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

Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number.

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>.

In the geometric sequence $\{a_n\}$, $S_4=1$, $S_8=3$, then the value of $a_{17}+a_{18}+a_{19}+a_{20}$ is.

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>.

Calculate $[(6^{6} \div 6^{5})^3 \cdot 8^3] \div 4^3$.

Answer: $1728$


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

How many multiples of 5 are there between 105 and 500?

Answer: $79$


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

A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?

Answer: $99$


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

For real numbers $t,$ consider the point of intersection of the triplet of lines $3x - 2y = 8t - 5$, $2x + 3y = 6t + 9$, and $x + y = 2t + 1$. All the plotted points lie on a line. Find the slope of this line.

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


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

Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number?

Answer: $143$


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

$ABC$  is a triangle with  $AB = 33$ ,  $AC = 21$  and  $BC = m$ , an integer. There are points  $D$ ,  $E$  on the sides  $AB$ ,  $AC$  respectively such that  $AD = DE = EC = n$ , an integer. Find  $m$ .

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>.

Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers?

Answer: $17$


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

Let the coefficient of $x^{-4}$ in the expansion of $\left(1- \frac {1}{x^{2}}\right)^{n}$ (where $n\in\mathbb{N}_{+}$) be denoted as $a_{n}$. Calculate the value of $$\frac {1}{a_{2}}+ \frac {1}{a_{3}}+…+ \frac {1}{a_{2015}}$$.

Answer: $\frac {4028}{2015}$


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

The area of the base of a hemisphere is $144\pi$. A cylinder of the same radius as the hemisphere and height equal to the radius of the hemisphere is attached to its base. What is the total surface area of the combined solid (hemisphere + cylinder)?

Answer: $576\pi$


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>.

Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction.

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


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

Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a student's score ranks among the top 9100 in the province. Then, the student's math score will not be less than ______ points. (Reference data: $P(\mu -\sigma\ \ \lt X \lt \mu +\sigma )=0.6827, P(\mu -2\sigma\ \ \lt X \lt \mu +2\sigma )=0.9545$)

Answer: $118$


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

Given the circle $x^{2}-2x+y^{2}-2y+1=0$, find the cosine value of the angle between the two tangents drawn from the point $P(3,2)$.

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>.

A classroom has 9 desks arranged in a row. Two students, Alex and Jessie, choose their desks at random. What is the probability that they do not sit next to each other?

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


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>.

Calculate the surface integrals of the first kind:
a) \(\iint_{\sigma}|x| dS\), where \(\sigma\) is defined by \(x^2 + y^2 + z^2 = 1\), \(z \geqslant 0\).

b) \(\iint_{\sigma} (x^2 + y^2) dS\), where \(\sigma\) is defined by \(x^2 + y^2 = 2z\), \(z = 1\).

c) \(\iint_{\sigma} (x^2 + y^2 + z^2) dS\), where \(\sigma\) is the part of the cone defined by \(z^2 - x^2 - y^2 = 0\), \(z \geqslant 0\), truncated by the cylinder \(x^2 + y^2 - 2x = 0\).

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


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>.

During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the "Qingming Sacrifice to the Martyrs" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students can only choose one way to participate. It is known that the ratio of the number of students in the three grades of the middle school, Grade 7, Grade 8, and Grade 9, is $4:5:6$. In order to understand the way students participate in the "Qingming Sacrifice to the Martyrs" activity, a stratified sampling method is used for investigation, and the following data is obtained:

| Grade and Participation Method | Grade 7 | Grade 8 | Grade 9 |
|-------------------------------|---------|---------|---------|
| Visit Memorial Hall            | $2a-1$  | $8$     | $10$    |
| Online Participation           | $a$     | $b$     | $2$     |

$(1)$ Find the values of $a$ and $b$;
$(2)$ From the students surveyed in each grade of the school who chose to participate in the "Qingming Sacrifice to the Martyrs" activity online, randomly select two students. Find the probability that these two students are from the same grade.

