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

In a right triangle PQR with right angle at P, suppose $\sin Q = 0.6$. If the length of QP is 15, what is the length of QR?

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>.

Given that Liliane has $30\%$ more cookies than Jasmine and Oliver has $10\%$ less cookies than Jasmine, and the total number of cookies in the group is $120$, calculate the percentage by which Liliane has more cookies than Oliver.

Answer: $44.44\%$


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

Rodney is now guessing a secret number based on these clues:
- It is a two-digit integer.
- The tens digit is even.
- The units digit is odd.
- The number is greater than 50.

Answer: $\frac{1}{10}$


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

Find the area of quadrilateral \(ABCD\) if \(AB = BC = 3\sqrt{3}\), \(AD = DC = \sqrt{13}\), and vertex \(D\) lies on a circle of radius 2 inscribed in the angle \(ABC\), where \(\angle ABC = 60^\circ\).

Answer: $3\sqrt{3}$


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

Find the sum of the distinct prime factors of $7^7 - 7^4$.

Answer: $31$


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

Four friends initially plan a road trip and decide to split the fuel cost equally. However, 3 more friends decide to join at the last minute. Due to the increase in the number of people sharing the cost, the amount each of the original four has to pay decreases by $\$$8. What was the total cost of the fuel?

Answer: $74.67$


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

Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$

Answer: $2520$


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

Sara conducted a survey among a group to find out how many people were aware that bats can transmit diseases. She found out that $75.3\%$ believed bats could transmit diseases. Among those who believed this, $60.2\%$ incorrectly thought that all bats transmit Zika virus, which amounted to 37 people. Determine how many people were surveyed in total by Sara.

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>.

In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given that $b=3a$ and $c=2$, find the area of $\triangle ABC$ when angle $A$ is at its maximum value.

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>.

An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).

Answer: $10$


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

Alice's favorite number has the following properties:

- It has 8 distinct digits.
- The digits are decreasing when read from left to right.
- It is divisible by 180.

What is Alice's favorite number?

*Author: Anderson Wang*

Answer: $97654320$


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

Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\]

Answer: $15.5$


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  $ABC$  be a scalene triangle whose side lengths are positive integers. It is called *stable* if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

*Proposed by Evan Chen*

Answer: $20$


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

What is the sum of the number of faces, edges, and vertices of a square pyramid that has a square base? Additionally, find the number of diagonals in the square base.

Answer: $18$


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

Given \\(a < 0\\), \\((3x^{2}+a)(2x+b) \geqslant 0\\) holds true over the interval \\((a,b)\\), then the maximum value of \\(b-a\\) is \_\_\_\_\_\_.

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


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

Vasya has:
a) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;
b) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;
c) a book of lyrical poems with a height of 40 cm, published by Vasya himself.

Vasya wants to arrange these books on a shelf so that his own work is in the center, and the books located at the same distance from it on both the left and the right have equal heights. In how many ways can this be done?
a) $3 \cdot 2! \cdot 4!$;

b) $2! \cdot 3!$;

c) $\frac{51}{3! \cdot 2!}$;

d) none of the above answers are correct.

Answer: $144$


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

The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \ldots\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square).  What is the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when the 1994th term of the sequence is divided by 1000?

Answer: $063$


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

Two circles are centered at \( (5,5) \) and \( (25,15) \) respectively, each tangent to the \( y \)-axis. Find the distance between the closest points of these two circles.

Answer: $10\sqrt{5} - 20$


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

In square  $ABCD$ ,  $\overline{AC}$  and  $\overline{BD}$  meet at point  $E$ .
Point  $F$  is on  $\overline{CD}$  and  $\angle CAF = \angle FAD$ .
If  $\overline{AF}$  meets  $\overline{ED}$  at point  $G$ , and if  $\overline{EG} = 24$  cm, then find the length of  $\overline{CF}$ .

