.5, 1.5)$. Let region $\mathcal{S}$ be the union of these ten circular regions. Line $m,$ with slope $-2$, divides $\mathcal{S}$ into two regions of equal area. Line $m$'s equation can be expressed in the form $px=qy+r$, where $p, q,$ and $r$ are positive integers whose greatest common divisor is 1. Find $p^2+q^2+r^2$.

Answer: $30$


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

Given the sequence $\{a\_n\}$, where $a\_1=1$, $a\_2=3$, and $2nS\_n=(n+1)S\_{n+1}+(n-1)S\_{n-1}$ ($n\geqslant 2,n\in N$), find $S\_{30}$.

Answer: $\frac{34}{5}$


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>.

Given an odd function defined on $\mathbb{R}$, when $x > 0$, $f(x)=x^{2}+2x-1$.
(1) Find the value of $f(-3)$;
(2) Find the analytic expression of the function $f(x)$.

Answer: $-14$


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

Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$.

Answer: $19$


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

In $\triangle ABC$, the sides opposite to angles A, B, and C are $a$, $b$, and $c$, respectively. Given the equation $$2b\cos A - \sqrt{3}c\cos A = \sqrt{3}a\cos C$$.
(1) Find the value of angle A;
(2) If $\angle B = \frac{\pi}{6}$, and the median $AM = \sqrt{7}$ on side $BC$, find the area of $\triangle ABC$.

Answer: $\sqrt{3}$


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>.

In the diagram below, the circle with center  $A$  is congruent to and tangent to the circle with center  $B$ .  A third circle is tangent to the circle with center  $A$  at point  $C$  and passes through point  $B$ .  Points  $C$ ,  $A$ , and  $B$  are collinear.  The line segment  $\overline{CDEFG}$  intersects the circles at the indicated points.  Suppose that  $DE = 6$  and  $FG = 9$ .  Find  $AG$ .
[asy]
unitsize(5);
pair A = (-9 sqrt(3), 0);
pair B = (9 sqrt(3), 0);
pair C = (-18 sqrt(3), 0);
pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);
pair E = (2 sqrt(3), 4 sqrt(6));
pair F = (7 sqrt(3), 5 sqrt(6));
pair G = (12 sqrt(3), 6 sqrt(6));
real r = 9sqrt(3);
draw(circle(A, r));
draw(circle(B, r));
draw(circle((B + C) / 2, 3r / 2));
draw(C -- D);
draw(" $6$ ", E -- D);
draw(E -- F);
draw(" $9$ ", F -- G);
dot(A);
dot(B);
label(" $A$ ", A, plain.E);
label(" $B$ ", B, plain.E);
label(" $C$ ", C, W);
label(" $D$ ", D, dir(160));
label(" $E$ ", E, S);
label(" $F$ ", F, SSW);
label(" $G$ ", G, N);
[/asy]

Answer: $9\sqrt{19}$


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

The maximum value of the function $f(x) = \frac{\frac{1}{6} \cdot (-1)^{1+ C_{2x}^{x}} \cdot A_{x+2}^{5}}{1+ C_{3}^{2} + C_{4}^{2} + \ldots + C_{x-1}^{2}}$ ($x \in \mathbb{N}$) is ______.

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>.

Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2.

Answer: $1100111_2$


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 an obtuse triangle \(ABC\) with obtuse angle \(C\). Points \(P\) and \(Q\) are marked on its sides \(AB\) and \(BC\) respectively, such that \(\angle ACP = CPQ = 90^\circ\). Find the length of segment \(PQ\) if it is known that \(AC = 25\), \(CP = 20\), and \(\angle APC = \angle A + \angle B\).

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>.

Given several numbers, one of them,  $a$ , is chosen and replaced by the three numbers  $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are  $1000$  ones, and we apply the process several times. A number  $m$  is called *good* if there are  $m$  or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.

Answer: $667$


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

Evaluate: $6 - 5\left[7 - (\sqrt{16} + 2)^2\right] \cdot 3.$

Answer: $-429$


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

Solve the equations:
$(1)x^{2}-4x-1=0$;
$(2)\left(x+3\right)^{2}=x+3$.

Answer: $-2$


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

Let  $ABCD$  be a rectangle with side lengths  $AB = CD = 5$  and  $BC = AD = 10$ .  $W, X, Y, Z$  are points on  $AB, BC, CD$  and  $DA$  respectively chosen in such a way that  $WXYZ$  is a kite, where  $\angle ZWX$  is a right angle. Given that  $WX = WZ = \sqrt{13}$  and  $XY = ZY$ , determine the length of  $XY$ .

Answer: $\sqrt{65}$


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

There are real numbers  $a, b, c, d$  such that for all  $(x, y)$  satisfying  $6y^2 = 2x^3 + 3x^2 + x$ , if  $x_1 = ax + b$  and  $y_1 = cy + d$ , then  $y_1^2 = x_1^3 - 36x_1$ . What is  $a + b + c + d$ ?