Answer: $\frac{5}{21}$


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 regular octagon's perimeter is given as $P=16\sqrt{2}$. If $R_i$ denotes the midpoint of side $V_iV_{i+1}$ (with $V_8V_1$ as the last side), calculate the area of the quadrilateral $R_1R_3R_5R_7$.
A) $8$
B) $4\sqrt{2}$
C) $10$
D) $8 + 4\sqrt{2}$
E) $12$

Answer: $8 + 4\sqrt{2}$


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>.

If the line $l_1: x + ay + 6 = 0$ is parallel to the line $l_2: (a-2)x + 3y + 2a = 0$, calculate the distance between lines $l_1$ and $l_2$.

Answer: $\frac{8\sqrt{2}}{3}$


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

Solve the equations:
(1) $(x-3)^2+2x(x-3)=0$
(2) $x^2-4x+1=0$.

Answer: $2-\sqrt{3}$


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

In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$.
$(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$.

Answer: $2 + \sqrt{3}$


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

In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$ respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at point $R$, with $BR = 8$ and $PR = 9$. If $\triangle BRP$ is a right triangle with $\angle BRP = 90^\circ$, what is the area of the square $ABCD$?
A) 144
B) 169
C) 225
D) 256
E) 289

Answer: $225$


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

Compute: $104 \times 96$.

Answer: $9984$


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

Suppose there are 15 dogs including Rex and Daisy. We need to divide them into three groups of sizes 6, 5, and 4. How many ways can we form the groups such that Rex is in the 6-dog group and Daisy is in the 4-dog group?

Answer: $72072$


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

Let \(A\) and \(B\) be two moving points on the ellipse \(x^2 + 3y^2 = 1\), and \(OA \perp OB\) (where \(O\) is the origin). Find the product of the maximum and minimum values of \( |AB| \).

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


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

In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively. The radius of the circumcircle of $\triangle ABC$ is $1$, and $b = acosC - \frac{{\sqrt{3}}}{6}ac$.
$(Ⅰ)$ Find the value of $a$;
$(Ⅱ)$ If $b = 1$, find the area of $\triangle ABC$.

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


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>.

Given that the function $f(x)$ defined on $\mathbb{R}$ satisfies $f(4)=2-\sqrt{3}$, and for any $x$, $f(x+2)=\frac{1}{-f(x)}$, find $f(2018)$.

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


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

Let O be the center of the square ABCD. If 3 points are chosen from O, A, B, C, and D at random, find the probability that the 3 points are collinear.

Answer: $\frac{1}{5}$


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>.

An experimenter is conducting an experiment that involves implementing five procedures in sequence. Procedure A must only occur either as the first or the last step, and procedures C and D must be implemented consecutively. The number of possible arrangements for the sequence of these procedures is _______.

Answer: $24$


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

Given an ellipse $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left focus $F_{1}$ and right focus $F_{2}$, and the focal distance $F_{1}F_{2}$ is $2$. A line passing through $F_{1}$ intersects the ellipse $E$ at points $A$ and $B$, and the perimeter of $\triangle ABF_{2}$ is $4\sqrt{3}$.
$(1)$ Find the equation of the ellipse $E$;
$(2)$ If the slope of line $AB$ is $2$, find the area of $\triangle ABF_{2}$.

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


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

Given a geometric sequence $\{a_{n}\}$, the sum of the first n terms is $S_{n}$, $S_{2}=7$ and $S_{6}=91$. Calculate $S_{4}$.

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>.

Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$.

Answer: $-4$


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

A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation?

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


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

The number 2015 can be represented as a sum of consecutive integers in several ways, for example, $2015 = 1007 + 1008$ or $2015 = 401 + 402 + 403 + 404 + 405$. How many ways can this be done?

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>.

Ranu starts with one standard die on a table.  At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step.  After 2013 such steps, let  $D$  be the number of dice on the table.  What is the expected value (average value) of  $6^D$ ?

