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

In triangle $PQR,$ $S$ is on $\overline{PQ}$ such that $PS:SQ = 4:1,$ and $T$ is on $\overline{QR}$ such that $QT:TR = 4:1.$ If lines $ST$ and $PR$ intersect at $U,$ then find $\frac{ST}{TU}.$

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


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

How many numbers should there be in a lottery for the probability of getting an ambo to be $\frac{5}{473}$, when drawing five numbers?

Answer: $44$


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

Find the positive integer  $n$  such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where  $f(n)$  denotes the  $n$ th positive integer which is not a perfect square.

*Proposed by David Stoner*

Answer: $6077248$


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

For  $k\ge 1$ , define  $a_k=2^k$ . Let   $$ S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right). $$   Compute  $\lfloor 100S\rfloor$ .

Answer: $157$


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

The area of each of Art's cookies is given by the formula for the area of a trapezoid: A = 1/2h(b1 + b2). The total area of dough used by Art to make a batch of cookies is 240 square inches, and this is equivalent to the number of cookies multiplied by the area of each cookie. Let n represent the number of cookies in a batch. The equation is thus expressed as: 240 = n * (1/2 * 4 * (4 + 6)). Solve for n.

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>.

From the consecutive natural numbers 1, 2, 3, …, 2014, select $n$ numbers such that for any two numbers chosen, one is not five times the other. Find the maximum value of $n$ and explain the reason.

Answer: $1665$


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

Is it possible to divide an equilateral triangle into 1,000,000 convex polygons such that any straight line intersects no more than 40 of them?

Answer: $1707$


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

The sequences where each term is a real number are denoted as $\left\{a_{n}\right\}$, with the sum of the first $n$ terms recorded as $S_{n}$. Given that $S_{10} = 10$ and $S_{30} = 70$, what is the value of $S_{40}$?

Answer: $150$


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

Twenty tiles are numbered 1 through 20 and are placed into box $C$. Twenty other tiles numbered 15 through 34 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 18 and the tile from box $D$ is either odd or greater than 30? Express your answer as a common fraction.

Answer: $\frac{17}{40}$


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>.

(1) In an arithmetic sequence $\{ a_n \}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.

(2) Determine the interval of decrease for the function $y = \log_{\frac{1}{2}}(x^2 + 2x - 3)$.

(3) In triangle $ABC$, given $a = \sqrt{3}$, $b = 1$, and $\angle A = \frac{\pi}{3}$, find the value of $\cos B$.

(4) Point $A$ lies on circle $O$ with diameter $BC$ and is distinct from points $B$ and $C$. Point $P$ is outside the plane $ABC$ and plane $PBC$ is perpendicular to plane $ABC$. Given $BC = 3$, $PB = 2\sqrt{2}$, and $PC = \sqrt{5}$, find the surface area of the circumscribed sphere of the tetrahedron $P-ABC$.

Answer: $10\pi$


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

Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. Calculate the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $O_1O_2O_3O_4$.

Answer: $3/2$


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

Given the equation $2x + 3k = 1$ with $x$ as the variable, if the solution for $x$ is negative, then the range of values for $k$ is ____.

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


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

Find the last two digits (in order) of \( 7^{30105} \).

Answer: $43$


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

Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probability that both groups will stop flipping their coins on the same round.

Answer: $\frac{1}{702}$


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

A regular decagon $B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}$ is inscribed in a circle of area $1$ square units. Point $Q$ lies inside the circle such that the region bounded by $\overline{QB_1},\overline{QB_2},$ and the minor arc $\widehat{B_1B_2}$ of the circle has area $\tfrac{1}{10},$ while the region bounded by $\overline{QB_3},\overline{QB_4},$ and the minor arc $\widehat{B_3B_4}$ of the circle has area $\tfrac{1}{12}.$ There is a positive integer $m$ such that the area of the region bounded by $\overline{QB_7},\overline{QB_8},$ and the minor arc $\widehat{B_7B_8}$ of the circle is equal to $\tfrac{1}{11}-\tfrac{\sqrt{3}}{m}.$ Find $m.$

Answer: $110\sqrt{3}$


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

Find the greatest root of the equation $g(x) = 21x^4 - 20x^2 + 3$.

