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

Maya owns 16 pairs of shoes, consisting of 8 identical black pairs, 4 identical brown pairs, 3 identical grey pairs, and 1 pair of white shoes. If Maya randomly picks two shoes, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe?

Answer: $\frac{45}{248}$


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

A cube has a square pyramid placed on one of its faces. Determine the sum of the combined number of edges, corners, and faces of this new shape.

Answer: $34$


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

Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$.

Answer: $\frac{9\pi - 18}{2}$


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 $0 < \alpha < \pi$, $\tan\alpha = -2$.
(1) Find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$;
(2) Calculate the value of $$\frac{2\cos\left(\frac{\pi}{2} + \alpha\right) - \cos(\pi - \alpha)}{\sin\left(\frac{\pi}{2} - \alpha\right) - 3\sin(\pi + \alpha)};$$
(3) Simplify $2\sin^2\alpha - \sin\alpha\cos\alpha + \cos^2\alpha$.

Answer: $\frac{11}{5}$


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

Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial.

Answer: $8\sqrt{6}$


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

In diagram square $ABCD$, four triangles are removed resulting in rectangle $PQRS$. Two triangles at opposite corners ($SAP$ and $QCR$) are isosceles with each having area $120 \text{ m}^2$. The other two triangles ($SDR$ and $BPQ$) are right-angled at $D$ and $B$ respectively, each with area $80 \text{ m}^2$. What is the length of $PQ$, in meters?

Answer: $4\sqrt{15}$


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

Carefully observe the arrangement pattern of the following hollow circles ($○$) and solid circles ($●$): $○●○○●○○○●○○○○●○○○○○●○○○○○○●…$. If this pattern continues, a series of $○$ and $●$ will be obtained. The number of $●$ in the first $100$ of $○$ and $●$ is $\_\_\_\_\_\_\_$.

Answer: $12$


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

How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.)

Answer: $100$


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

We seek the true statement. There are one hundred statements written in a notebook:

1) There is exactly one false statement in this notebook.
2) There are exactly two false statements in this notebook.
...
100) There are exactly one hundred false statements in this notebook.

Which one of these statements is true, given that only one is true?

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>.

A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display?

Answer: $n = 9$


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

Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline.

Answer: $22.7$


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

Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$.

Answer: $15$


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

Given $f(x)= \sqrt {3}\sin x\cos (x+ \dfrac {π}{6})+\cos x\sin (x+ \dfrac {π}{3})+ \sqrt {3}\cos ^{2}x- \dfrac { \sqrt {3}}{2}$.
(I) Find the range of $f(x)$ when $x\in(0, \dfrac {π}{2})$;
(II) Given $\dfrac {π}{12} < α < \dfrac {π}{3}$, $f(α)= \dfrac {6}{5}$, $- \dfrac {π}{6} < β < \dfrac {π}{12}$, $f(β)= \dfrac {10}{13}$, find $\cos (2α-2β)$.

Answer: $-\dfrac{33}{65}$


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>.

A triangular lattice is formed with seven points arranged as follows: six points form a regular hexagon and one point is at the center. Each point is one unit away from its nearest neighbor. Determine how many equilateral triangles can be formed where all vertices are on this lattice. Assume the points are numbered 1 to 7, with 1 to 6 being the peripheral points and 7 being the center.

Answer: $12$


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

(1) Evaluate: $\sin^2 120^\circ + \cos 180^\circ + \tan 45^\circ - \cos^2 (-330^\circ) + \sin (-210^\circ)$;
(2) Determine the monotonic intervals of the function $f(x) = \left(\frac{1}{3}\right)^{\sin x}$.

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


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

Diagonals \( AC \) and \( BD \) of the cyclic quadrilateral \( ABCD \) are perpendicular and intersect at point \( M \). It is known that \( AM = 3 \), \( BM = 4 \), and \( CM = 6 \). Find \( CD \).

Answer: $10.5$


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>.

Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. How many points did Lily score?