Answer: $10071$


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

For a positive integer  $n>1$ , let  $g(n)$  denote the largest positive proper divisor of  $n$  and  $f(n)=n-g(n)$ . For example,  $g(10)=5, f(10)=5$  and  $g(13)=1,f(13)=12$ . Let  $N$  be the smallest positive integer such that  $f(f(f(N)))=97$ . Find the largest integer not exceeding  $\sqrt{N}$

Answer: $19$


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

A pen costs  $\mathrm{Rs.}\, 13$  and a note book costs  $\mathrm{Rs.}\, 17$ . A school spends exactly  $\mathrm{Rs.}\, 10000$  in the year  $2017-18$  to buy  $x$  pens and  $y$  note books such that  $x$  and  $y$  are as close as possible (i.e.,  $|x-y|$  is minimum). Next year, in  $2018-19$ , the school spends a little more than  $\mathrm{Rs.}\, 10000$  and buys  $y$  pens and  $x$  note books. How much **more** did the school pay?

Answer: $40$


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

The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape?

Answer: $14$


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

A projectile is launched with an initial velocity of $u$ at an angle of $\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:
\[
x = ut \cos \phi,
\]
\[
y = ut \sin \phi - \frac{1}{2} gt^2,
\]
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant but $\phi$ varies from $0^\circ$ to $180^\circ$. As $\phi$ changes, the highest points of the trajectories trace a closed curve. The area enclosed by this curve can be expressed as $d \cdot \frac{u^4}{g^2}$. Find the value of $d$.

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


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

Tio Mané has two boxes, one with seven distinct balls numbered from 1 to 7 and another with eight distinct balls numbered with all prime numbers less than 20. He draws one ball from each box.

Calculate the probability that the product is odd. What is the probability that the product of the numbers on the drawn balls is even?

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>.

What is the result of subtracting $7.305$ from $-3.219$?

Answer: $-10.524$


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

A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$?

Answer: $0.19$


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>.

Determine the fifth-largest divisor of 2,500,000,000.

Answer: $156,250,000$


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

What is the total volume and the total surface area in square feet of three cubic boxes if their edge lengths are 3 feet, 5 feet, and 6 feet, respectively?

Answer: $420$


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

Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$.

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


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

Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+4\cos^{2}\theta}=2a$ ($a>0$).

1. Find the Cartesian equation of curve $C$.
2. Given point $P(0,4)$, line $l$ intersects curve $C$ at points $M$ and $N$. If $|PM|\cdot|PN|=14$, find the value of $a$.

Answer: $\frac{2\sqrt{21}}{3}$


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

Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$.

Answer: $-4$


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>.

Compute \[ \left\lfloor \dfrac {1007^3}{1005 \cdot 1006} - \dfrac {1005^3}{1006 \cdot 1007} + 5 \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$

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>.

Given an isosceles triangle $ABC$ with $AB = AC = 40$ units and $BC = 24$ units, let $CX$ be the angle bisector of $\angle BCA$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Provide your answer as a simplified fraction.

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 $\sqrt[3]{2.37} \approx 1.333$ and $\sqrt[3]{23.7} \approx 2.872$, determine the approximate value of $\sqrt[3]{2370}$.

Answer: $13.33$


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

For all positive reals  $ a$ ,  $ b$ , and  $ c$ , what is the value of positive constant  $ k$  satisfies the following inequality?
 $ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$  .

Answer: $6020$


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

The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$

Answer: $105$


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

If the function $f(x)=\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+ax+4$ is strictly decreasing on the interval $[-1,4]$, then the value of the real number $a$ is ______.

Answer: $-4$


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

In the expression $10 \square 10 \square 10 \square 10 \square 10$, fill in the four spaces with each of the operators "+", "-", "×", and "÷" exactly once. The maximum possible value of the resulting expression is:

Answer: $109$


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

Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps.

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>.

In a sequence of positive integers starting from 1, certain numbers are painted red according to the following rules: First paint 1, then the next 2 even numbers $2, 4$; then the next 3 consecutive odd numbers after 4, which are $5, 7, 9$; then the next 4 consecutive even numbers after 9, which are $10, 12, 14, 16$; then the next 5 consecutive odd numbers after 16, which are $17, 19, 21, 23, 25$. Following this pattern, we get a red subsequence $1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, \cdots$. What is the 2003rd number in this red subsequence?

Answer: $3943$


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

Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order).

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>.

