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

In a right square prism \( P-ABCD \) with side edges and base edges both equal to 4, determine the total length of all the curves formed on its surface by points that are 3 units away from vertex \( P \).

Answer: $6\pi$


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>.

Of the natural numbers greater than 1000 that are composed of the digits $0, 1, 2$ (where each digit can be used any number of times or not at all), in ascending order, what is the position of 2010?

Answer: $30$


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

Find all irreducible positive fractions which increase threefold if both the numerator and the denominator are increased by 12.

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


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

Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour.

Answer: $14$


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

In the Cartesian coordinate system $xOy$, the coordinates of points $E'$ and $F'$ are $(0, \sqrt{3})$ and $(0, -\sqrt{3})$, respectively. A moving point $G$ satisfies: the product of the slopes of line $E'G$ and line $F'G$ is $-\frac{3}{4}$.
(1) Find the equation of the trajectory of the moving point $G$;
(2) Draw two perpendicular rays from point $O$ that intersect the trajectory from (1) at points $A$ and $B$, respectively. Find the minimum value of the area of $\triangle OAB$.

Answer: $\frac{12}{7}$


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

The perimeter of a triangle is 30, and all sides are different integers. There are a total of     triangles.

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>.

Let the function $y=f(k)$ be a monotonically increasing function defined on $N^*$, and $f(f(k))=3k$. Find the value of $f(1)+f(9)+f(10)$.

Answer: $39$


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

In how many distinct ways can I arrange six keys on a keychain, if I want to keep my house key adjacent to my car key and my office key adjacent to my mailbox key? Two arrangements are not considered different if the keys can be rotated or reflected into the same order.

Answer: $24$


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

Given the values $1256, 2561, 5612, 6125$, calculate the sum.

Answer: $15554$


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

Compute the definite integral:

$$
\int_{0}^{1} \frac{x^{4}}{\left(2-x^{2}\right)^{3 / 2}} \, dx
$$

Answer: $\frac{5}{2} - \frac{3\pi}{4}$


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>.

In tetrahedron \(ABCD\), it is known that \(\angle ADB = \angle BDC = \angle CDA = 60^\circ\), \(AD = BD = 3\), and \(CD = 2\). Find the radius of the circumscribed sphere of tetrahedron \(ABCD\).

Answer: $\sqrt{3}$


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

O is the center of square ABCD, and M and N are the midpoints of BC and AD, respectively. Points \( A', B', C', D' \) are chosen on \( \overline{AO}, \overline{BO}, \overline{CO}, \overline{DO} \) respectively, so that \( A' B' M C' D' N \) is an equiangular hexagon. The ratio \(\frac{[A' B' M C' D' N]}{[A B C D]}\) can be written as \(\frac{a+b\sqrt{c}}{d}\), where \( a, b, c, d \) are integers, \( d \) is positive, \( c \) is square-free, and \( \operatorname{gcd}(a, b, d)=1 \). Find \( 1000a + 100b + 10c + d \).

Answer: $8634$


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

Find the smallest positive integer \( n \) that satisfies the following condition: If the set \( A = \{1, 2, 3, \ldots, n\} \) is arbitrarily divided into 63 pairwise disjoint subsets \( A_1, A_2, A_3, \ldots, A_{63} \) (which are non-empty and their union is the set \( A \)), then there always exist two positive integers \( x \) and \( y \) that belong to the same subset \( A_i \) (\( 1 \leq i \leq 63 \)) such that \( x > y \) and \( 31x \leq 32y \).

Answer: $2016$


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

We measured the elevation angles of a tower standing on a horizontal plane from points $50 \mathrm{~m}$ and $100 \mathrm{~m}$ away from the base of the tower. The sum of the measured angles is $45^{\circ}$. How tall is the tower?

Answer: $28.08$


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

The image shows a grid consisting of 25 small equilateral triangles. How many rhombuses can be formed from two adjacent small triangles?

Answer: $30$


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

Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set {1,2,3,4,5,6}.
2. No digit occurs more than once in the same integer.
3. The digits in each integer are in increasing order.
4. Any two integers have at least one digit in common (possibly at different positions).
5. There is no digit which appears in all the integers.

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>.