Answer: $40$


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

Let $m \in \mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and a line passing through a fixed point $B$ is given by $mx-y-m+3=0$. These two lines intersect at point $P(x, y)$. Find the maximum value of $|PA|+|PB|$.

Answer: $2\sqrt{5}$


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

Solve the equation \[-2x^2 = \frac{4x + 2}{x + 2}.\]

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>.

How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square?

Answer: $343$


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

Let triangle  $ABC$  with incenter  $I$  and circumcircle  $\Gamma$  satisfy  $AB = 6\sqrt{3}, BC = 14,$  and  $CA = 22$ . Construct points  $P$  and  $Q$  on rays  $BA$  and  $CA$  such that  $BP = CQ = 14$ . Lines  $PI$  and  $QI$  meet the tangents from  $B$  and  $C$  to  $\Gamma$ , respectively, at points  $X$  and  $Y$ . If  $XY$  can be expressed as  $a\sqrt{b} - c$  for positive integers  $a,b,c$  with  $c$  squarefree, find  $a + b + c$ .

*Proposed by Andrew Wu*

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>.

Let $a$ and $b$ be positive integers such that $a$ has $4$ factors and $b$ has $2a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$?

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>.

There are two ways of choosing six different numbers from the list \( 1,2,3,4,5,6,7,8,9 \) so that the product of the six numbers is a perfect square. Suppose that these two perfect squares are \( m^{2} \) and \( n^{2} \), with \( m \) and \( n \) positive integers and \( m \neq n \). What is the value of \( m+n \)?

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>.

One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), it could be that she is in the cafe (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has a one in two chance of not finding her. If she is on one of the courts, there is another one in three chance of missing her, but if she went to the cafe, François will definitely find her: he knows which cafe Béatrice usually enjoys her ice cream. François visited all three possible meeting places but still did not find Béatrice.

What is the probability that Béatrice was on the beach, assuming she did not change locations while François was searching for her?

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


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

Suppose we wish to divide 12 dogs into three groups, one with 4 dogs, one with 6 dogs, and one with 2 dogs. We need to form these groups such that Rex is in the 4-dog group, Buddy is in the 6-dog group, and Bella and Duke are both in the 2-dog group. How many ways can we form these groups under these conditions?

Answer: $56$


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

Given a sequence of numbers arranged according to a certain rule: $-\frac{3}{2}$, $1$, $-\frac{7}{10}$, $\frac{9}{17}$, $\ldots$, based on the pattern of the first $4$ numbers, the $10$th number is ____.

Answer: $\frac{21}{101}$


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>.

Given a circle $C: (x-1)^2+(y-2)^2=25$, and a line $l: (2m+1)x+(m+1)y-7m-4=0$. If the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is \_\_\_\_\_\_.

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


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

Let  $ABCD$  be a rectangle with  $AB=10$  and  $BC=26$ . Let  $\omega_1$  be the circle with diameter  $\overline{AB}$  and  $\omega_2$  be the circle with diameter  $\overline{CD}$ . Suppose  $\ell$  is a common internal tangent to  $\omega_1$  and  $\omega_2$  and that  $\ell$  intersects  $AD$  and  $BC$  at  $E$  and  $F$  respectively. What is  $EF$ ?

[asy]
        size(10cm);
        draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
        draw((1,0)--(25,10));
        draw(circle((0,5),5));
        draw(circle((26,5),5));
        dot((1,0));
        dot((25,10));
        label(" $E$ ",(1,0),SE);
        label(" $F$ ",(25,10),NW);
        label(" $A$ ", (0,0), SW);
        label(" $B$ ", (0,10), NW);
        label(" $C$ ", (26,10), NE);
        label(" $D$ ", (26,0), SE);
        dot((0,0));
        dot((0,10));
        dot((26,0));
        dot((26,10));
[/asy]

*Proposed by Nathan Xiong*

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>.