A community plans to organize three activities, "Book Club," "Fun Sports," and "Environmental Theme Painting," to enrich the lives of residents. A total of 120 people have signed up for the activities, with each resident participating in at most two activities. It is known that 80 people participate in the "Book Club," 50 people in "Fun Sports," and 40 people in "Environmental Theme Painting." Additionally, 20 people participate in both the "Book Club" and "Fun Sports," and 10 people participate in both "Fun Sports" and "Environmental Theme Painting." Find the number of people who participate in both the "Book Club" and "Environmental Theme Painting."

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>.

Given the parameter equation of line $l$ and the equation of circle $C$ in the polar coordinate system, find the rectangular coordinate equation of circle $C$ and the minimum value of $\frac{1}{|PA|} + \frac{1}{|PB|}$, where $P(1, 2)$ and $A$, $B$ are the intersection points of line $l$ and circle $C$.

The parameter equation of line $l$ in the rectangular coordinate system is $\begin{cases} x = 1 + t\cos\alpha\\ y = 2 + t\sin\alpha \end{cases}$ ($t$ is the parameter), and the equation of circle $C$ in the polar coordinate system (with the same unit length and origin as the rectangular coordinate system, and the positive $x$-axis as the polar axis) is $\rho = 6\sin\theta$.

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


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>.

In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\tan A = (2c-b)\tan B$.
$(1)$ Find angle $A$;
$(2)$ If $\overrightarrow{m}=(0,-1)$ and $\overrightarrow{n}=(\cos B, 2\cos^2\frac{C}{2})$, find the minimum value of $|\overrightarrow{m}+\overrightarrow{n}|$.

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>.

During the November monthly exam at our school, approximately 1,000 science students participated, with mathematics scores distributed normally as $\xi \sim N(100, a^2), (a > 0)$, and a full score of 150. The statistics showed that about 60% of the students scored between 80 and 120 points. Therefore, approximately \_\_\_\_\_\_ students scored no less than 120 points in this monthly exam.

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>.

Given that $A$ is an interior angle of $\triangle ABC$, when $x= \frac {5\pi}{12}$, the function $f(x)=2\cos x\sin (x-A)+\sin A$ attains its maximum value. The sides opposite to the angles $A$, $B$, $C$ of $\triangle ABC$ are $a$, $b$, $c$ respectively.
$(1)$ Find the angle $A$;
$(2)$ If $a=7$ and $\sin B + \sin C = \frac {13 \sqrt {3}}{14}$, find the area of $\triangle ABC$.

Answer: $10\sqrt{3}$


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

Let point P be a fixed point inside a circle ⊙O with a radius of 5, and OP=4. The sum of all possible integer values of the chord lengths passing through point P is.

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>.

A school wishes to understand the psychological state of learning among its senior students and adopts a systematic sampling method to select 40 students out of 800 for a test. The students are randomly assigned numbers from 1 to 800 and then grouped. In the first group, number 18 is selected through simple random sampling. Among the 40 selected students, those with numbers in the range [1, 200] take test paper A, numbers in the range [201, 560] take test paper B, and the remaining students take test paper C. Calculate the number of students who take test paper C.

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>.

Sarah is leading a class of  $35$  students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands  $3$  times, what is the greatest possible number of students that could have been standing before her third wave?

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>.

If a sequence's sum of the first $n$ terms, $S_n = 1 - 5 + 9 - \ldots + (-1)^{n+1}(4n - 3)$ for $n \in \mathbb{N}^*$, find the value of $S_{15} - S_{22} + S_{31}$.

Answer: $134$


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

Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A Story About Friendship and Non-friendship" are next to each other (in any order).
In how many ways can Vasya do this?

Choose the correct answer:
a) \(4 \cdot 7!\);
b) \(9!\);
c) \(\frac{9!}{4!}\);
d) \(4! \cdot 7!\);
e) another answer.

Answer: $4 \cdot 7!$


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>.

Find the result of $46_8 - 63_8$ and express your answer in base 10.

Answer: $-13$


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

Given that $F$ is the focus of the parabola $x^{2}=8y$, $P$ is a moving point on the parabola, and the coordinates of $A$ are $(0,-2)$, find the minimum value of $\frac{|PF|}{|PA|}$.

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>.

Let \( x \) be a real number such that \( x + \frac{1}{x} = 5 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \).