Answer: $48$


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

After the appearance of purple sand flowerpots in the Ming and Qing dynasties, their development momentum was soaring, gradually becoming the collection target of collectors. With the development of pot-making technology, purple sand flowerpots have been integrated into the daily life of ordinary people. A certain purple sand product factory is ready to mass-produce a batch of purple sand flowerpots. The initial cost of purchasing equipment for the factory is $10,000. In addition, $27 is needed to produce one purple sand flowerpot. When producing and selling a thousand purple sand flowerpots, the total sales of the factory is given by $P(x)=\left\{\begin{array}{l}5.7x+19,0＜x⩽10,\\ 108-\frac{1000}{3x},x＞10.\end{array}\right.($unit: ten thousand dollars).
$(1)$ Find the function relationship of total profit $r\left(x\right)$ (unit: ten thousand dollars) with respect to the output $x$ (unit: thousand pieces). (Total profit $=$ total sales $-$ cost)
$(2)$ At what output $x$ is the total profit maximized? And find the maximum value of the total profit.

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>.

Alice and Bob live on the same road. At time  $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at  $3:19\text{pm}$ , and Bob arrived at Alice's house at  $3:29\text{pm}$ . Charlie, who was driving by, noted that Alice and Bob passed each other at  $3:11\text{pm}$ . Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day.

*Proposed by Kevin You*

Answer: $179$


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

On the surface of a sphere with a radius of $2$, there is a triangular prism with an equilateral triangle base and lateral edges perpendicular to the base. All vertices of the prism are on the sphere's surface. Determine the maximum lateral area of this triangular prism.

Answer: $12\sqrt{3}$


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

In the quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(K\). Points \(L\) and \(M\) are the midpoints of sides \(BC\) and \(AD\), respectively. Segment \(LM\) contains point \(K\). The quadrilateral \(ABCD\) is such that a circle can be inscribed in it. Find the radius of this circle, given that \(AB = 3\), \(AC = \sqrt{13}\), and \(LK: KM = 1: 3\).

Answer: $3/2$


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

Given that \(2 \cdot 50N\) is an integer and its representation in base \(b\) is 777, find the smallest positive integer \(b\) such that \(N\) is a fourth power of an integer.

Answer: $18$


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

To stabilize housing prices, a local government decided to build a batch of affordable housing for the community. The plan is to purchase a piece of land for 16 million yuan and build a residential community with 10 buildings on it. Each building has the same number of floors, and each floor has a construction area of $1,000 m^{2}$. The construction cost per square meter is related to the floor level, with the cost for the $x^{th}$ floor being $(kx+800)$ yuan (where $k$ is a constant). It has been calculated that if each building has 5 floors, the average comprehensive cost per square meter for the community is 1,270 yuan.

Note: The average comprehensive cost per square meter $= \dfrac{\text{land purchase cost} + \text{all construction costs}}{\text{total construction area}}$.

$(1)$ Find the value of $k$;

$(2)$ To minimize the average comprehensive cost per square meter for the community, how many floors should the 10 buildings have? What is the average comprehensive cost per square meter at this time?

Answer: $1225$


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

A zoo has six pairs of different animals (one male and one female for each species). The zookeeper wishes to begin feeding the male lion and must alternate between genders each time. Additionally, after feeding any lion, the next animal cannot be a tiger. How many ways can he feed all the animals?

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 tetrahedron $P-ABC$ with its circumscribed sphere's center $O$ on $AB$, and $PO \perp$ plane $ABC$, $2AC = \sqrt{3}AB$. If the volume of the tetrahedron $P-ABC$ is $\frac{3}{2}$, find the volume of the sphere.

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


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>.

Trapezoid $EFGH$ has base $EF = 15$ units and base $GH = 25$ units. Diagonals $EG$ and $FH$ intersect at $Y$. If the area of trapezoid $EFGH$ is $200$ square units, what is the area of triangle $FYG$?

Answer: $46.875$


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

Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the maximum whole number of minutes Petya's sleep could have lasted?

Answer: $361$


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

A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?

Answer: $338350$


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

I have a bag with $8$ marbles numbered from $1$ to $8.$ Mathew has a bag with $16$ marbles numbered from $1$ to $16.$ Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such that the sum of the numbers on my marbles is a prime number and equals the number on his marble?