Identical regular pentagons are arranged in a ring. Each of the regular pentagons has a perimeter of 65. The regular polygon formed as the inner boundary of the ring has a perimeter of \( P \). What is the value of \( P \)?

Answer: $130$


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

What is the least integer whose square is 12 more than three times the number?

Answer: $-3$


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

Given a right triangle \(ABC\) with a right angle at \(A\). On the leg \(AC\), a point \(D\) is marked such that \(AD:DC = 1:3\). Circles \(\Gamma_1\) and \(\Gamma_2\) are then drawn with centers at \(A\) and \(C\) respectively, both passing through point \(D\). \(\Gamma_2\) intersects the hypotenuse at point \(E\). Another circle \(\Gamma_3\) with center at \(B\) and radius \(BE\) intersects \(\Gamma_1\) inside the triangle at a point \(F\) such that \(\angle AFB\) is a right angle. Find \(BC\), given that \(AB = 5\).

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>.

Two circles touch each other internally at point K. Chord \( AB \) of the larger circle touches the smaller circle at point L, with \( A L = 10 \). Find \( B L \) given that \( A K : B K = 2 : 5 \).

Answer: $25$


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

In the $xOy$ coordinate plane, the curve
$$
y=(3 x-1)\left(\sqrt{9 x^{2}-6 x+5}+1\right)+(2 x-3)\left(\sqrt{4 x^{2}-12 x+13}+1\right)
$$

intersects the $x$-axis at the point $\qquad$ .

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


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 triangular prism \( P-ABC \), \( PA \perp \) plane \( ABC \), and \( AC \perp BC \). Given \( AC = 2 \), the dihedral angle \( P-BC-A \) is \( 60^\circ \), and the volume of the triangular prism \( P-ABC \) is \( \frac{4\sqrt{6}}{3} \). Find the sine value of the angle between line \( PB \) and plane \( PAC \).

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


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

Given that point P is on the curve \( y = -x^{2} - 1 \) and point Q is on the curve \( x = 1 + y^{2} \), find the minimum value of \( |PQ| \).

Answer: $\frac{3\sqrt{2}}{4}$


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

Mike and Harry play a game on an 8x8 board. Mike chooses \( k \) squares and writes an \( M \) in each of them. Harry then chooses \( k+1 \) squares and writes an \( H \) in each of them. After Harry is done, Mike wins if there is a sequence of letters forming " \( H M M \) " or " \( M M H \) ", when read either horizontally or vertically, and Harry wins otherwise. Determine the smallest value of \( k \) for which Mike has a winning strategy.

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>.

The surface area ratio of an equilateral cone to a sphere is 3:1. How many times larger is the volume of the equilateral cone compared to the volume of the sphere?

Answer: $2\sqrt{3}$


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

Let  $AB$  be a segment of unit length and let  $C, D$  be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set  $\{A,B,C,D\}.$

Answer: $\frac{\sqrt{5}}{125}$


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

Car A departs from point $A$ heading towards point $B$ and returns; Car B departs from point $B$ at the same time heading towards point $A$ and returns. After the first meeting, Car A continues for 4 hours to reach $B$, and Car B continues for 1 hour to reach $A$. If the distance between $A$ and $B$ is 100 kilometers, what is Car B's distance from $A$ when Car A first arrives at $B$?

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>.

The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid.

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>.

In the permutation \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of \(1, 2, 3, 4, 5\), how many permutations are there that satisfy \(a_{1} < a_{2}, a_{2} > a_{3}, a_{3} < a_{4}, a_{4} > a_{5}\)?

Answer: $16$


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

In the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots\). Find \(a_{1999}\).

Answer: $3331$


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

To the eight-digit number 20222023, append one digit to the left and one digit to the right so that the resulting ten-digit number is divisible by 72. Determine all possible solutions.

Answer: $3202220232$


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

Given the six whole numbers 10-15, compute the largest possible value for the sum, S, of the three numbers on each side of the triangle.

Answer: $39$


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

Divide the numbers 1 through 10 into three groups such that the difference between any two numbers within the same group does not appear in that group. If one of the groups is {2, 5, 9}, then the sum of all the numbers in the group that contains 10 is $\qquad$.