Answer: $\frac{\sqrt{21}}{7}$


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>.

There are three cards with numbers on both the front and back sides: one with 0 and 1, another with 2 and 3, and a third with 4 and 5. A student uses these cards to form a three-digit even number. How many different three-digit even numbers can the student make?

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>.

A class has  $25$  students. The teacher wants to stock  $N$  candies, hold the Olympics and give away all  $N$  candies for success in it (those who solve equally tasks should get equally, those who solve less  get less, including, possibly, zero candies). At what smallest  $N$  this will be possible, regardless of the number of tasks on Olympiad and the student successes?

Answer: $600$


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

Consider the function
\[ f(x) = \max \{-8x - 29, 3x + 2, 7x - 4\} \] defined for all real $x$. Let $q(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $a_1$, $a_2$, $a_3$. Find $a_1 + a_2 + a_3$.

Answer: $-\frac{163}{22}$


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

The coordinates of vertex \( C(x, y) \) of triangle \( \triangle ABC \) satisfy the inequalities \( x^{2}+y^{2} \leq 8+2y \) and \( y \geq 3 \). The side \( AB \) is on the x-axis. Given that the distances from point \( Q(0,1) \) to the lines \( AC \) and \( BC \) are both 1, find the minimum area of \( \triangle ABC \).

Answer: $6 \sqrt{2}$


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

A larger square is constructed, and another square is formed inside it by connecting the midpoints of each side of the larger square. If the area of the larger square is 100, what is the area of the smaller square formed inside?

Answer: $25$


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

Find the value of \( k \) such that, for all real numbers \( a, b, \) and \( c \),

$$
(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \cdot abc
$$

Answer: $-2$


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

Vasya wrote natural numbers on the pages of an 18-page notebook. On each page, he wrote at least 10 different numbers, and on any consecutive three pages, there are no more than 20 different numbers in total. What is the maximum number of different numbers Vasya could have written on the pages of the notebook?

Answer: $190$


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

Let  $\triangle{ABC}$  be a triangle with  $AB = 10$  and  $AC = 11$ . Let  $I$  be the center of the inscribed circle of  $\triangle{ABC}$ . If  $M$  is the midpoint of  $AI$  such that  $BM = BC$  and  $CM = 7$ , then  $BC$  can be expressed in the form  $\frac{\sqrt{a}-b}{c}$  where  $a$ ,  $b$ , and  $c$  are positive integers. Find  $a+b+c$ .

Note that this problem is null because a diagram is impossible.

*Proposed by Andy Xu*

Answer: $622$


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

The wizard revealed the secret of wisdom. For those who wanted to know precisely where the secret is hidden, he left a clue in his magic book:

\[ 5 \cdot \text{BANK} = 6 \cdot \text{SAD} \]

Each letter in this clue represents a certain digit. Find these digits and substitute them into the GPS coordinates:

\[ (\mathrm{C} \cdot 6+1) . (\mathrm{C} \cdot 6)(\mathrm{6} \cdot 6) \, \text{DB(K:2)(H-1)} \]

\[ (K: 2)(H-1) \cdot (\text{D}-1)(H-1)(\text{C} \cdot 9+\text{D}) \text{A(K:2)} \]

Where exactly is the secret revealed by the wizard hidden?

Each expression in parentheses and each letter represents a number. There are no symbols between them other than those explicitly stated.

The coordinate format is a decimal fraction with a dot as the separator.

Answer: $55.543065317$


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

How many distinct equilateral triangles can be constructed by connecting three different vertices of a regular dodecahedron?

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>.

Polly has three circles cut from three pieces of colored card. She originally places them on top of each other as shown. In this configuration, the area of the visible black region is seven times the area of the white circle.

Polly moves the circles to a new position, as shown, with each pair of circles touching each other. What is the ratio between the areas of the visible black regions before and after?

Answer: $7:6$


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

Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed.

Answer: $43$


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

The picture shows several circles connected by segments. Tanya chooses a natural number \( n \) and places different natural numbers in the circles so that the following property holds for all these numbers:

If numbers \( a \) and \( b \) are not connected by a segment, then the sum \( a^2 + b^2 \) must be coprime with \( n \). If they are connected, then the numbers \( a^2 + b^2 \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?