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>.

Given the function $f(x) = \frac{x}{\ln x}$, and $g(x) = f(x) - mx (m \in \mathbb{R})$,

(I) Find the interval of monotonic decrease for function $f(x)$.

(II) If function $g(x)$ is monotonically decreasing on the interval $(1, +\infty)$, find the range of the real number $m$.

(III) If there exist $x_1, x_2 \in [e, e^2]$ such that $m \geq g(x_1) - g'(x_2)$ holds true, find the minimum value of the real number $m$.

Answer: $\frac{1}{2} - \frac{1}{4e^2}$


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>.

Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?

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>.

A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters?

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>.

It is planned to establish an additional channel for exchanging stereo audio signals (messages) for daily reporting communication sessions between two working sites of a deposit. Determine the required bandwidth of this channel in kilobits, considering that the sessions will be conducted for no more than 51 minutes. The requirements for a mono signal per second are given below:

- Sampling rate: 63 Hz
- Sampling depth: 17 bits
- Metadata volume: 47 bytes for every 5 kilobits of audio

Answer: $2.25$


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

Let  $A_1A_2A_3A_4A_5$  be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral  $A_2A_5B_2B_5$  (shaded in the picture to the right) to the area of the entire 10-sided polygon.

[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705);
filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));
draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);

pair O = (A_1+A_2+A_3+A_4+A_5)/5;
label(" $A_1$ ",A_1, 2dir(A_1-O));
label(" $A_2$ ",A_2, 2dir(A_2-O));
label(" $A_3$ ",A_3, 2dir(A_3-O));
label(" $A_4$ ",A_4, 2dir(A_4-O));
label(" $A_5$ ",A_5, 2dir(A_5-O));
label(" $B_1$ ",B_1, 2dir(B_1-O));
label(" $B_2$ ",B_2, 2dir(B_2-O));
label(" $B_3$ ",B_3, 2dir(B_3-O));
label(" $B_4$ ",B_4, 2dir(B_4-O));
label(" $B_5$ ",B_5, 2dir(B_5-O));
[/asy]

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


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

Given the polynomial $$Q(x) = \left(1 + x + x^2 + \ldots + x^{20}\right)^2 - x^{20},$$ find the sum $$\beta_1 + \beta_2 + \beta_6$$ where the complex zeros of $Q(x)$ are written in the form, $\beta_k=r_k[\cos(2\pi\beta_k)+i\sin(2\pi\beta_k)]$, with $0<\beta_1\le\beta_2\le\ldots\le\beta_{41}<1$ and $r_k>0$.

Answer: $\frac{3}{7}$


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

Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and passing through the point $P(0, \sqrt{5})$, with an eccentricity of $\frac{2}{3}$, and $A$ being a moving point on the line $x=4$.
- (I) Find the equation of the ellipse $C$;
- (II) Point $B$ is on the ellipse $C$, satisfying $OA \perpendicular OB$, find the minimum length of segment $AB$.

Answer: $\sqrt{21}$


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>.

A mathematical demonstration showed that there were distinct positive integers such that $97^4 + 84^4 + 27^4 + 3^4 = m^4$. Calculate the value of $m$.

Answer: $108$


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

We denote by  gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd $(4, 6, 8)=2$  and gcd  $(12, 15)=3$ .) Suppose that positive integers  $a, b, c$  satisfy the following four conditions: $\bullet$  gcd  $(a, b, c)=1$ , $\bullet$  gcd  $(a, b + c)>1$ , $\bullet$  gcd  $(b, c + a)>1$ , $\bullet$  gcd  $(c, a + b)>1$ .
a) Is it possible that  $a + b + c = 2015$ ?
b) Determine the minimum possible value that the sum  $a+ b+ c$  can take.

Answer: $30$


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

A rectangular piece of paper $A B C D$ is folded and flattened such that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ landing on side $A B$. Given that $\angle 1 = 22^{\circ}$, find $\angle 2$.

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>.

Given a triangle $\triangle ABC$ with its three interior angles $A$, $B$, and $C$ satisfying: $$A+C=2B, \frac {1}{\cos A}+ \frac {1}{\cos C}=- \frac { \sqrt {2}}{\cos B}$$, find the value of $$\cos \frac {A-C}{2}$$.

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


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

What is the smallest positive integer with exactly 16 positive divisors?

Answer: $384$


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

A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the table, without slipping, and stops once the dot returns to the top. Find the length of the path followed by the dot in terms of $\pi$.

Answer: $2\pi$


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>.

The height $BD$ of the acute-angled triangle $ABC$ intersects with its other heights at point $H$. Point $K$ lies on segment $AC$ such that the angle $BKH$ is maximized. Find $DK$ if $AD = 2$ and $DC = 3$.

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>.

On a luxurious ocean liner, 3000 adults consisting of men and women embark on a voyage. If 55% of the adults are men and 12% of the women as well as 15% of the men are wearing sunglasses, determine the total number of adults wearing sunglasses.