Answer: $22$


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

Let triangle  $ABC$  with incenter  $I$  satisfy  $AB = 10$ ,   $BC = 21$ , and   $CA = 17$ . Points  $D$  and E lie on side  $BC$  such that  $BD = 4$ ,  $DE = 6$ , and  $EC = 11$ . The circumcircles of triangles  $BIE$  and  $CID$  meet again at point  $P$ , and line  $IP$  meets the altitude from  $A$  to  $BC$  at  $X$ . Find  $(DX \cdot EX)^2$ .

Answer: $85$


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

An ant situated at point \( A \) decides to walk 1 foot east, then \( \frac{1}{2} \) foot northeast, then \( \frac{1}{4} \) foot east, then \( \frac{1}{8} \) foot northeast, then \( \frac{1}{16} \) foot east, and so on (that is, the ant travels alternately between east and northeast, and the distance travelled is decreased by half every time the ant changes its direction). The ant eventually reaches a certain point \( B \). Determine the distance between the ant's initial and final positions.

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


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

Car A and Car B start simultaneously from points $A$ and $B$ respectively, traveling towards each other. The initial speed ratio of car A to car B is 5:4. Shortly after departure, car A has a tire blowout, stops to replace the tire, and then resumes the journey, increasing its speed by $20\%$. They meet at the midpoint between $A$ and $B$ after 3 hours. After meeting, car B continues forward while car A turns back. When car A returns to point $A$ and car B reaches the position where car A had the tire blowout, how many minutes did car A spend replacing the tire?

Answer: $52$


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

Find all values of \(a\) such that the roots \(x_1, x_2, x_3\) of the polynomial \(x^3 - 6x^2 + ax + a\) satisfy \((x_1 - 3)^2 + (x_2 - 3)^3 + (x_3 - 3)^3 = 0\).

Answer: $-9$


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

In triangle \(ABC\), the angle bisector of \(\angle ABC\) intersects side \(AC\) at point \(K\). It is given that \(BC = 2\), \(KC = 1\), and \(BK = \frac{3\sqrt{2}}{2}\). Find the area of triangle \(ABC\).

Answer: $\frac{15 \sqrt{7}}{16}$


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>.

We write the following equation: \((x-1) \ldots (x-2020) = (x-1) \ldots (x-2020)\). What is the minimal number of factors that need to be erased so that there are no real solutions?

Answer: $1010$


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

A large rectangular garden contains two flower beds in the shape of congruent isosceles right triangles and a trapezoidal playground. The parallel sides of the trapezoid measure $30$ meters and $46$ meters. Determine the fraction of the garden occupied by the flower beds.
A) $\frac{1}{10}$
B) $\frac{1}{11}$
C) $\frac{4}{23}$
D) $\frac{1}{6}$

Answer: $\frac{4}{23}$


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

The clock runs 3 minutes fast every 24 hours. The clock was set accurately. After what minimum time will the clock show the correct time again?

Answer: $240$


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

At McDonald's restaurants, we can order Chicken McNuggets in packages of 6, 9, or 20 pieces. (For example, we can order 21 pieces because $21=6+6+9$, but there is no way to get 19 pieces.) What is the largest number of pieces that we cannot order?

Answer: $43$


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

In a $7 \times 7$ grid, choose $k$ cells such that the centers of any 4 chosen cells do not form the vertices of a rectangle. Find the maximum value of $k$ that satisfies this condition.

Answer: $21$


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

The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.

$$
AB = 18, EO = 7, AE = 2 BE, R = ?
$$

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 sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{1}=1\), and for each \(n \in \mathbf{N}^{*}\), \(a_{n}\) and \(a_{n+1}\) are the two roots of the equation \(x^{2}+3nx+b_{n}=0\). Find \(\sum_{k=1}^{20} b_{k}\).

Answer: $6385$


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

Three vertices of a rectangle are at points $(2, 7)$, $(13, 7)$, and $(13, -6)$. What is the area of the intersection between this rectangle and the circular region described by equation $(x - 2)^2 + (y + 6)^2 = 25$?

Answer: $\frac{25}{4}\pi$


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>.

If three different lines $x+y=1$, $x-y=1$, and $ax+y=1$ cannot form a triangle, then the value of the real number $a$ is.

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>.

For a $5 \times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this?