Every day, Xiaoming goes to school along a flat road \(AB\), an uphill road \(BC\), and a downhill road \(CD\) (as shown in the diagram). Given that \(AB : BC : CD = 1 : 2 : 1\) and that Xiaoming's speeds on flat, uphill, and downhill roads are in the ratio 3 : 2 : 4, respectively, find the ratio of the time Xiaoming takes to go to school to the time he takes to come home.

Answer: $19:16$


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 subset  $M$  of  $\{1, 2, . . . , 2006\}$  has the property that for any three elements  $x, y, z$  of  $M$  with  $x < y < z$ ,  $x+ y$  does not divide  $z$ . Determine the largest possible size of  $M$ .

Answer: $1004$


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

In triangle $XYZ,$ $\angle X = 60^\circ,$ $\angle Y = 75^\circ,$ and $YZ = 6.$  Find $XZ.$

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


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

What is the maximum number of queens that can be placed on the black squares of an $8 \times 8$ chessboard so that each queen is attacked by at least one of the others?

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>.

Find the sum of $10_{10} + 23_{10}$, first by converting each number to base 3, performing the addition in base 3, and then converting the result back to base 10.

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>.

A certain bookstore currently has $7700$ yuan in funds, planning to use all of it to purchase a total of $20$ sets of three types of books, A, B, and C. Among them, type A books cost $500$ yuan per set, type B books cost $400$ yuan per set, and type C books cost $250$ yuan per set. The bookstore sets the selling prices of type A, B, and C books at $550$ yuan per set, $430$ yuan per set, and $310$ yuan per set, respectively. Let $x$ represent the number of type A books purchased by the bookstore and $y$ represent the number of type B books purchased. Answer the following questions:
$(1)$ Find the functional relationship between $y$ and $x$ (do not need to specify the range of the independent variable);
$(2)$ If the bookstore purchases at least one set each of type A and type B books, how many purchasing plans are possible?
$(3)$ Under the conditions of $(1)$ and $(2)$, based on market research, the bookstore decides to adjust the selling prices of the three types of books as follows: the selling price of type A books remains unchanged, the selling price of type B books is increased by $a$ yuan (where $a$ is a positive integer), and the selling price of type C books is decreased by $a$ yuan. After selling all three types of books, the profit obtained is $20$ yuan more than the profit from one of the plans in $(2)$. Write down directly which plan the bookstore followed and the value of $a$.

Answer: $10$


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

10 pairs of distinct shoes are mixed in a bag. If 4 shoes are randomly drawn, how many possible outcomes are there for the following results?
(1) None of the 4 shoes form a pair;
(2) Among the 4 shoes, there is one pair, and the other two shoes do not form a pair;
(3) The 4 shoes form exactly two pairs.

Answer: $45$


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

Given an ellipse $C$: $\frac{{x}^{2}}{3}+\frac{{y}^{2}}{{b}^{2}}=1\left(b \gt 0\right)$ with the right focus $F$, two lines passing through $F$ are tangent to the circle $O$: $x^{2}+y^{2}=r^{2}(r \gt 0)$ at points $A$ and $B$, forming a right triangle $\triangle ABF$. It is also known that the maximum distance between a point on ellipse $C$ and a point on circle $O$ is $\sqrt{3}+1$.
$(1)$ Find the equations of ellipse $C$ and circle $O$;
$(2)$ If a line $l: y=kx+m$ ($k \lt 0, m \gt 0$) that does not pass through point $F$ is tangent to circle $O$ and intersects ellipse $C$ at points $P$ and $Q$, find the perimeter of $\triangle FPQ$.

Answer: $2\sqrt{3}$


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

How many integers are there from 1 to 16500 that

a) are not divisible by 5;

b) are not divisible by either 5 or 3;

c) are not divisible by 5, 3, or 11?

Answer: $8000$


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

Given the ellipse $C$: $\begin{cases}x=2\cos θ \\\\ y=\sqrt{3}\sin θ\end{cases}$, find the value of $\frac{1}{m}+\frac{1}{n}$.

