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

A straight line $l$ passes through a vertex and a focus of an ellipse. If the distance from the center of the ellipse to $l$ is one quarter of its minor axis length, calculate the eccentricity of the ellipse.

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


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

A school plans to purchase two brands of soccer balls, brand A and brand B. It is known that the unit price of brand A soccer balls is $30 less than the unit price of brand B soccer balls. The quantity of brand A soccer balls that can be purchased with $1000 is the same as the quantity of brand B soccer balls that can be purchased with $1600.
$(1)$ Find the unit prices of brand A and brand B soccer balls.
$(2)$ The school plans to purchase a total of 80 soccer balls of both brand A and brand B. Let $a$ be the number of brand A soccer balls to be purchased, where the quantity of brand A soccer balls is at least 30 but not more than 3 times the quantity of brand B soccer balls. Determine how many brand A soccer balls should be purchased to minimize the total cost $W$.

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 that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students.

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>.

Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is along side $EG$ of the triangle, and side $AB$ is now one-third the length of side $AD$. The altitude from $F$ to side $EG$ is 12 inches, and side $EG$ is 15 inches. Determine the area of rectangle $ABCD$.

Answer: $\frac{10800}{289}$


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

Square \(ABCD\) has sides of length 14. A circle is drawn through \(A\) and \(D\) so that it is tangent to \(BC\). What is the radius of the circle?

Answer: $8.75$


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

In the third season of "China Poetry Conference", there were many highlights under the theme of "Life has its own poetry". In each of the ten competitions, there was a specially designed opening poem recited in unison by a hundred people under the coordination of lights and dances. The poems included "Changsha Spring in the Jingyuan Garden", "The Road to Shu is Difficult", "The Song of Xi'le", "The Ballad of the Wanderer", "Moon Over the Guan Mountains", and "Six Plates Mountain in the Qingping Melody". The first six competitions were arranged in a certain order, with "The Road to Shu is Difficult" before "The Ballad of the Wanderer", and neither "Changsha Spring in the Jingyuan Garden" nor "Six Plates Mountain in the Qingping Melody" were in the last competition or adjacent to each other. The number of arrangements is $\_\_\_\_\_\_$. (Answer with a number.)

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>.

In triangle \(XYZ,\) \(XY = 5,\) \(XZ = 7,\) \(YZ = 9,\) and \(W\) lies on \(\overline{YZ}\) such that \(\overline{XW}\) bisects \(\angle YXZ.\) Find \(\cos \angle YXW.\)

Answer: $\frac{3\sqrt{5}}{10}$


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>.

The perpendicular bisectors of the sides of triangle $DEF$ meet its circumcircle at points $D'$, $E'$, and $F'$, respectively. If the perimeter of triangle $DEF$ is 42 and the radius of the circumcircle is 10, find the area of hexagon $DE'F'D'E'F$.

Answer: $105$


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

The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs?

Answer: $75$


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

Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack?

Answer: $\frac{17}{442}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 $(0, 0), (3030, 0), (3030, 3031), and (0, 3031)$. What is the probability that $x > 3y$? Express your answer as a common fraction.

Answer: $\frac{505}{3031}$


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

If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$.

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


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

Given that the magnitude of vector $\overrightarrow {a}$ is 1, the magnitude of vector $\overrightarrow {b}$ is 2, and the magnitude of $\overrightarrow {a}+ \overrightarrow {b}$ is $\sqrt {7}$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.

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


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

Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A.

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


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>.

If the line $(m+2)x+3y+3=0$ is parallel to the line $x+(2m-1)y+m=0$, then the real number $m=$ \_\_\_\_\_\_.

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>.

Consider all the positive integers  $N$  with the property that all of the divisors of  $N$  can be written as  $p-2$  for some prime number  $p$ . Then, there exists an integer  $m$  such that  $m$  is the maximum possible number of divisors of all
numbers  $N$  with such property. Find the sum of all possible values of  $N$  such that  $N$  has  $m$  divisors.

*Proposed by **FedeX333X***

Answer: $135$


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

In a certain high school physical examination for seniors, the heights (in centimeters) of 12 students are $173$, $174$, $166$, $172$, $170$, $165$, $165$, $168$, $164$, $173$, $175$, $178$. Find the upper quartile of this data set.

Answer: $173.5$


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

Given a region bounded by a larger quarter-circle with a radius of $5$ units, centered at the origin $(0,0)$ in the first quadrant, a smaller circle with radius $2$ units, centered at $(0,4)$ that lies entirely in the first quadrant, and the line segment from $(0,0)$ to $(5,0)$, calculate the area of the region.

Answer: $\frac{9\pi}{4}$


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>.

Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\cdots+1998+1999-2000-2001$.

Answer: $2001$


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

Neznaika is drawing closed paths inside a $5 \times 8$ rectangle, traveling along the diagonals of $1 \times 2$ rectangles. In the illustration, an example of a path passing through 12 such diagonals is shown. Help Neznaika draw the longest possible path.

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>.

Points $P$ and $Q$ are on a circle with radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of line segment $PR$.

Answer: $\sqrt{32}$


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>.