Answer: $1440$


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

Given \(0 < a < \sqrt{3} \cos \theta\), \(\theta \in \left[-\frac{\pi}{4}, \frac{\pi}{3}\right]\), find the minimum value of \(f(a, \theta) = \cos^3 \theta + \frac{4}{3a \cos^2 \theta - a^3}\).

Answer: $\frac{17 \sqrt{2}}{4}$


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

Calculate: $5 \times 13 \times 31 \times 73 \times 137$

Answer: $20152015$


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

Angles $C$ and $D$ are supplementary. If the measure of angle $C$ is $12$ times angle $D$, what is the measure of angle $C$?

Answer: $166.15$


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>.

Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of the original square larger than the area of the small square?

Answer: $81$


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

For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary.

Answer: $0.79$


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

In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Furthermore, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?

Answer: $176$


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

Let \(\left\{a_{n}\right\}\) be a sequence of positive integers such that \(a_{1}=1\), \(a_{2}=2009\) and for \(n \geq 1\), \(a_{n+2} a_{n} - a_{n+1}^{2} - a_{n+1} a_{n} = 0\). Determine the value of \(\frac{a_{993}}{100 a_{991}}\).

Answer: $89970$


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

In how many ways can 10 fillér and 50 fillér coins be placed side by side (with all centers on a straight line) to cover a $1 \mathrm{~m}$ long segment (not more), using at least 50 coins, and considering the order of the two types of coins? (Coins of the same value are not distinguished. The diameter of the 10 fillér is $19 \mathrm{~mm}$, and the diameter of the 50 fillér is $22 \mathrm{~mm}$.)

Answer: $270725$


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

A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road, which received the mail and turned back immediately without delay. The Moskvich arrived at the post office 20 minutes earlier than usual. How many minutes earlier than the scheduled time did the plane land?

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>.

In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $c=4$. Point $D$ is on $CD\bot AB$, and $c\cos C\cos \left(A-B\right)+4=c\sin ^{2}C+b\sin A\sin C$. Find the maximum value of the length of segment $CD$.

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>.

The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.

How many fruits in total grew on the miracle tree?

Answer: $30$


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

In the quadrilateral \(ABCD\), it is known that \(\angle ABD = \angle ACD = 45^\circ\), \(\angle BAC = 30^\circ\), and \(BC = 1\). Find \(AD\).

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>.

Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer.

Answer: $1159$


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

Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$.

Answer: $61$


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

Calculate the definite integral:

$$
\int_{0}^{\sqrt{2} / 2} \frac{x^{4} \cdot d x}{\sqrt{\left(1-x^{2}\right)^{3}}}
$$

Answer: $\frac{5}{4} - \frac{3\pi}{8}$


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

There are 3 screw-in light bulbs and 5 bayonet light bulbs in the box, and light bulbs are randomly drawn without replacement until the 5th light bulb is drawn to have all the screw-in light bulbs. Calculate the probability of this event.

Answer: $\frac{3}{28}$


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>.

When \( N \) takes all the values from 1, 2, 3, \ldots, 2015, how many numbers of the form \( 3^{n} + n^{3} \) are divisible by 7?

Answer: $288$


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

Given two integers $a$ and $b$, if they are not coprime and neither is a multiple of the other, they are called a "league" pair. Let $A$ be an $n$-element subset of the set $M = \{1, 2, \cdots, 2014\}$ such that every pair of numbers in $A$ is a league pair. Determine the maximum value of $n$.

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>.

Find the value of \(\cos ^{5} \frac{\pi}{9}+\cos ^{5} \frac{5 \pi}{9}+\cos ^{5} \frac{7 \pi}{9}\).

Answer: $\frac{15}{32}$


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>.

In an arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$, and $a_{11} > |a_{10}|$. The maximum negative value of the partial sum $S_n$ of the first $n$ terms of the sequence $\{a_n\}$ is the sum of the first ______ terms.

Answer: $19$


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

Given a sequence of length 15 consisting of zeros and ones, find the number of sequences that have either all zeros consecutive or all ones consecutive, but not both.

Answer: $238$


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

Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games, meaning into unordered pairs of unordered pairs of people?