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


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

A circle of radius $15$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $BC$ extended through $C$ intersects the circle at point $E$. Find the number of degrees in angle $AEC$.

Answer: $90$


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

Given the vector $A$ with components $(1, -1, -3, -4)$,

Answer: $3\sqrt{3}$


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 function $f(x)=\sin (2x+ \frac {π}{3})$, for any $x\_1$, $x\_2$, $x\_3$, and $0\leqslant x\_1 < x\_2 < x\_3\leqslant π$, the equation $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_3)|\leqslant m$ holds true. Find the minimum value of the real number $m$.

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


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

Given an arithmetic sequence $\{a_{n}\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\{b_{n}\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\in{N}^{*}$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;
$(2)$ Let the sum of the first $n$ terms of the sequence $\{b_{n}\}$ be $B_{n}$, the set $P=\{n|n\leqslant 100$ and ${B}_{n}≤100,n∈{N}^{*}\}$. Find the sum $S$ of all elements in $P$.

Answer: $2520$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $543_7$, $65_7$, and $6_7$ in base $7$.

Answer: $650_7$


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>.

Two positive integers  $m$  and  $n$  are both less than  $500$  and  $\text{lcm}(m,n) = (m-n)^2$ . What is the maximum possible value of  $m+n$ ?

Answer: $840$


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

On the display of a certain instrument, each indicator light can show different signals by lighting up in red, yellow, or blue. It is known that there are 6 indicator lights in a row, and each time 3 of them light up, with only 2 being adjacent, then the total number of different signals that can be displayed is ______.

Answer: $324$


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

The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence plus the square of the first term. What is the value of the first term which exceeds 10000?

Answer: $16384$


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

Determine the total number of real solutions for the equation
\[
\frac{x}{50} = \sin x.
\]

Answer: $32$


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

$(MON 1)$  Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

Answer: $1944$


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

How many distinct lines pass through the point  $(0, 2016)$  and intersect the parabola  $y = x^2$  at two lattice points?  (A lattice point is a point whose coordinates are integers.)

Answer: $36$


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

Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$. The measures of arcs are $\stackrel \frown {AB} = 80^\circ$ and $\stackrel \frown {BC} = 100^\circ$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Find the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$.

Answer: $10$


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

Given a point P on the parabola $y^2=4x$, let the distance from point P to the directrix of this parabola be $d_1$, and the distance from point P to the line $x+2y-12=0$ be $d_2$.

Answer: $\frac{11 \sqrt{5}}{5}$


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  $a$  and  $b$  be positive integers such that  $2a - 9b + 18ab = 2018$ . Find  $b - a$ .

Answer: $223$


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

How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots?

Answer: $100$


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

From the 20 natural numbers 1, 2, 3, ..., 20, if three numbers are randomly selected and their sum is an even number greater than 10, then there are $\boxed{\text{answer}}$ such sets of numbers.

Answer: $563$


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

There is a rectangular field that measures  $20\text{m}$  by  $15\text{m}$ . Xiaoyu the butterfly is sitting at the perimeter of the field on one of the  $20\text{m}$  sides such that he is  $6\text{m}$  from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?

Answer: $17$


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

Given that three fair coins are tossed once and for each head, one fair die is rolled, with a second roll occurring if the first roll is 6, determine the probability that the sum of all die rolls is odd.

Answer: $\frac{7}{16}$


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>.

Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) such that \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \).

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>.

Binbin's height is 1.46 meters, his father is 0.32 meters taller than Binbin, and his mother's height is 1.5 meters.
(1) How tall is Binbin's father?
(2) How much shorter is Binbin's mother than his father?

Answer: $0.28$


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

Given that $\frac{{\cos 2\alpha}}{{\sin(\alpha+\frac{\pi}{4})}}=\frac{4}{7}$, find the value of $\sin 2\alpha$.

Answer: $\frac{41}{49}$


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

When $10^{95} - 95 - 2$ is expressed as a single whole number, calculate the sum of the digits.