Answer: $409$


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

$2.46\times 8.163\times (5.17+4.829)$ is approximately equal to what value?

Answer: $200$


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

How many distinct four-digit positive integers have a digit product equal to 18?

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>.

In the xy-plane, consider a right triangle $ABC$ with the right angle at $C$. The hypotenuse $AB$ is of length $50$. The medians through vertices $A$ and $B$ are described by the lines $y = x + 5$ and $y = 2x + 2$, respectively. Determine the area of triangle $ABC$.

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>.

Initially, the numbers 1 and 2 are written at opposite positions on a circle. Each operation consists of writing the sum of two adjacent numbers between them. For example, the first operation writes two 3's, and the second operation writes two 4's and two 5's. After each operation, the sum of all the numbers becomes three times the previous total. After sufficient operations, find the sum of the counts of the numbers 2015 and 2016 that are written.

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>.

The Wolf and the three little pigs wrote a detective story "The Three Little Pigs-2", and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remaining 2100 gold coins to the Wolf. The fee for each book is divided equally among its authors. How much money should the Wolf take for himself?

Answer: $700$


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

Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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>.

The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?

Answer: $59$


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

How many two-digit numbers have digits whose sum is a prime number?

Answer: $31$


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

Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order.

Answer: $26$


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

Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:
(1) $a_0+a_1+...+a_5$;
(2) $|a_0|+|a_1|+...+|a_5|$;
(3) $a_1+a_3+a_5$;
(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$.

Answer: $-243$


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

Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\]

Answer: $\frac{\sqrt{373}}{6}$


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>.

Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7.

Answer: $27$


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 the sequence $\{a\_n\}(n=1,2,3,...,2016)$, circle $C\_1$: $x^{2}+y^{2}-4x-4y=0$, circle $C\_2$: $x^{2}+y^{2}-2a_{n}x-2a_{2017-n}y=0$. If circle $C\_2$ bisects the circumference of circle $C\_1$, then the sum of all terms in the sequence $\{a\_n\}$ is $\_\_\_\_\_\_$.

Answer: $4032$


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

Let $\triangle ABC$ be a right triangle with $\angle ABC = 90^\circ$, and let $AB = 10\sqrt{21}$ be the hypotenuse. Point $E$ lies on $AB$ such that $AE = 10\sqrt{7}$ and $EB = 20\sqrt{7}$. Let $F$ be the foot of the altitude from $C$ to $AB$. Find the distance $EF$. Express $EF$ in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.

Answer: $31$


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

How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two conditions?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$.

Answer: $59$


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

Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge.

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


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

For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 4~.$$

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>.

Find the greatest common divisor of $8!$ and $(6!)^2.$

Answer: $5760$


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

A district in a city is laid out in an $11 \times 11$ grid. Every day, a sprinkler truck departs from the bottom-left corner $A(0,0)$ and travels along the streets to reach the top-right corner $B(10,10)$. At each intersection, the driver randomly chooses a direction, as long as it does not deviate from the shortest path. One day, the street from $(9,9)$ to $(10,9)$ is blocked due to an accident, but the driver is not aware of this at the time of departure. What is the probability that the sprinkler truck can still reach $B$ without any issues?

Answer: $1 - \frac{\binom{18}{9}}{\binom{20}{10}}$


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

Evaluate $|\omega^2 + 4\omega + 34|$ if $\omega = 5 + 3i$.

Answer: $\sqrt{6664}$


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

From the numbers 2, 3, 4, 5, 6, 7, 8, 9, two different numbers are selected to be the base and the exponent of a logarithm, respectively. How many different logarithmic values can be formed?

Answer: $52$


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

Let  $ABC$  be a triangle with side lengths  $AB=6, AC=7,$  and  $BC=8.$  Let  $H$  be the orthocenter of  $\triangle ABC$  and  $H'$  be the reflection of  $H$  across the midpoint  $M$  of  $BC.$   $\tfrac{[ABH']}{[ACH']}$  can be expressed as  $\frac{p}{q}$ . Find  $p+q$ .

*2022 CCA Math Bonanza Individual Round #14*

Answer: $251$


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

What is the smallest positive integer with exactly 12 positive integer divisors?

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>.

Let \( g_{1}(x) = \sqrt{4 - x} \), and for integers \( n \geq 2 \), define \[ g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right). \] Find the largest \( n \) (denote this as \( M \)) for which the domain of \( g_n \) is nonempty. For this value of \( M \), if the domain of \( g_M \) consists of a single point \( \{d\} \), compute \( d \).

Answer: $-589$


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

A rectangular table  $ 9$  rows  $ \times$   $ 2008$  columns is fulfilled with numbers  $ 1$ ,  $ 2$ , ..., $ 2008$  in a such way that each number appears exactly  $ 9$  times in table and difference between any two numbers from same column is not greater than  $ 3$ . What is maximum value of minimum sum in column (with minimal sum)?

Answer: $24$