Answer: $315$


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

Given a square \(ABCD\) with side length 2, \(E\) is the midpoint of \(AB\). The square is folded along lines \(EC\) and \(ED\) so that \(AE\) coincides with \(BE\), and point \(A\) coincides with point \(B\), named point \(O\). Calculate the volume of the tetrahedron \(O-CDE\).

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


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

Given the relationship $P={P}_{0}{e}^{-kt}$ between the concentration of toxic and harmful substances $P$ (unit: $mg/L$) in the exhaust gas and time $t$ (unit: $h$) during the filtration process, determine the percentage of the original toxic and harmful substances that will remain after $5$ hours, given that $20\%$ of the substances are eliminated after $2$ hours.

Answer: $57\%$


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

In an olympiad, 2006 students participated. It was found that a student, Vasia, solved only one out of the six problems. Additionally, the number of participants who solved at least 1 problem is 4 times greater than those who solved at least 2 problems;
the number who solved at least 2 problems is 4 times greater than those who solved at least 3 problems;
the number who solved at least 3 problems is 4 times greater than those who solved at least 4 problems;
the number who solved at least 4 problems is 4 times greater than those who solved at least 5 problems;
the number who solved at least 5 problems is 4 times greater than those who solved all 6.

How many students did not solve any problem?

Answer: $982$


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

A basketball championship has been played in a round-robin format, with each pair of teams playing twice and no ties (overtime is played until one team wins). The winner of a match receives 2 points, and the loser receives 1 point. At the end of the championship, the sum of the points obtained by all the teams except the champion is 2015 points. How many matches has the champion won?

Answer: $39$


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

Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM:MC = 1:2 \). The line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1.

Answer: $\frac{11}{24}$


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>.

There are 5 girls sitting in a row on five chairs, and opposite them, on five chairs, there are 5 boys sitting. It was decided that the boys would switch places with the girls. In how many ways can this be done?

Answer: $14400$


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

Given a line $l$ passing through point $A(1,1)$ with a slope of $-m$ ($m>0$) intersects the x-axis and y-axis at points $P$ and $Q$, respectively. Perpendicular lines are drawn from $P$ and $Q$ to the line $2x+y=0$, and the feet of the perpendiculars are $R$ and $S$. Find the minimum value of the area of quadrilateral $PRSQ$.

Answer: $3.6$


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

In rectangle $ABCD$, $AB=8$ and $BC=6$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF=3$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.

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>.

There are 55 points marked on a plane: the vertices of a regular 54-gon and its center. Petya wants to color a set of three marked points in red so that the colored points form the vertices of a regular triangle. In how many ways can Petya do this?

Answer: $72$


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

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$ respectively. Given that angle $A = \frac{\pi}{4}$, $\sin A + \sin (B - C) = 2\sqrt{2}\sin 2C$, and the area of triangle $ABC$ is $1$. Find the length of side $BC$.

Answer: $\sqrt{5}$


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

Find the smallest value of  $n$  for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds  $3^{2007}$ .

Answer: $2000$


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

The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration.

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 two groups of numerical sequences, each containing 15 arithmetic progressions with 10 terms each. The first terms of the progressions in the first group are $1, 2, 3, \ldots, 15$, and their differences are respectively $2, 4, 6, \ldots, 30$. The second group of progressions has the same first terms $1, 2, 3, \ldots, 15$, but the differences are respectively $1, 3, 5, \ldots, 29$. Find the ratio of the sum of all elements of the first group to the sum of all elements of the second group.

Answer: $160/151$


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

Points \( M \) and \( N \) are the midpoints of the sides \( AC \) and \( CB \) of the isosceles triangle \( ACB \). Point \( L \) lies on the median \( BM \) such that \( BL : BM = 4 : 9 \). A circle with center at point \( L \) is tangent to the line \( MN \) and intersects the line \( AB \) at points \( Q \) and \( T \). Find the perimeter of the triangle \( MNC \) if \( QT = 2 \) and \( AB = 8 \).

Answer: $2(2 + \sqrt{13})$


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

Triangle \( ABC \) has a right angle at \( B \). Point \( D \) lies on side \( BC \) such that \( 3 \angle BAD = \angle BAC \). Given \( AC = 2 \) and \( CD = 1 \), compute \( BD \).