Answer: $840$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 every unit square of a  $2018$  x  $2018$  board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.

Answer: $2 * (2018!)^2$


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 \( m \) and \( n \) be positive integers such that \( m > n \). If the last three digits of \( 2012^m \) and \( 2012^n \) are identical, find the smallest possible value of \( m+n \).

Answer: $104$


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

Given the function $f(x)=e^{ax}$, a line parallel to the $y$-axis is drawn through $A(a,0)$ and intersects the function $f(x)$ at point $P$. A tangent line to $f(x)$ at $P$ intersects the $x$-axis at point $B$. Find the minimum value of the area of $\triangle APB$.

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


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>.

Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$.

Answer: $26$


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

Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $(x^{2}- \dfrac {y^{2}}{24}=1)$, and $P$ is a point on the hyperbola, find the area of $\triangle F\_1PF\_2$.

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>.

An investor placed $\$15,000$ in a three-month term deposit that offered a simple annual interest rate of $8\%$. After three months, the total value of the deposit was reinvested in another three-month term deposit. After these three months, the investment amounted to $\$15,735$. Determine the annual interest rate, $s\%$, of the second term deposit.

Answer: $11.36$


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

Given that ${(1-2x)^{2016}}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+\cdots+a_{2015}(x-2)^{2015}+a_{2016}(x-2)^{2016}$ $(x\in\mathbb{R})$, find the value of $a_{1}-2a_{2}+3a_{3}-4a_{4}+\cdots+2015a_{2015}-2016a_{2016}$.

Answer: $4032$


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

Given points P and Q are on a circle of radius 7 and PQ = 8, find the length of the line segment PR, where R is the midpoint of the minor arc PQ.

Answer: $\sqrt{98 - 14\sqrt{33}}$


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>.

The inclination angle of the line $\sqrt{3}x + y - 1 = 0$ is ______.

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


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>.

There are 3 female and 2 male volunteers, a total of 5 volunteers, who need to be distributed among 3 communities to participate in volunteer services. Each community can have 1 to 2 people. Female volunteers A and B must be in the same community, and male volunteers must be in different communities. The number of different distribution methods is \_\_\_\_\_\_.

Answer: $12$


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

Alice and Bob each arrive at a meeting at a random time between 8:00 and 9:00 AM. If Alice arrives after Bob, what is the probability that Bob arrived before 8:45 AM?

Answer: $\frac{9}{16}$


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>.

Find the smallest integer starting with the digit 7 that becomes three times smaller when this digit is moved to the end. Find all such numbers.


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

Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that is reached according to the original planned time. What is the time of arrival at point \( B \)?

Answer: $11:40$


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>.

Let \( f(n) \) be the sum of the squares of the digits of positive integer \( n \) (in decimal). For example, \( f(123) = 1^{2} + 2^{2} + 3^{2} = 14 \). Define \( f_{1}(n) = f(n) \), and \( f_{k+1}(n) = f\left(f_{k}(n)\right) \) for \( k = 1, 2, 3, \ldots \). What is the value of \( f_{2005}(2006) \)?

Answer: $145$


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

For a positive number $x$, define $f(x)=\frac{x}{x+1}$. For example, $f(1)=\frac{1}{1+1}=\frac{1}{2}$, $f(2)=\frac{2}{2+1}=\frac{2}{3}$, $f(\frac{1}{2})=\frac{\frac{1}{2}}{\frac{1}{2}+1}=\frac{1}{3}$.
$(1)$ Find the value of: $f(3)+f(\frac{1}{3})=$______; $f(4)+f(\frac{1}{4})=$______.
$(2)$ Conjecture: $f(x)+f(\frac{1}{x})=$______.
$(3)$ Application: Based on the conclusion of $(2)$, calculate the value of the following expression: $f(2023)+f(2022)+f(2021)+\ldots +f(2)+f(1)+f(\frac{1}{2})+⋯+f(\frac{1}{2021})+f(\frac{1}{2022})+f(\frac{1}{2023})$.