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>.

Given that there are $m$ distinct positive even numbers and $n$ distinct positive odd numbers such that their sum is 2015. Find the maximum value of $20m + 15n$.

Answer: $1105$


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

Let $g(x) = dx^3 + ex^2 + fx + g$, where $d$, $e$, $f$, and $g$ are integers. Suppose that $g(1) = 0$, $70 < g(5) < 80$, $120 < g(6) < 130$, $10000m < g(50) < 10000(m+1)$ for some integer $m$. What is $m$?

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>.

Find the minimum value of the discriminant of a quadratic trinomial whose graph does not intersect the regions below the x-axis and above the graph of the function \( y = \frac{1}{\sqrt{1-x^2}} \).

Answer: $-4$


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

Given that in triangle $ABC$, $\overrightarrow{A B} \cdot \overrightarrow{B C} = 3 \overrightarrow{C A} \cdot \overrightarrow{A B}$, find the maximum value of $\frac{|\overrightarrow{A C}| + |\overrightarrow{A B}|}{|\overrightarrow{B C}|}$:

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>.

Find all \( t \) such that \( x-t \) is a factor of \( 10x^2 + 21x - 10 \).

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


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>.

How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit?

Answer: $296$


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

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $4\sin A\sin B-4\cos ^{2} \frac {A-B}{2}= \sqrt {2}-2$.

$(1)$ Find the magnitude of angle $C$;
$(2)$ Given $\frac {a\sin B}{\sin A}=4$ and the area of $\triangle ABC$ is $8$, find the value of side length $c$.

Answer: $c=4$


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 the product $(\frac{4}{8})(\frac{8}{12})(\frac{12}{16})\cdots(\frac{2016}{2020})$. Express your answer as a common fraction.

Answer: $\frac{2}{505}$


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

Given the function $f(x) = x^2 + (a+8)x + a^2 + a - 12$ ($a < 0$), and $f(a^2 - 4) = f(2a - 8)$, determine the minimum value of $\frac {f(n)-4a}{n+1}$ for $n \in \mathbb{N}^{+}$.

Answer: $\frac {37}{4}$


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

The smallest possible value of $m$ for which Casper can buy exactly $10$ pieces of strawberry candy, $18$ pieces of lemon candy, and $20$ pieces of cherry candy, given that each piece of orange candy costs $15$ cents.

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>.

The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \( P(-2,0) \) to its asymptote is \( \frac{2 \sqrt{6}}{3} \). If a line with slope \( \frac{\sqrt{2}}{2} \) passes through point \( P \) and intersects the hyperbola at points \( A \) and \( B \), intersects the y-axis at point \( M \), and \( P M \) is the geometric mean of \( P A \) and \( P B \), then the half focal length of the hyperbola is what?

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>.

Given points A, B, and C on the surface of a sphere O, the height of the tetrahedron O-ABC is 2√2 and ∠ABC=60°, with AB=2 and BC=4. Find the surface area of the sphere O.

Answer: $48\pi$


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

Let $a$, $b$, and $c$ be positive real numbers. Find the minimum value of
\[
\frac{(a^2 + 4a + 4)(b^2 + 4b + 4)(c^2 + 4c + 4)}{abc}.
\]

Answer: $64$


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

On a mathematics quiz, there were $6x$ problems. Lucky Lacy missed $2x$ of them. What percent of the problems did she get correct?

Answer: $66.67\%$


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

A bagel is cut into sectors. A total of 10 cuts were made. How many pieces were formed?

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>.

A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads?

Answer: $\frac{5}{8}$


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

A cube with edge length 2 cm has a dot marked at the center of the top face. The cube is on a flat table and rolls without slipping, making a full rotation back to its initial orientation, with the dot back on top. Calculate the length of the path followed by the dot in terms of $\pi. $

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


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

Let  $ABCD$  be a quadrilateral with  $\overline{AB}\parallel\overline{CD}$ ,  $AB=16$ ,  $CD=12$ , and  $BC<AD$ . A circle with diameter  $12$  is inside of  $ABCD$  and tangent to all four sides. Find  $BC$ .