Answer: $65$


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

What is the coefficient of $x^3$ when
$$x^5 - 4x^3 + 3x^2 - 2x + 5$$
is multiplied by
$$3x^2 - 2x + 4$$
and further multiplied by
$$1 - x$$
and the like terms are combined?

Answer: $-3$


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 \( p \) and \( q \) be the two distinct solutions to the equation
\[ (x-6)(3x+10) = x^2 - 19x + 50. \]
What is \( (p + 2)(q + 2) \)?

Answer: $108$


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

The Grunters play the Screamers 6 times. The Grunters have a 60% chance of winning any given game. If a game goes to overtime, the probability of the Grunters winning changes to 50%. There is a 10% chance that any game will go into overtime. What is the probability that the Grunters will win all 6 games, considering the possibility of overtime?

Answer: $\frac{823543}{10000000}$


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

The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches 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 that the function $f(x) = \frac{1}{2}\sin(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) is an even function, and points P and Q are the highest and lowest points respectively on the graph of $y = f(x)$ such that $| \overrightarrow{PQ} | = \sqrt{2}$,
(1) Find the explicit formula of the function $f(x)$;
(2) In triangle $\triangle ABC$, where $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, given that $a=1, b= \sqrt{2}$, and $f\left(\frac{A}{\pi}\right) = \frac{\sqrt{3}}{4}$, determine the size of angle $C$.

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


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

A line $l$ passes through two points $P(-1,2)$ and $Q(2,-2)$, and intersects the hyperbola $(y-2)^{2}-x^{2}=1$ at two points $A$ and $B$.
$(1)$ Write the parametric equation of $l$ as required by the question;
$(2)$ Find the distance between the midpoint $M$ of $AB$ and point $P$.

Answer: $5 \sqrt {65}$


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

(1) Given $\cos(15°+\alpha) = \frac{15}{17}$, with $\alpha \in (0°, 90°)$, find the value of $\sin(15°-\alpha)$.
(2) Given $\cos\alpha = \frac{1}{7}$, $\cos(\alpha-\beta) = \frac{13}{14}$, and $0 < \beta < \alpha < \frac{\pi}{2}$, find the value of $\beta$.

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


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

A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 6 that is not a lucky integer?

Answer: $114$


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

Each of the squares of an  $8 \times 8$  board can be colored white or black. Find the number of colorings of the board such that every  $2 \times 2$  square contains exactly 2 black squares and 2 white squares.

Answer: $8448$


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

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

Answer: $6560$


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

Given 100 real numbers, with their sum equal to zero. What is the minimum number of pairs that can be selected from them such that the sum of the numbers in each pair is non-negative?

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>.

Each brick in the pyramid contains one number. Whenever possible, the number in each brick is the least common multiple of the numbers of the two bricks directly above it.

What number could be in the bottom brick? Determine all possible options.

(Hint: What is the least common multiple of three numbers, one of which is a divisor of another?)

Answer: $2730$


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

The points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( S A B, S A C, S B C, \) and \( A B C \) of the tetrahedron \( S A B C \). It is known that \( A B = S C = 5 \), \( A C = S B = 7 \), \( B C = S A = 8 \). Find the volume of the tetrahedron \( K L M N \). If necessary, round your answer to two decimal places.

Answer: $0.66$


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

The carbon dioxide emissions in a certain region reach a peak of a billion tons (a > 0) and then begin to decline. The relationship between the carbon dioxide emissions S (in billion tons) and time t (in years) satisfies the function S = a · b^t. If after 7 years, the carbon dioxide emissions are (4a)/5 billion tons, determine the time it takes to achieve carbon neutrality, where the region offsets its own carbon dioxide emissions by (a)/4 billion tons.

Answer: $42$


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

Gabriela found an encyclopedia with  $2023$  pages, numbered from  $1$  to  $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?

Answer: $44$


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

Let  $ K$  be the curved surface obtained by rotating the parabola  $ y \equal{} \frac {3}{4} \minus{} x^2$  about the  $ y$ -axis.Cut  $ K$  by the plane  $ H$  passing through the origin and forming angle  $ 45^\circ$  for the axis. Find the volume of the solid surrounded by  $ K$  and  $ H.$

Note that you are not allowed to use Double Integral for the problem.