Answer: $2022.5$


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

Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\frac{\pi}{6}$, and the distance from origin to this line is $\frac{\sqrt{3}}{2}$.

(1) Find the equation of the ellipse.
(2) Suppose a line with a positive slope passes through point D($-1, 0$) and intersects the ellipse at points E and F. If $\overrightarrow{ED} = 2\overrightarrow{DF}$, find the equation of line EF.
(3) Is there a real number $k$ such that the line $y = kx + 2$ intersects the ellipse at points P and Q, and the circle with diameter PQ passes through point D($-1, 0$)? If it exists, find the value of $k$; if not, explain why.

Answer: $k = \frac{7}{6}$


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>.

Given that the general term of the sequence $\{a_n\}$ is $a_n=2^{n-1}$, and the general term of the sequence $\{b_n\}$ is $b_n=3n$, let set $A=\{a_1,a_2,\ldots,a_n,\ldots\}$, $B=\{b_1,b_2,\ldots,b_n,\ldots\}$, $n\in\mathbb{N}^*$. The sequence $\{c_n\}$ is formed by arranging the elements of set $A\cup B$ in ascending order. Find the sum of the first 28 terms of the sequence $\{c_n\}$, denoted as $S_{28}$.

Answer: $820$


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

Given the curve $C$ represented by the equation $\sqrt {x^{2}+2 \sqrt {7}x+y^{2}+7}+ \sqrt {x^{2}-2 \sqrt {7}x+y^{2}+7}=8$, find the distance from the origin to the line determined by two distinct points on the curve $C$.

Answer: $\dfrac {12}{5}$


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 a hypothetical scenario, a small country is planning an international event in 2023. Let \( A \), \( B \), and \( C \) be distinct positive integers such that their product \( A \cdot B \cdot C = 2023 \). Determine the largest possible value of the sum \( A + B + C \).

Answer: $297$


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

Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction.

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


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

Let $f(x) = 2a^{x} - 2a^{-x}$ where $a > 0$ and $a \neq 1$.
 $(1)$ Discuss the monotonicity of the function $f(x)$;
 $(2)$ If $f(1) = 3$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \in [0,3]$, find the minimum value of $g(x)$.

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>.

Given the function $f(x)=\tan(\omega x + \phi)$ $(\omega \neq 0, \left|\phi\right| < \frac{\pi}{2})$, points $\left(\frac{2\pi}{3}, 0\right)$ and $\left(\frac{7\pi}{6}, 0\right)$ are two adjacent centers of symmetry for $f(x)$, and the function is monotonically decreasing in the interval $\left(\frac{2\pi}{3}, \frac{4\pi}{3}\right)$, find the value of $\phi$.

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


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

For a finite sequence $B = (b_1, b_2, \dots, b_m)$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + T_2 + \dots + T_m}{m},\]
where $T_k = b_1 + b_2 + \dots + b_k$ for $1 \leq k \leq m$.

If the Cesaro sum of the 50-term sequence $(b_1, b_2, \dots, b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$?

Answer: $492$


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

Find both the sum and the product of the coordinates of the midpoint of the segment with endpoints $(8, 15)$ and $(-2, -3)$.

Answer: $18$


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

Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator?

Answer: $11$


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

In the $xy$-plane, a triangle has vertices with coordinates $(x, y)$, where $x$ and $y$ are integers satisfying $1 \leqslant x \leqslant 4$ and $1 \leqslant y \leqslant 4$. How many such triangles are there?

(Source: 44th American High School Mathematics Exam, 1993)

Answer: $516$


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

Given that the heights of the first ten students formed a geometric sequence, with the fourth student 1.5 meters tall and the tenth student 1.62 meters tall, determine the height of the seventh student.

Answer: $\sqrt{2.43}$


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 area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\frac{4}{3}$.