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>.

Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face.

Answer: $10$


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

What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \( d \) contained in this segment, there are no more than \( 1 + 1000 d^2 \) points?

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>.

A school club designed an unconventional rectangular checkerboard that measures 10 by 6. Calculate the probability that a randomly chosen unit square does not touch any edge of the board or is not adjacent to any edge.

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


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

How many natural numbers greater than 10 but less than 100 are relatively prime to 21?

Answer: $51$


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

Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0), (2010,0), (2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$?

Answer: $\frac{335}{2011}$


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

Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements.

Answer: $216$


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

Evaluate the expression $\dfrac{\sqrt[6]{5}}{\sqrt[4]{5}}$. What power of 5 does this expression represent?

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


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

A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both these circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.

Answer: $18$


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

Given the numbers 1, 2, 3, 4, 5, there are $5!$ permutations $a_1, a_2, a_3, a_4, a_5$. Find the number of distinct permutations where $a_k \geq k - 2$ for all $k = 1, 2, 3, 4, 5$.

Answer: $54$


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

In triangle $XYZ,$ angle bisectors $\overline{XU}$ and $\overline{YV}$ intersect at $Q.$  If $XY = 8,$ $XZ = 6,$ and $YZ = 4,$ find $\frac{YQ}{QV}.$

Answer: $1.5$


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

Determine the value of:
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).\]

Answer: $9006.5$


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>.

There is a hemispherical raw material. If this material is processed into a cube through cutting, the maximum value of the ratio of the volume of the obtained workpiece to the volume of the raw material is ______.

Answer: $\frac { \sqrt {6}}{3\pi }$


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>.

For how many four-digit whole numbers does the sum of the digits equal $30$?

Answer: $20$


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

A and B plays a game on a pyramid whose base is a  $2016$ -gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the  $k$  colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of  $k$  for which  $B$  can guarantee that all sides are colored.

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>.

Let $\phi$ be the smallest acute angle for which $\cos \phi,$ $\cos 2\phi,$ $\cos 4\phi$ form an arithmetic progression, in some order. Find $\sin \phi.$

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


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

The numbers $1, 2, \dots, 16$ are randomly placed into the squares of a $4 \times 4$ grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is even.

Answer: $\frac{36}{20922789888000}$


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

If 1983 were expressed as a sum of distinct powers of 2, with the requirement that at least five distinct powers of 2 are used, what would be the least possible sum of the exponents of these powers?

Answer: $55$


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

Let $\{b_k\}$ be a sequence of integers where $b_1 = 2$ and $b_{m+n} = b_m + b_n + m^2 + n^2$ for all positive integers $m$ and $n$. Find $b_{12}$.

Answer: $160$


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

Define a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are spacy?

Answer: $60$


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

Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \le i \le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists.

Answer: $64$


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

Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing at his original speed. By the time the document is completed, A and B have typed an equal number of characters. What is the total number of characters in the document?

Answer: $18000$


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

Given that $\sin \alpha = 2 \cos \alpha$, find the value of $\cos ( \frac {2015\pi}{2}-2\alpha)$.

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>.

Circles  $C_1$  and  $C_2$  intersect at points  $X$  and  $Y$  . Point  $A$  is a point on  $C_1$  such that the tangent line with respect to  $C_1$  passing through  $A$  intersects  $C_2$  at  $B$  and  $C$ , with  $A$  closer to  $B$  than  $C$ , such that  $2016 \cdot AB = BC$ . Line  $XY$  intersects line  $AC$  at  $D$ . If circles  $C_1$  and  $C_2$  have radii of  $20$  and  $16$ , respectively, find  $\sqrt{1+BC/BD}$ .

Answer: $2017$


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

Let  $\triangle ABC$  have  $\angle ABC=67^{\circ}$ . Point  $X$  is chosen such that  $AB = XC$ ,  $\angle{XAC}=32^\circ$ , and  $\angle{XCA}=35^\circ$ . Compute  $\angle{BAC}$  in degrees.

*Proposed by Raina Yang*

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>.