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


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

In triangle ABC, point D is on line segment AB such that AD bisects $\angle CAB$. Given that $BD = 36$, $BC = 45$, and $AC = 27$, find the length of segment $AD$.

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>.

A square piece of paper has a side length of 1. It is folded such that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Given $C'D = \frac{1}{4}$, find the perimeter of triangle $\bigtriangleup AEC'$.
**A)** $\frac{25}{12}$
**B)** $\frac{33}{12}$
**C)** $\frac{10}{3}$
**D)** $\frac{8}{3}$
**E)** $\frac{9}{3}$

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


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

Let \(a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of all positive integers that are relatively prime to 75, where \(a_{1}<a_{2}<a_{3}<\cdots\). (The first five terms of the sequence are: \(a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8\).) Find the value of \(a_{2008}\).

Answer: $3764$


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

In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those who did not solve problem A, the number of people who solved problem B is twice the number of people who solved problem C. Additionally, the number of people who only solved problem A is equal to the combined number of people who only solved problem B and those who only solved problem C. What is the maximum number of people who solved problem A?

Answer: $23$


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

Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously?

Answer: $35$


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

Find all values of \( a \) for which the equation \( x^{2} + 2ax = 8a \) has two distinct integer roots. In your answer, record the product of all such values of \( a \), rounding to two decimal places if necessary.

Answer: $506.25$


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

Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \).

Answer: $1987$


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

Positive real numbers \( x, y, z \) satisfy: \( x^{4} + y^{4} + z^{4} = 1 \). Find the minimum value of the algebraic expression \( \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} \).

Answer: $\frac{9 \sqrt[4]{3}}{8}$


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>.

Given a set \( A = \{0, 1, 2, \cdots, 9\} \), and a family of non-empty subsets \( B_1, B_2, \cdots, B_j \) of \( A \), where for \( i \neq j \), \(\left|B_i \cap B_j\right| \leqslant 2\), determine the maximum value of \( k \).

Answer: $175$


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

Radovan read an interesting book. Yesterday, he read 15 pages, and today he read 12 pages. He realized with surprise that the sum of the page numbers he read yesterday is the same as the sum of the page numbers he read today. On which page will he start reading tomorrow? (Radovan does not skip any pages or read any page twice. He never ends his daily reading in the middle of a page.)

Answer: $74$


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

Given that $\frac{a}{36-a}+\frac{b}{48-b}+\frac{c}{72-c}=9$, evaluate $\frac{4}{36-a}+\frac{6}{48-b}+\frac{9}{72-c}$.

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


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

Let \( r_{1}, r_{2}, \cdots, r_{20} \) be the roots of the polynomial \( x^{20}-7x^{3}+1 \). If \(\frac{1}{r_{1}^{2}+1}+\frac{1}{r_{2}^{2}+1}+\cdots+\frac{1}{r_{20}^{2}+1} \) can be expressed in the form \( \frac{m}{n} \) (with \( m \) and \( n \) coprime), find the value of \( m+n \).

Answer: $240$


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

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

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


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

In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\alpha$ through $AM$ parallel to $BC$, and denote $\angle PAM$ as $\theta_1$. The acute dihedral angle between $\alpha$ and the base plane $ABC$ is denoted as $\theta_2$. When $\theta_1$ is maximized, find the value of $\tan \theta_2$.

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


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

From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board?

Answer: $23$


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

Let \( x, y, z, w \) be four consecutive vertices of a regular \( A \)-gon. If the length of the line segment \( xy \) is 2 and the area of the quadrilateral \( xyzw \) is \( a + \sqrt{b} \), find the value of \( B = 2^a \cdot 3^b \).

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>.

A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?

Answer: $4 - 2\sqrt{3}$


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

In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maintenance cost for one orphan is 200 florins per year?

Answer: $1896$


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

Every day from Monday to Friday, an old man went to the blue sea and cast his net. Each day, the number of fish caught in the net was not greater than the number caught the previous day. Over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three days - Monday, Wednesday, and Friday?

Answer: $50$


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

Given an equilateral triangle of side 10, divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part. Repeat this step for each side of the resulting polygon. Find \( S^2 \), where \( S \) is the area of the region obtained by repeating this procedure infinitely many times.