Answer: $12098$


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

Inside the square \(ABCD\), points \(K\) and \(M\) are marked (point \(M\) is inside triangle \(ABD\), point \(K\) is inside \(BMC\)) such that triangles \(BAM\) and \(DKM\) are congruent \((AM = KM, BM = MD, AB = KD)\). Find \(\angle KCM\) if \(\angle AMB = 100^\circ\).

Answer: $35$


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

In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class.

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Score (\%)}&\textbf{Number of Students}\\\hline
95&12\\\hline
85&24\\\hline
75&30\\\hline
65&20\\\hline
55&18\\\hline
45&10\\\hline
35&6\\\hline
\end{tabular}

Answer: $69.83$


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

Given the equations $z^2 = 1 + 3\sqrt{10}i$ and $z^2 = 2 - 2\sqrt{2}i$, where $i = \sqrt{-1}$, find the vertices formed by the solutions of these equations on the complex plane and compute the area of the quadrilateral they form.
A) $17\sqrt{6} - 2\sqrt{2}$
B) $18\sqrt{6} - 3\sqrt{2}$
C) $19\sqrt{6} - 2\sqrt{2}$
D) $20\sqrt{6} - 2\sqrt{2}$

Answer: $19\sqrt{6} - 2\sqrt{2}$


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

For real numbers \( x \) and \( y \), simplify the equation \(\cfrac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} + 2\frac{1}{y}} = 4\) and express it as \(\frac{x+y}{x+2y}\).

Answer: $\frac{4}{11}$


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

Determine the minimum number of digits to the right of the decimal point required to express the fraction $\frac{987654321}{2^{30} \cdot 5^5}$ as a decimal.

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>.

Out of 100 externally identical marbles, one is radioactive, but I don't know which one it is. A friend of mine would buy only non-radioactive marbles from me, at a price of 1 forint each. Another friend of mine has an instrument that can determine whether or not there is a radioactive marble among any number of marbles. He charges 1 forint per measurement, but if there is a radioactive marble among those being measured, all of the marbles in the measurement will become radioactive.

What is the maximum profit I can absolutely achieve?

Answer: $92$


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

Given the equation of line $l$ is $ax+by+c=0$, where $a$, $b$, and $c$ form an arithmetic sequence, the maximum distance from the origin $O$ to the line $l$ is ______.

Answer: $\sqrt{5}$


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

In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), calculate the ratio of the area of the shaded region to the area of the unshaded region.

Answer: $\frac{12}{13}$


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

Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Find the value of $(1+p)(1+q)(1+r)$.

Answer: $51$


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

Consider finding the result when we compute the series $$1^3 + 2^3 + 3^3 + \dots + 49^3 + 50^3$$ and the series $$(-1)^3 + (-2)^3 + (-3)^3 + \dots + (-49)^3 + (-50)^3,$$ then subtract the second series' result from the first series' result. What is the sum?

Answer: $3251250$


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

Calculate the probability of selecting the letter "$s$" in the word "statistics".

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


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

A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$.

Answer: $0.3$


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

Let   $T  =  TNFTPP$ .  As   $n$   ranges  over  the  integers,  the  expression   $n^4  -  898n^2  +  T  -  2160$   evaluates  to just  one  prime  number.  Find  this  prime.

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, here's the value of T$T=2161$.

Answer: $1801$


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

Given that the sum of the first $n$ terms ($S_n$) of the sequence $\{a_n\}$ satisfies $S_n = 2a_n - 1$ ($n \in \mathbb{N}^*$).

(1) Find the general term formula of the sequence $\{a_n\}$;

(2) If the sequence $\{b_n\}$ satisfies $b_n = 1 + \log_2 a_n$,

(I) Find the sum of the first $n$ terms ($T_n$) of the sequence $\{a_n b_n\}$;

(II) Find the minimum value of $\frac{b_n^2 + 9}{(\log_2 a_n) + 2}$.

Answer: $\frac{13}{3}$


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

Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros.

Answer: $1968$


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

$F$ is the right focus of the hyperbola $C: \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \left(a > 0, b > 0\right)$. A perpendicular line is drawn from point $F$ to asymptote $C$, with the foot of the perpendicular denoted as $A$, intersecting another asymptote at point $B$. If $2\overrightarrow {AF} = \overrightarrow {FB}$, then find the eccentricity of $C$.