The greatest common divisor of 30 and some number between 70 and 80 is 10. What is the number, if the least common multiple of these two numbers is also between 200 and 300?

Answer: $80$


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

Suppose $b$ and $c$ are constants such that the quadratic equation $2ax^2 + 15x + c = 0$ has exactly one solution. If the value of $c$ is 9, find the value of $a$ and determine the unique solution for $x$.

Answer: $-\frac{12}{5}$


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

 $a,b,c$  - are sides of triangle  $T$ . It is known, that if we increase any one  side by  $1$ , we get new
a) triangle
b)acute triangle

Find minimal possible area of triangle  $T$  in case of a) and in case b)

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


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

The angle of inclination of the line $x-y-1=0$ is ____.

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


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

Let an ordered pair of positive integers  $(m, n)$  be called *regimented* if for all nonnegative integers  $k$ , the numbers  $m^k$  and  $n^k$  have the same number of positive integer divisors. Let  $N$  be the smallest positive integer such that  $\left(2016^{2016}, N\right)$  is regimented. Compute the largest positive integer  $v$  such that  $2^v$  divides the difference  $2016^{2016}-N$ .

*Proposed by Ashwin Sah*

Answer: $10086$


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

If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth.

Answer: $0.16$


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 number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:

1. The Falcons won more games than the Hawks.
2. The Warriors won more games than the Knights but fewer than the Royals.
3. The Knights won more than 25 games.

How many games did the Warriors win? The wins recorded are 20, 26, 30, 35, 40, and 45.

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>.

Let \( p, q, r \) be the roots of the polynomial \( x^3 - 8x^2 + 14x - 2 = 0 \). Define \( t = \sqrt{p} + \sqrt{q} + \sqrt{r} \). Find \( t^4 - 16t^2 - 12t \).

Answer: $-8$


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

Find a three-digit number equal to the sum of the tens digit, the square of the hundreds digit, and the cube of the units digit.

Find the number \(\overline{abcd}\) that is a perfect square, if \(\overline{ab}\) and \(\overline{cd}\) are consecutive numbers, with \(\overline{ab} > \(\overline{cd}\).

Answer: $357$


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

Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], determine the range of the function g(x) on the interval [-2020,2020].

Answer: $[-4043,4041]$


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

The minimum positive period of the function $f(x)=2\sin\left(\frac{x}{3}+\frac{\pi}{5}\right)-1$ is ______, and the minimum value is ______.

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 \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \).

Answer: $\frac{1}{4}$


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

A school organized a trip to the Expo Park for all third-grade students and rented some large buses. Initially, the plan was to have 28 people on each bus. After all the students boarded, it was found that 13 students could not get on the buses. So, they decided to have 32 people on each bus, and this resulted in 3 empty seats on each bus. How many third-grade students does this school have? How many large buses were rented?

Answer: $125$


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

Let  $A$  be the set  $\{k^{19}-k: 1<k<20, k\in N\}$ . Let  $G$  be the GCD of all elements of  $A$ .
Then the value of  $G$  is?

Answer: $798$


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

Simplify: $\frac{{x}^{2}-4}{{x}^{2}-4x+4}+\frac{x}{{x}^{2}-x}\div \frac{x-2}{x-1}$, then choose a value of $x$ from the integers in the range $-1\leqslant x\leqslant 3$ to substitute and evaluate.

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


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

Let  $p$ ,  $q$ ,  $r$ , and  $s$  be 4 distinct primes such that  $p+q+r+s$  is prime, and the numbers  $p^2+qr$  and  $p^2+qs$  are both perfect squares. What is the value of  $p+q+r+s$ ?

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  $x,y,z$  be  positive reals , such that  $x+y+z=1399$  find the
   $$ \max( [x]y + [y]z + [z]x )  $$
(  $[a]$  is the biggest integer not exceeding  $a$ )

Answer: $652400$


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

Given $\alpha \in \left(0,\pi \right)$, $sin\alpha+cos\alpha=\frac{\sqrt{3}}{3}$, find $\cos 2\alpha$.