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


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

You are given the digits $0$, $1$, $2$, $3$, $4$, $5$. Form a four-digit number with no repeating digits.

(I) How many different four-digit numbers can be formed?

(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit?

Answer: $100$


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

A rectangle has side lengths  $6$  and  $8$ . There are relatively prime positive integers  $m$  and  $n$  so that  $\tfrac{m}{n}$  is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find  $m + n$ .

Answer: $11$


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

The cars Niva and Toyota are driving on a ring track of a test site, a quarter of which is on a dirt road and the remaining part is on asphalt. The speed of the Niva on the dirt road is 80 km/h, and on asphalt it is 90 km/h. The speed of the Toyota on the dirt road is 40 km/h, and on asphalt it is 120 km/h. The cars start simultaneously at the beginning of the dirt road section and first drive through this dirt part. On which lap will one of the cars overtake the other for the first time?

Answer: $11$


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

Three dice are thrown, and the sums of the points that appear on them are counted. In how many ways can you get a total of 5 points and 6 points?

Answer: $10$


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

Given the circle with radius $6\sqrt{2}$, diameter $\overline{AB}$, and chord $\overline{CD}$ intersecting $\overline{AB}$ at point $E$, where $BE = 3\sqrt{2}$ and $\angle AEC = 60^{\circ}$, calculate $CE^2+DE^2$.

Answer: $216$


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

27 people went to a mall to buy water to drink. There was a promotion in the mall where three empty bottles could be exchanged for one bottle of water. The question is: For 27 people, the minimum number of bottles of water that need to be purchased so that each person can have one bottle of water to drink is $\boxed{18}$ bottles.

Answer: $18$


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

Given a geometric sequence $\{a_n\}$ where each term is positive, and $a_1a_3=4$, $a_7a_9=25$, find $a_5$.

Answer: $\sqrt{10}$


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>.

A group of one hundred friends, including Petya and Vasya, live in several cities. Petya found the distance from his city to the city of each of the other 99 friends and summed these 99 distances, obtaining a total of 1000 km. What is the maximum possible total distance that Vasya could obtain using the same method? Assume cities are points on a plane and if two friends live in the same city, the distance between their cities is considered to be zero.

Answer: $99000$


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

Given the function $f(x)=2\cos^2\frac{x}{2}+\sin x-1$. Find:
- $(Ⅰ)$ The minimum positive period, monotonic decreasing interval, and symmetry center of $f(x)$.
- $(Ⅱ)$ When $x\in \left[-\pi ,0\right]$, find the minimum value of $f(x)$ and the corresponding value of $x$.

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


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>.

Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2023,0),(2023,2024),$ and $(0,2024)$. What is the probability that $x > 9y$? Express your answer as a common fraction.

Answer: $\frac{2023}{36432}$


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

Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points?

Answer: $16$


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

Given an arithmetic sequence $\{a_n\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for which $S_n > 0$ holds is ______.

Answer: $2011$


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

Inside a circle \(\omega\) there is a circle \(\omega_{1}\) touching \(\omega\) at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that the points \(K, L,\) and \(M\) are collinear. Find the radius of circle \(\omega\), given that the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively.

Answer: $11$


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

Alice has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal rounded to the nearest thousandth.

Answer: $0.171$


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

After a track and field event, each athlete shook hands once with every athlete from every other team, but not with their own team members. Afterwards, two coaches arrived, each only shaking hands with each athlete from their respective teams. If there were a total of 300 handshakes at the event, what is the fewest number of handshakes the coaches could have participated in?

Answer: $20$


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

A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Determine the radius of the sphere if it can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)?

Answer: $12.5$


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>.

For a real number  $a$  and an integer  $n(\geq 2)$ , define  $$ S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} $$  Find every value of  $a$  s.t. sequence  $\{S_n(a)\}_{n\geq 2}$  converges to a positive real.