Answer: $\frac{9\pi}{256}$


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

How many solutions of the equation $\tan x = \tan 2x$ are in the interval $0 \le x \le \tan^{-1} 500$? Assume $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}$.

Answer: $159$


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

Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is \(5:4\). Find the minimum possible value of their common perimeter.

Answer: $524$


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>.

Given the function $f(2x+1)=x^{2}-2x$, determine the value of $f(\sqrt{2})$.

Answer: $\frac{5-4\sqrt{2}}{4}$


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

8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.

a) How many four-digit numbers are liked by both Andrey and Tanya?

b) Find the total sum of the digits of all such four-digit numbers.

Answer: $14580$


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

How many distinct, positive factors does $1320$ have?

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>.

Alice and Bob each draw one number from 50 slips of paper numbered from $1$ to $50$ placed in a hat. Alice says, "I can't tell who has the larger number." Bob then says, "I am certain who has the larger number." Understanding Bob's certainty, Alice asks Bob if his number is prime. Bob answers, "Yes." Alice then says, "In that case, when I multiply your number by $100$ and add my number, the result is a perfect square. What could my number possibly be?"
**A)** $24$
**B)** $61$
**C)** $56$
**D)** $89$

Answer: $61$


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

Given the function $f(x)=\sin 2x$, its graph intersects with the line $2kx-2y-kπ=0 (k > 0)$ at exactly three common points. The x-coordinates of these points in ascending order are $x_{1}$, $x_{2}$, $x_{3}$. Calculate the value of $(x_{1}-x_{3})\tan (x_{2}-2x_{3})$.

Answer: $-1$


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

A rectangular piece of paper, PQRS, has PQ = 20 and QR = 15. The piece of paper is glued flat on the surface of a large cube so that Q and S are at vertices of the cube. The shortest distance from P to R, as measured through the cube, can be calculated using the 3D geometry of the cube.

Answer: $18.4$


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

Define the sequence $b_1, b_2, b_3, \ldots$ by $b_n = \sum\limits_{k=1}^n \cos{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $b_n < 0$.

Answer: $632$


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

Let \( a_{1}, a_{2}, \cdots, a_{n} \) be an arithmetic sequence, and it is given that
$$
\sum_{i=1}^{n}\left|a_{i}+j\right|=2028 \text{ for } j=0,1,2,3.
$$
Find the maximum value of the number of terms \( n \).

Answer: $52$


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

An infinite geometric series has a first term of \( 416 \) and a sum of \( 3120 \). What is its common ratio?

Answer: $\frac{84}{97}$


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

The famous German mathematician Dirichlet made significant achievements in the field of mathematics. He was the first person in the history of mathematics to pay attention to concepts and consciously "replace intuition with concepts." The function named after him, $D\left(x\right)=\left\{\begin{array}{l}{1, x \text{ is rational}}\\{0, x \text{ is irrational}}\end{array}\right.$, is called the Dirichlet function. Now, a function similar to the Dirichlet function is defined as $L\left(x\right)=\left\{\begin{array}{l}{x, x \text{ is rational}}\\{0, x \text{ is irrational}}\end{array}\right.$. There are four conclusions about the Dirichlet function and the $L$ function:
$(1)D\left(1\right)=L\left(1\right)$;
$(2)$ The function $L\left(x\right)$ is an even function;
$(3)$ There exist four points $A$, $B$, $C$, $D$ on the graph of the $L$ function such that the quadrilateral $ABCD$ is a rhombus;
$(4)$ There exist three points $A$, $B$, $C$ on the graph of the $L$ function such that $\triangle ABC$ is an equilateral triangle.
The correct numbers of the conclusions are ____.

Answer: $(1)(4)$


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

Find $\left(\frac{2}{3}\right)^{6} \cdot \left(\frac{5}{6}\right)^{-4}$.

Answer: $\frac{82944}{456375}$


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

Given vectors $a$ and $b$ that satisfy $(a+2b)\cdot(5a-4b)=0$, and $|a|=|b|=1$, find the angle $\theta$ between $a$ and $b$.