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


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 the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2)$.

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>.

Given vectors $\overrightarrow{a} =(\cos x,\sin x)$, $\overrightarrow{b} =(3,-\sqrt{3} )$, with $x\in[0,\pi]$.
$(1)$ If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $x$;
$(2)$ Let $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$, find the maximum and minimum values of $f(x)$ and the corresponding values of $x$.

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 $T$ be a subset of $\{1,2,3,...,40\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$?

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 regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.

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


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

Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed.

Answer: $10$


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

What is the total number of digits used when the first 2500 positive even integers are written?

Answer: $9448$


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

Given the pattern of positive odd numbers shown below, find the 6th number from the left in the 21st row.

Answer: $811$


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

Given an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m,0)$ ($m > \frac{3}{4}$) and intersects the ellipse $E$ at points $A$ and $B$. Point $P(\frac{5}{4},0)$ is given, and $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is a constant.
- (Ⅰ) Find the equation of the ellipse $E$.
- (Ⅱ) Find the maximum area of $\triangle OAB$.

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>.

Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms?

Answer: $51$


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

In a regular octagon, there are two types of diagonals - one that connects alternate vertices (shorter) and another that skips two vertices between ends (longer). What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest form.

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>.

A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is found that if the price of each shirt is reduced by $1, the store can sell an additional 2 shirts per day on average. If the store makes an average daily profit of $2000, what should be the selling price of each shirt?

Answer: $120$


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

Function  $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$  satisfies the conditions:

(i)  $f(1, 1) =1$ ,

(ii)  $f(p + 1, q) + f(p, q + 1) = f(p, q)$  for all  $p, q \in  \mathbb N$ , and

(iii)  $qf(p + 1, q) = pf(p, q + 1)$  for all  $p, q \in  \mathbb N$ .

Find  $f(1990, 31).$

Answer: $\frac{30! \cdot 1989!}{2020!}$


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

How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$.

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>.

A cone with a base radius of $15$ cm and a height of $30$ cm has a sphere inscribed within it. The radius of the sphere can be expressed in the form $b\sqrt{d} - b$ cm. Determine the values of $b$ and $d$.

Answer: $12.5$


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

If two 4'' by 4'' squares are added at each successive stage, what will be the area of the rectangle at Stage 4, in square inches?

Answer: $128$


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

A point \((x, y)\) is a distance of 14 units from the \(x\)-axis. It is a distance of 8 units from the point \((1, 8)\). Given that \(x > 1\), what is the distance \(n\) from this point to the origin?

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>.

On a street, there are 10 lamps numbered 1, 2, 3, …, 10. Now, we need to turn off four of them, but we cannot turn off two or three adjacent lamps, nor can we turn off the two lamps at both ends. How many ways are there to turn off the lamps under these conditions?

Answer: $20$


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

If non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}| = |\overrightarrow{b}|$ and $(\sqrt{3}\overrightarrow{a} - 2\overrightarrow{b}) \cdot \overrightarrow{a} = 0$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________.

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


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

If $a$ and $b$ are positive integers and the equation \( ab - 8a + 7b = 395 \) holds true, what is the minimal possible value of \( |a - b| \)?

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>.

For some positive integer $n$, the number $120n^3$ has $120$ positive integer divisors, including $1$ and the number $120n^3$. How many positive integer divisors does the number $64n^4$ have?

Answer: $375$


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

Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)?

Answer: $20$


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

What is the smallest positive integer with exactly 20 positive divisors?

Answer: $144$


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

Three not necessarily integer numbers are given. If each of these numbers is increased by 1, the product of the numbers also increases by 1. If each of these numbers is increased by 2, their product also increases by 2. Find these numbers.

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>.

Let  $\Omega$  be a unit circle and  $A$  be a point on  $\Omega$ . An angle  $0 < \theta < 180^\circ$  is chosen uniformly at random, and  $\Omega$  is rotated  $\theta$  degrees clockwise about  $A$ . What is the expected area swept by this rotation?