Answer: $4800$


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

Let non-negative real numbers \(a_1, a_2, \ldots, a_{100}\) satisfy: \( a_i + a_{i+1} + a_{i+2} \leq 1 \) for \( 1 \leq i \leq 100 \), where \(a_{101} = a_1\) and \(a_{102} = a_2\). Find the maximum value of \(\sum_{i=1}^{100} a_i a_{i+2}\).

Answer: $25/2$


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>.

Two people, A and B, start from the same point on a 300-meter circular track and run in opposite directions. A runs at 2 meters per second, and B runs at 4 meters per second. When they first meet, A turns around and runs back. When A and B meet again, B turns around and runs back. Following this pattern, after how many seconds will the two people meet at the starting point for the first time?

Answer: $250$


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

In triangle \(ABC\), sides \(AB\) and \(BC\) are equal, \(AC = 2\), and \(\angle ACB = 30^\circ\). From vertex \(A\), the angle bisector \(AE\) and the median \(AD\) are drawn to the side \(BC\). Find the area of triangle \(ADE\).

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


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

If point \( P \) is on the curve \( y=\frac{1}{2} e^{x} \) and point \( Q \) is on the curve \( y=\ln (2 x) \), then the minimum value of \( |PQ| \) is \( \qquad \).

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


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

A solid cube of side length $4$ has $12$ edges, and a solid cube of side length $2$ has $12$ edges. The remaining solid is created by removing $8$ solid cubes of side length $2$ from the corners of the solid cube of side length $4$. Calculate the total number of edges of the remaining solid.

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>.

I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of the ticket, given that my neighbor easily solved this problem.

Answer: $99999$


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

Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length  $1$ , what is the total area of all the individual circles?

*2015 CCA Math Bonanza Lightning Round #4.4*

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


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

Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BPQ\) given \(BH:CH = 4:9\).

Answer: $24$


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

Given that \( n \) is a positive integer, \( P \) is a prime number, and \( pn \) has exactly 8 positive divisors, arrange them in ascending order as \( 1=d_{1}<d_{2}< \cdots <d_{8}=pn \). Additionally, let \( d_{17p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13p\right) \). Find \( n \).

Answer: $2021$


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

Out of the 200 natural numbers between 1 and 200, how many numbers must be selected to ensure that there are at least 2 numbers among them whose product equals 238?

Answer: $198$


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

A circle has 2017 distinct points $A_{1}, \ldots, A_{2017}$ marked on it, and all possible chords connecting pairs of these points are drawn. A line is drawn through the point $A_{1}$, which does not pass through any of the points $A_{2}, \ldots A_{2017}$. Find the maximum possible number of chords that can intersect this line in at least one point.

Answer: $1018080$


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

Let $f(x) = \frac{x^6 - 2}{4}$. Find $f^{-1}\left(-\frac{1}{8}\right)$.

Answer: $\left(\frac{3}{2}\right)^{\frac{1}{6}}$


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

In the quadrilateral \(ABCD\), it is given that \(\cos \angle BAD = \frac{3}{4}\), \(\angle BAC = \angle DAC\), \(AD < AB\), and \(AB = 5\), \(AC = BD = \sqrt{14}\). If \(\overrightarrow{AC} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AD}\) (\(\lambda, \mu \in \mathbf{R}\)), find \(\lambda + \mu\).

Answer: $\frac{7}{5}$


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 \( m \) and \( n \) (with \( m > n \)) be positive integers such that \( 70^2 \) divides \( 2023^m - 2023^n \). What is the smallest value of \( m+n \)?

Answer: $24$


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

Form natural numbers without repeating digits using the digits 0, 1, and 2, and calculate the total number of such natural numbers.

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>.

Vovochka adds three-digit numbers in a column in the following way: he does not carry over tens, but writes the sum of pairs of digits in the same place value position under each pair of digits, even if the result is a two-digit number. For example, for the sum \(248 + 208\), he would get the result 4416. Find the smallest possible positive difference between Vovochka's answer and the correct answer.

Answer: $1800$


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

The center of a circle with a radius of 5, circumscribed around an isosceles trapezoid, lies on the longer base, and the shorter base is equal to 6. Find the area of the trapezoid.

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>.

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

Answer: $13$