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


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

In how many ways can 13 bishops be placed on an $8 \times 8$ chessboard such that:
(i) a bishop is placed on the second square in the second row,
(ii) at most one bishop is placed on each square,
(iii) no bishop is placed on the same diagonal as another bishop,
(iv) every diagonal contains a bishop?
(For the purposes of this problem, consider all diagonals of the chessboard to be diagonals, not just the main diagonals).

Answer: $1152$


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

In a new configuration, six circles with a radius of 5 units intersect at a single point. What is the number of square units in the area of the shaded region? The region is formed similarly to the original problem where the intersections create smaller sector-like areas. Express your answer in terms of $\pi$.

Answer: $75\pi - 25\sqrt{3}$


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>.

Fill the numbers 1 to 6 into the six boxes in the image. The smallest result you can get is ______

Answer: $342$


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

Given that Bob was instructed to subtract 5 from a certain number and then divide the result by 7, but instead subtracted 7 and then divided by 5, yielding an answer of 47, determine what his answer would have been had he worked the problem correctly.

Answer: $33$


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

Isosceles right triangle $PQR$ (with $\angle PQR = \angle PRQ = 45^\circ$ and hypotenuse $\overline{PQ}$) encloses a right triangle $ABC$ (hypotenuse $\overline{AB}$) as shown. Given $PC = 5$ and $BP = CQ = 4$, compute $AQ$.

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


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>.

Define
\[
d_k = k + \cfrac{1}{3k + \cfrac{1}{3k + \cfrac{1}{3k + \dotsb}}}.
\]
Calculate $\sum_{k = 1}^{10} (d_k^2 + 2)$.

Answer: $405$


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

Three cones are placed on a table on their bases, touching each other. The radii of their bases are 1, 12, and 12, and the apex angles of the cones are $-4 \operatorname{arctg} \frac{1}{3}$, $4 \operatorname{arctg} \frac{2}{3}$, and $4 \operatorname{arctg} \frac{2}{3}$ respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is placed on the table, touching all the cones. Find the radius of the sphere.

Answer: $40/21$


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>.

How many of the 256 smallest positive integers written in base 8 use the digit 6 or 7 (or both)?

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>.

If
\[x + \sqrt{x^2 - 4} + \frac{1}{x - \sqrt{x^2 - 4}} = 10,\]
then find
\[x^2 + \sqrt{x^4 - 4} + \frac{1}{x^2 + \sqrt{x^4 - 4}}.\]

Answer: $\frac{841}{100}$


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

Given the function $f(x)=\ln (ax+1)+ \frac {x^{3}}{3}-x^{2}-ax(a∈R)$,
(1) Find the range of values for the real number $a$ such that $y=f(x)$ is an increasing function on $[4,+∞)$;
(2) When $a\geqslant \frac {3 \sqrt {2}}{2}$, let $g(x)=\ln [x^{2}(ax+1)]+ \frac {x^{3}}{3}-3ax-f(x)(x > 0)$ and its two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$ are exactly the zeros of $φ(x)=\ln x-cx^{2}-bx$, find the minimum value of $y=(x_{1}-x_{2})φ′( \frac {x_{1}+x_{2}}{2})$.

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


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>.

Inside a square of side length 1, four quarter-circle arcs are traced with the edges of the square serving as the radii. These arcs intersect pairwise at four distinct points, forming the vertices of a smaller square. This process is repeated for the smaller square, and continuously for each subsequent smaller square. What is the sum of the areas of all squares formed in this manner?

Answer: $\frac{2}{1 - (2 - \sqrt{3})}$


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>.

Suppose a number $m$ is randomly selected from the set $\{7, 9, 12, 18, 21\}$, and a number $n$ is randomly selected from $\{2005, 2006, 2007, \ldots, 2025\}$. Calculate the probability that the last digit of $m^n$ is $6$.

Answer: $\frac{8}{105}$


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>.

Find the largest positive integer  $n>10$  such that the residue of  $n$  when divided by each perfect square between  $2$  and  $\dfrac n2$  is an odd number.

Answer: $505$


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

A youth radio station, to attract listeners' attention, gives away gifts and grand prizes among them. Gifts are given away hourly over sixteen hours (one gift per hour), and grand prizes are given away during four evening hours (one grand prize per hour). The probability that listeners win a prize is 0.3, and a grand prize is 0.02.