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


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

The increasing sequence $1, 3, 4, 9, 10, 12, 13, \dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\text{th}}$ term of this sequence.

Answer: $2280$


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

Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \( \frac{48}{\sqrt{K}} \). What is \( K \)?

Answer: $\frac{36}{\pi^3}$


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>.

Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$.

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


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>.

Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = -a_n - \left(\frac{1}{2}\right)^{n-1} + 2$, and $(1) b_n = 2^n a_n$, find the general term formula for $\{b_n\}$. Also, $(2)$ find the maximum term of $\{a_n\}$.

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>.

What is the greatest integer less than or equal to \[\frac{5^{80} + 3^{80}}{5^{75} + 3^{75}}?\]

Answer: $3124$


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

Given that one of the children is a boy and the probability of having a boy or a girl is equal, calculate the probability that the other child is a girl.

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


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

Given $x \gt 0$, $y \gt 0$, and $x+y=1$, find the minimum value of $\frac{2{x}^{2}-x+1}{xy}$.

Answer: $2\sqrt{2}+1$


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>.

Given the arithmetic sequence $\{a_n\}$, $S_n$ denotes the sum of its first $n$ terms. Given that $a_4 + a_8 = 4$, find the value of $S_{11} + a_6$.

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>.

In Dr. Strange's laboratory, there are some bacteria. Each day, 11 bacteria are eliminated, and each night, 5 bacteria are added. If there are 50 bacteria on the morning of the first day, on which day will all the bacteria be eliminated?

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>.

Solve the system of equations:
\begin{cases}
\frac{m}{3} + \frac{n}{2} = 1 \\
m - 2n = 2
\end{cases}

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


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

Given the function \(\frac{(4^t - 2t)t}{16^t}\), find the maximum value for real values of \(t\).

Answer: $\frac{1}{8}$


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

The diagram shows an arc \( PQ \) of a circle with center \( O \) and radius 8. Angle \( QOP \) is a right angle, the point \( M \) is the midpoint of \( OP \), and \( N \) lies on the arc \( PQ \) so that \( MN \) is perpendicular to \( OP \). Which of the following is closest to the length of the perimeter of triangle \( PNM \)?

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>.

A sports league has 20 teams split into four divisions of 5 teams each. How many games are there in a complete season if each team plays every other team in its own division three times and every team in the other divisions once?

Answer: $270$


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

Determine the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_8 \le 1023$ such that $a_i-i$ is even for $1\le i \le 8$. The answer can be expressed as $\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided by 1000.

Answer: $515$


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

Point $C(0,p)$ lies on the $y$-axis between $Q(0,15)$ and $O(0,0)$. Point $B$ has coordinates $(15,0)$. Determine an expression for the area of $\triangle COB$ in terms of $p$, and compute the length of segment $QB$. Your answer should be simplified as much as possible.

Answer: $15\sqrt{2}$


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

In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

Answer: $\frac{2\sqrt{3}+1}{3}$


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>.

Two sides of a right triangle have the lengths 8 and 15. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth.

Answer: $215.7$


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>.

How many  $6$ -tuples  $(a, b, c, d, e, f)$  of natural numbers are there for which  $a>b>c>d>e>f$  and  $a+f=b+e=c+d=30$  are simultaneously true?

Answer: $364$


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

Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. Given that $a_1 + a_2 + a_3 = 3$ and $a_4 + a_5 + a_6 = 6$, calculate the value of $S_{12}$.

Answer: $45$


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

Given the function $f(x)=e^{x}(ax+b)-x^{2}-4x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$ is $y=4x+4$.

(1) Find the values of $a$ and $b$;

(2) Determine the intervals of monotonicity for $f(x)$ and find the maximum value of $f(x)$.

Answer: $4(1-e^{-2})$


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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $\angle A=45^{\circ}$, $a=6$.
(1) If $\angle C=105^{\circ}$, find $b$;
(2) Find the maximum area of $\triangle ABC$.

Answer: $9(1+\sqrt{2})$


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

Let  $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form  $\frac{b}{2^a}$ , where  $a$  is a non-negative integer and  $b$  is an odd integer (not necessarily positive). Find  $a+b$ .