Answer: $2\pi$


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

Mike and Alain play a game in which each player is equally likely to win. The first player to win three games becomes the champion, and no further games are played. If Mike has won the first game, what is the probability that Mike becomes the champion?

Answer: $\frac{11}{16}$


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

Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \geq 2$, find the minimum value of $x+ \sqrt {3}y$.

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


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

An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are positive integers with no common divisor larger than 1. What is the value of \( p+q \)?

(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \( r \), height \( h \), and slant height \( s \) has lateral surface area equal to \( \pi r s \).)

Answer: $59$


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

Define a function $f$ from nonnegative integers to real numbers, with $f(1) = 1$ and the functional equation:
\[ f(m+n) + f(m-n) = 3(f(m) + f(n)) \]
for all nonnegative integers $m \ge n$. Determine $f(10)$.

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 equation
\[(x - \sqrt[3]{7})(x - \sqrt[3]{29})(x - \sqrt[3]{61}) = \frac{1}{5}\]
has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$

Answer: $97.6$


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>.

Two concentric squares share the same center $O$, each with sides of length 2. The length of the line segment $\overline{AB}$ is $\frac{1}{3}$, and the area of the octagon $ABCDEFGH$ formed by the intersection of lines drawn from the vertices of the inner square to the midpoints of sides of the outer square is sought. Express this area as a fraction $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy] //code taken from thread for problem
real alpha = 15;
pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin;
pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z;
draw(W--X--Y--Z--cycle^^w--x--y--z--cycle);
pair A=intersectionpoint(Y--Z, y--z),  C=intersectionpoint(Y--X, y--x),  E=intersectionpoint(W--X, w--x),  G=intersectionpoint(W--Z, w--z),  B=intersectionpoint(Y--Z, y--x),  D=intersectionpoint(Y--X, w--x),  F=intersectionpoint(W--X, w--z),  H=intersectionpoint(W--Z, y--z);
dot(O);
label("$O$", O, SE);
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$E$", E, dir(O--E));
label("$F$", F, dir(O--F));
label("$G$", G, dir(O--G));
label("$H$", H, dir(O--H));[/asy]

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>.

Given the parabola \(\Gamma: y^{2}=8 x\) with focus \(F\), a line \(l\) passing through \(F\) intersects parabola \(\Gamma\) at points \(A\) and \(B\). Tangents to parabola \(\Gamma\) at \(A\) and \(B\) intersect the \(y\)-axis at points \(P\) and \(Q\) respectively. Find the minimum area of the quadrilateral \(APQB\).

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>.

Hari is obsessed with cubics.  He comes up with a cubic with leading coefficient 1, rational coefficients and real roots  $0 < a < b < c < 1$ .  He knows the following three facts:  $P(0) = -\frac{1}{8}$ , the roots form a geometric progression in the order  $a,b,c$ , and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2} \]  The value  $a + b + c$  can be expressed as  $\frac{m}{n}$ , where  $m,n$  are relatively prime positive integers.  Find  $m + n$ .

*Proposed by Akshar Yeccherla (TopNotchMath)*

Answer: $19$


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

In the rectangular coordinate system $(xOy)$, the curve $C\_1$: $ \begin{cases} x=a\cos φ \ y=b\sin φ\end{cases}(φ)$ is a parameter, where $(a > b > 0)$, and in the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the curve $C\_2$: $ρ=2\cos θ$, the ray $l$: $θ=α(ρ≥0)$, intersects the curve $C\_1$ at point $P$, and when $α=0$, the ray $l$ intersects the curve $C\_2$ at points $O$ and $Q$, $(|PQ|=1)$; when $α= \dfrac {π}{2}$, the ray $l$ intersects the curve $C\_2$ at point $O$, $(|OP|= \sqrt {3})$.
(I) Find the general equation of the curve $C\_1$;
(II) If the line $l′$: $ \begin{cases} x=-t \ y= \sqrt {3}t\end{cases}(t)$ is a parameter, $t≠0$, intersects the curve $C\_2$ at point $R$, and $α= \dfrac {π}{3}$, find the area of $△OPR$.

Answer: $\dfrac {3 \sqrt {30}}{20}$