Find the probability that over 30 days:

a) Listeners will win three grand prizes;

b) Listeners will win between 130 to 160 prizes.

Answer: $0.862$


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

Given a triangle \( ABC \) with sides \( AB=13 \), \( BC=20 \), and \( AC=21 \). Point \( K \) is on side \( AB \), point \( L \) is on side \( AC \), and point \( N \) is on side \( BC \). It is known that \( AK=4 \), \( CN=1 \), and \( CL=\frac{20}{21} \). A line through point \( K \) parallel to \( NL \) intersects side \( AC \) at point \( M \). Find the area of the quadrilateral \( NLMK \).

Answer: $41$


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

If there are 150 seats in a row, calculate the fewest number of seats that must be occupied so the next person to be seated must sit next to someone.

Answer: $37$


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

Given that the total number of units produced by the workshops A, B, C, and D is 2800, and workshops A and C together contributed 60 units to the sample, determine the total number of units produced by workshops B and D.

Answer: $1600$


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

In triangle ABC, let the lengths of the sides opposite to angles A, B, and C be a, b, and c respectively, and b = 3, c = 1, A = 2B. Find the value of a.

Answer: $\sqrt{19}$


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

Find the minimum value of
\[3x^2 + 3xy + y^2 - 3x + 3y + 9\]
over all real numbers $x$ and $y.$

Answer: $\frac{45}{8}$


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>.

An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times.

Answer: $\frac{11}{2048}$


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

Given the digits $5,$ $6,$ $7,$ and $8,$ used exactly once to form four-digit integers, list these integers from least to greatest. For numbers starting with $7$ or $8,$ reverse the order of the last two digits. What is the $20^{\text{th}}$ integer in the list?

Answer: $7865$


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

Given $a=1$, $b=2$, $C=\frac{2π}{3}$ in triangle $\triangle ABC$, calculate the value of $c$.

Answer: $\sqrt{9}$


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

A pedestrian crossing signal at an intersection alternates between red and green lights, with the red light lasting for $30$ seconds. The probability that Little Ming, upon arriving at the intersection and encountering a red light, will have to wait at least $10$ seconds before the green light appears is _______.

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


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

If $x+y=10$ and $xy=12$, what is the value of $x^3-y^3$?

Answer: $176\sqrt{13}$


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

If we write $\sqrt{8} + \frac{1}{\sqrt{8}} + \sqrt{9} + \frac{1}{\sqrt{9}}$ in the form $\dfrac{a\sqrt{8} + b\sqrt{9}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$?

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>.

A 6x6x6 cube is formed by assembling 216 unit cubes. Ten unit squares are painted on each of the six faces of the cube, leaving some rows and columns unpainted. Specifically, two non-adjacent columns and two non-adjacent rows on each face are left unpainted. How many of the 216 unit cubes have no paint on them?

Answer: $168$


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

In an organization, there are five leaders and a number of regular members. Each year, the leaders are expelled, followed by each regular member recruiting three new members to become regular members. After this, five new leaders are elected from outside the organization. Initially, the organisation had twenty people total. How many total people will be in the organization six years from now?

Answer: $10895$


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

Using the vertices of a single rectangular solid (cuboid), how many different pyramids can be formed?

Answer: $106$


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

Given the vertices of a regular 100-sided polygon \( A_{1}, A_{2}, A_{3}, \ldots, A_{100} \), in how many ways can three vertices be selected such that they form an obtuse triangle?

Answer: $117600$


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

Compute the expressions
\[ C = 3 \times 4 + 5 \times 6 + 7 \times 8 + \cdots + 43 \times 44 + 45 \]
and
\[ D = 3 + 4 \times 5 + 6 \times 7 + \cdots + 42 \times 43 + 44 \times 45 \]
and find the positive difference between integers $C$ and $D$.

Answer: $882$


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

For finite sets  $A$  and  $B$ , call a function  $f: A \rightarrow B$  an \emph{antibijection} if there does not exist a set  $S \subseteq A \cap B$  such that  $S$  has at least two elements and, for all  $s \in S$ , there exists exactly one element  $s'$  of  $S$  such that  $f(s')=s$ . Let  $N$  be the number of antibijections from  $\{1,2,3, \ldots 2018 \}$  to  $\{1,2,3, \ldots 2019 \}$ . Suppose  $N$  is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that  $N=12=2\times 2\times 3$ , then the answer would be  $2+2+3=7$ .)