*2017 CCA Math Bonanza Tiebreaker Round #3*

Answer: $1441$


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

A circle has a radius of 3 units. A line segment of length 3 units is tangent to the circle at its midpoint. Determine the area of the region consisting of all such line segments.
A) $1.5\pi$
B) $2.25\pi$
C) $3\pi$
D) $4.5\pi$

Answer: $2.25\pi$


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>.

Given that 3 females and 2 males participate in a performance sequence, and the 2 males cannot appear consecutively, and female A cannot be the first to appear, determine the total number of different performance sequences.

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>.

Given an arithmetic sequence $\{a_n\}$ with a positive common difference $d > 0$, and $a_2$, $a_5 - 1$, $a_{10}$ form a geometric sequence. If $a_1 = 5$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, determine the minimum value of $$\frac{2S_{n} + n + 32}{a_{n} + 1}$$

Answer: $\frac{20}{3}$


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>.

Which of the following five values of \( n \) is a counterexample to the statement: For a positive integer \( n \), at least one of \( 6n-1 \) and \( 6n+1 \) is prime?

Answer: $20$


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

In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

Answer: $\frac{2\sqrt{3} + 1}{3}$


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>.

Given triangle ABC, the sides opposite to angles A, B, and C are denoted as a, b, and c, respectively, and a = 6. Find the maximum value of the area of triangle ABC given that $\sqrt{7}bcosA = 3asinB$.

Answer: $9\sqrt{7}$


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

The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?

Answer: $2444$


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

Given \( a_{n} = 4^{2n - 1} + 3^{n - 2} \) (for \( n = 1, 2, 3, \cdots \)), where \( p \) is the smallest prime number dividing infinitely many terms of the sequence \( a_{1}, a_{2}, a_{3}, \cdots \), and \( q \) is the smallest prime number dividing every term of the sequence, find the value of \( p \cdot q \).

Answer: $5 \times 13$


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

In a school, 40 students are enrolled in both the literature and science classes. Ten students received an A in literature and 18 received an A in science, including six who received an A in both subjects. Determine how many students did not receive an A in either subject.

Answer: $18$


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

A 10-digit arrangement  $ 0,1,2,3,4,5,6,7,8,9$  is called *beautiful* if (i) when read left to right,  $ 0,1,2,3,4$  form an increasing sequence, and  $ 5,6,7,8,9$  form a decreasing sequence, and (ii)  $ 0$  is not the leftmost digit. For example,  $ 9807123654$  is a beautiful arrangement. Determine the number of beautiful arrangements.

Answer: $126$


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

The inclination angle of the line $x-y+1=0$ can be calculated.

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


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

Let the hyperbola $C:\frac{x^2}{a^2}-y^2=1\;(a>0)$ intersect the line $l:x+y=1$ at two distinct points $A$ and $B$.
$(1)$ Find the range of real numbers for $a$.
$(2)$ If the intersection point of the line $l$ and the $y$-axis is $P$, and $\overrightarrow{PA}=\frac{5}{12}\overrightarrow{PB}$, find the value of the real number $a$.

Answer: $a = \frac{17}{13}$


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

Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$.

Answer: $17$


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

Given two non-zero planar vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy the condition: for any $λ∈R$, $| \overrightarrow{a}-λ \overrightarrow{b}|≥slant | \overrightarrow{a}- \frac {1}{2} \overrightarrow{b}|$, then:
$(①)$ If $| \overrightarrow{b}|=4$, then $\overrightarrow{a}· \overrightarrow{b}=$ _______ ;
$(②)$ If the angle between $\overrightarrow{a}, \overrightarrow{b}$ is $\frac {π}{3}$, then the minimum value of $\frac {|2 \overrightarrow{a}-t· \overrightarrow{b}|}{| \overrightarrow{b}|}$ is _______ .

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>.

Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. Find the maximum value of $\sqrt{a}+ \sqrt{b}+ \sqrt{3}c$.

Answer: $\frac{\sqrt{21}}{3}$


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>.

Six distinct integers are picked at random from  $\{1,2,3,\ldots,10\}$ .  What is the probability that, among those selected, the second smallest is  $3$ ?

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