Answer: $2019$


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

A marine biologist interested in monitoring a specific fish species population in a coastal area. On January 15, he captures and tags 80 fish, then releases them back into the water. On June 15, he captures another sample of 100 fish, finding that 6 of them are tagged. He assumes that 20% of the tagged fish have died or migrated out of the area by June 15, and also that 50% of the fish in the June sample are recent additions due to birth or migration. How many fish were in the coastal area on January 15, based on his assumptions?

Answer: $533$


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

In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is

Answer: $60$


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

A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. Determine the total number of different flags that can be made.

Answer: $260$


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

Given that in $\triangle ABC$, $C = 2A$, $\cos A = \frac{3}{4}$, and $2 \overrightarrow{BA} \cdot \overrightarrow{CB} = -27$.

(I) Find the value of $\cos B$;

(II) Find the perimeter of $\triangle ABC$.

Answer: $15$


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

Let points $P$ and $Q$ be moving points on the curve $f(x)=x^{2}-\ln x$ and the line $x-y-2=0$, respectively. The minimum value of the distance between points $P$ and $Q$ is _______.

Answer: $\sqrt{2}$


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

When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?

Answer: $2395$


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

Eight congruent disks are arranged around a square with side length 2 in such a manner that each disk is tangent to two others, with four disks centered on the midpoints of the square’s sides and four at the corners. The disks completely cover the boundary of the square. Calculate the total area of the eight disks.

Answer: $4\pi$


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>.

In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). Determine the ratio $PA:AB$.

Answer: $2:1$


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

Given the sets $A=\{x|x=2n-1,n\in\mathbb{N}^*\}$ and $B=\{x|x=2^n,n\in\mathbb{N}^*\}$. Arrange all elements of $A\cup B$ in ascending order to form a sequence $\{a_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{a_n\}$. Find the smallest value of $n$ such that $S_n > 12a_{n+1}$.

Answer: $27$


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

In a particular right triangle, the lengths of the two legs are 30 inches and 24 inches. Calculate both the area of the triangle and the length of its hypotenuse.

Answer: $\sqrt{1476}$


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

Given the functions $f(x)=2\sin \left(\pi x+\frac{\pi }{3}\right)$ and $g(x)=2\cos \left(\pi x+\frac{\pi }{3}\right)$, find the area of the triangle formed by the intersection points of the two functions within the interval $\left[-\frac{4}{3},\frac{7}{6}\right]$.

Answer: $2\sqrt{2}$


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

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers  $1$  to  $16$ ?

Answer: $11$


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

Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent vertices, signified as vertex B. Calculate the probability that vertex B is one of the five bottom vertices of the dodecahedron. Express your answer as a common fraction.

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>.

For every non-empty subset of the natural number set $N^*$, we define the "alternating sum" as follows: arrange the elements of the subset in descending order, then start with the largest number and alternately add and subtract each number. For example, the alternating sum of the subset $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$. Then, the total sum of the alternating sums of all non-empty subsets of the set $\{1, 2, 3, 4, 5, 6, 7\}$ is

Answer: $448$


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

In a store where all items are priced in whole numbers of rubles, there are two special offers:
1) A customer who buys at least three items at once can select one item for free, provided that its price does not exceed the minimum of the paid items' prices.
2) A customer who buys exactly one item priced at no less than \(N\) rubles receives a 20% discount on their next purchase (which can include any number of items).

A customer, visiting this store for the first time, wants to buy exactly four items with a total price of 1000 rubles, where the cheapest item is priced at no less than 99 rubles. Determine the maximum \(N\) such that the second offer is more beneficial for the customer than the first one.

Answer: $504$


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

Out of a batch of 5000 electric lamps, a self-random non-repeating sample of 300 was selected. The average burning time of the lamps in the sample was found to be 1450 hours, and the variance was 40,000. What is the probability that the average burning time of the lamps in the entire batch falls within the range from 1410 to 1490 hours?

Answer: $0.99964$


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

For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \leq 5$. Find the $15^{\text{th}}$ entry in an ordered list of all such $n$.

Answer: $21$