*Proposed by Ankit Bisain*

Answer: $1363641$


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

What is the area of the quadrilateral formed by the points of intersection of the circle \(x^2 + y^2 = 16\) and the ellipse \((x-3)^2 + 4y^2 = 36\).

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>.

Using a $4 \times 4$ grid where points are spaced equally at 1 unit apart both horizontally and vertically, how many rectangles are there whose four vertices are points on this grid?

Answer: $101$


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

Nina and Tadashi play the following game. Initially, a triple  $(a, b, c)$  of nonnegative integers with  $a+b+c=2021$  is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer  $k$  and one of the three entries on the board; then the player increases the chosen entry by  $k$  and decreases the other two entries by  $k$ . A player loses if, on their turn, some entry on the board becomes negative.

Find the number of initial triples  $(a, b, c)$  for which Tadashi has a winning strategy.

Answer: $6561$


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

Rectangular prism P Q R S W T U V has a square base P Q R S. Point X is on the face T U V W so that P X = 12, Q X = 10, and R X = 8. Determine the maximum possible area of rectangle P Q U T.

Answer: $67.82$


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

Given that the ratio of bananas to yogurt to honey is 3:2:1, and that Linda has 10 bananas, 9 cups of yogurt, and 4 tablespoons of honey, determine the maximum number of servings of smoothies Linda can make.

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>.

John is planning to fence a rectangular garden such that the area is at least 150 sq. ft. The length of the garden should be 20 ft longer than its width. Additionally, the total perimeter of the garden must not exceed 70 ft. What should the width, in feet, be?

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


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

Determine the volume of the right rectangular parallelepiped whose edges are formed by the distances from the orthocenter to the vertices of a triangle, where the radius of the circumcircle $r = 2.35$ and the angles are: $\alpha = 63^{\circ} 18^{\prime} 13^{\prime \prime}, \beta = 51^{\circ} 42^{\prime} 19^{\prime \prime}$.

Answer: $12.2$


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 $\frac{x}{2} = y^2$ and $\frac{x}{5} = 3y$, solve for $x$.

Answer: $112.5$


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

Let \( n \) be a two-digit number such that the square of the sum of the digits of \( n \) is equal to the sum of the digits of \( n^2 \). Find the sum of all possible values of \( n \).

Answer: $139$


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

For any positive integers \( m \) and \( n \), define \( r(m, n) \) as the remainder of \( m \div n \) (for example, \( r(8,3) \) represents the remainder of \( 8 \div 3 \), so \( r(8,3)=2 \)). What is the smallest positive integer solution satisfying the equation \( r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4 \)?

Answer: $120$


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

In triangle \(ABC\), angle \(B\) is \(120^\circ\), and \(AB = 2BC\). The perpendicular bisector of side \(AB\) intersects \(AC\) at point \(D\). Find the ratio \(AD:DC\).

Answer: $3/2$


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

The equation $y = -16t^2 + 34t + 25$ describes the height (in feet) of a ball thrown upwards at $34$ feet per second from $25$ feet above the ground. Determine the time (in seconds) when the ball will hit the ground.

Answer: $\frac{25}{8}$


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

In the isosceles triangle \( ABC \) (\( AB = BC \)), medians \( AD \) and \( EC \) intersect at point \( O \). The ratio of the radius of the circle inscribed in triangle \( AOC \) to the radius of the circle inscribed in quadrilateral \( ODBE \) is \(\frac{2}{3}\). Find the ratio \(\frac{AC}{BC}\).

Answer: $20/17$


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>.

An ellipse \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) has chords passing through the point \( C = (2, 2) \). If \( t \) is defined as
\[
t = \frac{1}{AC} + \frac{1}{BC}
\]
where \( AC \) and \( BC \) are distances from \( A \) and \( B \) (endpoints of the chords) to \( C \). Find the constant \( t \).

Answer: $\frac{4\sqrt{5}}{5}$


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

Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 18$ and $X$ is an integer, what is the smallest possible value of $X$?

Answer: $6172839500$


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

At time  $0$ , an ant is at  $(1,0)$  and a spider is at  $(-1,0)$ .  The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the  $x$ -axis.  It so happens that the ant's horizontal speed is always half the spider's.  What will the shortest distance ever between the ant and the spider be?

Answer: $\frac{\sqrt{14}}{4}$


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 an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) with left and right foci \( F_{1}, F_{2} \) respectively, let \( P \) be a point on the ellipse \( C \) in the first quadrant. The extended lines \( P F_{1}, P F_{2} \) intersect the ellipse \( C \) at points \( Q_{1}, Q_{2} \) respectively. Find the maximum value of the difference of areas of triangles \( \triangle P F_{1} Q_{2} \) and \( \triangle P F_{2} Q_{1} \).

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


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

A square is inscribed in the ellipse whose equation is $x^2 + 3y^2 = 3$. One vertex of the square is at $(0, 1)$, and one diagonal of the square lies along the y-axis. Determine the square of the length of each side of the square.

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


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

Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram.

Answer: $9\sqrt{41}$


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

1. There are 5 different books, and we need to choose 3 books to give to 3 students, one book per student. There are a total of     different ways to do this.
2. There are 5 different books, and we want to buy 3 books to give to 3 students, one book per student. There are a total of     different ways to do this.

Answer: $125$


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

Given that the cosine value of the vertex angle of an isosceles triangle equals $\dfrac{4}{5}$, calculate the sine value of the base angle of this triangle.

Answer: $\dfrac{2\sqrt{3}}{5}$


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 the inequality $x\ln x - kx > 3$, which holds true for any $x > 1$, determine the maximum value of the integer $k$.

Answer: $-3$


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 slopes of lines $l_1$ and $l_2$ are the two roots of the equation $6x^2+x-1=0$, respectively. The angle between lines $l_1$ and $l_2$ is __________.

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>.

Find the square root of $\dfrac{9!}{210}$.

Answer: $216\sqrt{3}$


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

Given the equations $3x + 2y = 6$ and $2x + 3y = 7$, find $14x^2 + 25xy + 14y^2$.

Answer: $85$


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

Initially, a prime number is displayed on the computer screen. Every second, the number on the screen is replaced with a number obtained by adding the last digit of the previous number, increased by 1. After how much time, at the latest, will a composite number appear on the screen?

Answer: $996$


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

In a regular quadrilateral frustum with lateral edges \(A A_{1}, B B_{1}, C C_{1}, D D_{1}\), the side length of the upper base \(A_{1} B_{1} C_{1} D_{1}\) is 1, and the side length of the lower base is 7. A plane passing through the edge \(B_{1} C_{1}\) perpendicular to the plane \(A D_{1} C\) divides the frustum into two equal-volume parts. Find the volume of the frustum.

Answer: $\frac{38\sqrt{5}}{5}$


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

In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.

Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round to whole degrees. What is the maximum possible deviation of the obtained value from the original temperature in Celsius degrees?

Answer: $13/18$


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>.

Given a circle and  $2006$  points lying on this circle. Albatross colors these  $2006$  points in  $17$  colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?

Answer: $117$


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

Given that $x$ is the geometric mean of $4$ and $16$, find the value of $x$.

Answer: $-8$


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

A circle inscribed in triangle \( ABC \) divides median \( BM \) into three equal parts. Find the ratio \( BC: CA: AB \).

Answer: $5:10:13$


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

The sum of all roots of the equation $2x^2-3x-5=0$ and the equation $x^2-6x+2=0$ is equal to $\boxed{\text{blank}}$, and the product of all roots is equal to $\boxed{\text{blank}}$.

Answer: $7\frac{1}{2}$


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>.

Let \( A = (-4, 0) \), \( B = (-1, 2) \), \( C = (1, 2) \), and \( D = (4, 0) \). Suppose that point \( P \) satisfies
\[ PA + PD = 10 \quad \text{and} \quad PB + PC = 10. \]
Find the \( y \)-coordinate of \( P \), when simplified, which can be expressed in the form \( \frac{-a + b \sqrt{c}}{d} \), where \( a, b, c, d \) are positive integers. Find \( a + b + c + d \).

Answer: $35$