Let $T$ be the set of ordered triples $(x,y,z)$ of real numbers where
\[\log_{10}(2x+2y) = z \text{ and } \log_{10}(x^{2}+2y^{2}) = z+2.\]
Find constants $c$ and $d$ such that for all $(x,y,z) \in T$, the expression $x^{3} + y^{3}$ equals $c \cdot 10^{3z} + d \cdot 10^{z}.$ What is the value of $c+d$?
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{5}{16}$
D) $\frac{1}{4}$
E) $\frac{1}{2}$

Answer: $\frac{5}{16}$


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 the sequence $\{a\_n\}$ satisfies $a\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, then the minimum value of $\{\frac{S_{n}+4}{n}\}$ is ___.

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


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>.

There is a group of monkeys transporting peaches from location $A$ to location $B$. Every 3 minutes a monkey departs from $A$ towards $B$, and it takes 12 minutes for a monkey to complete the journey. A rabbit runs from $B$ to $A$. When the rabbit starts, a monkey has just arrived at $B$. On the way, the rabbit encounters 5 monkeys walking towards $B$, and continues to $A$ just as another monkey leaves $A$. If the rabbit's running speed is 3 km/h, find the distance between locations $A$ and $B$.

Answer: $300$


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

Given $\left(a + \frac{1}{a}\right)^2 = 5$, find the value of $a^3 + \frac{1}{a^3}$.

Answer: $2\sqrt{5}$


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

Naomi has three colors of paint which she uses to paint the pattern below.  She paints each region a solid color, and each of the three colors is used at least once.  If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?

[asy]
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
[/asy]

Answer: $540$


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

Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. What time does Leo think it is?

A) 11:30
B) 11:45
C) 12:00
D) 12:30
E) 12:45

Answer: $12:30$


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

In the tetrahedron $P-ABC$, $\Delta ABC$ is an equilateral triangle, and $PA=PB=PC=3$, $PA \perp PB$. The volume of the circumscribed sphere of the tetrahedron $P-ABC$ is __________.

Answer: $\frac{27\sqrt{3}\pi}{2}$


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

In rectangle $ABCD$, $\overline{CE}$ bisects angle $C$ (no trisection this time), where $E$ is on $\overline{AB}$, $F$ is still on $\overline{AD}$, but now $BE=10$, and $AF=5$. Find the area of $ABCD$.

Answer: $200$


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

Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, $a_1=15$, and it satisfies $\frac{a_{n+1}}{2n-3} = \frac{a_n}{2n-5}+1$, knowing $n$, $m\in\mathbb{N}$, and $n > m$, find the minimum value of $S_n - S_m$.

Answer: $-14$


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

In a bag, there are $4$ red balls, $m$ yellow balls, and $n$ green balls. Now, two balls are randomly selected from the bag. Let $\xi$ be the number of red balls selected. If the probability of selecting two red balls is $\frac{1}{6}$ and the probability of selecting one red and one yellow ball is $\frac{1}{3}$, then $m-n=$____, $E\left(\xi \right)=$____.

Answer: $\frac{8}{9}$


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

In $\triangle ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$, and $A$, $B$, $C$ form an arithmetic sequence.

(I) If $b=7$ and $a+c=13$, find the area of $\triangle ABC$.

(II) Find the maximum value of $\sqrt{3}\sin A + \sin(C - \frac{\pi}{6})$ and the size of angle $A$ when the maximum value is reached.

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


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

Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.

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>.

A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receiving the task, the club members immediately started producing helium gas at a rate of $xL$ per day. It is known that the cost of raw materials for producing $1L$ of helium gas is $100$ yuan. If the daily production of helium gas is less than $4L$, the additional cost per day is $W_1=4x^2+16$ yuan. If the daily production of helium gas is greater than or equal to $4L$, the additional cost per day is $W_2=17x+\frac{9}{x}-3$ yuan. The production cost consists of raw material cost and additional cost.

$(1)$ Write the relationship between the total cost $W$ (in yuan) and the daily production $x$ (in $L$).

$(2)$ When the club produces how many liters of helium gas per day, the total cost is minimized? What is the minimum cost?

Answer: $4640$


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

Given 6 persons, with the restriction that person A and person B cannot visit Paris, calculate the total number of distinct selection plans for selecting 4 persons to visit Paris, London, Sydney, and Moscow, where each person visits only one city.

Answer: $240$


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

Given $α-β=\frac{π}{3}$ and $tanα-tanβ=3\sqrt{3}$, calculate the value of $\cos \left(\alpha +\beta \right)$.

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


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

A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percentage of the original list price should the marked price be?

Answer: $107\%$


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

In a village, a plot of land is shaped as a right triangle, where one of the legs measures 5 units and another measures 12 units. Farmer Euclid decides to leave a small unplanted square at the vertex of this right angle, and the shortest distance from this unplanted square to the hypotenuse is 3 units. Determine the fraction of the plot that is planted.
A) $\frac{412}{1000}$
B) $\frac{500}{1000}$
C) $\frac{290}{1000}$
D) $\frac{145}{1000}$
E) $\frac{873}{1000}$

Answer: $\frac{412}{1000}$


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

In Pascal's Triangle, we know each number is the combination of two numbers just above it. What is the sum of the middle three numbers in each of Rows 5, 6, and 7?

Answer: $157$


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

In the $4 \times 5$ grid shown, six of the $1 \times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \times 10$ grid are drawn, how many of the $1 \times 1$ squares are not intersected by either diagonal?

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>.

Find the smallest positive integer  $n$  with the property that in the set  $\{70, 71, 72,... 70 + n\}$  you can choose two different numbers whose product is the square of an integer.

Answer: $28$


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

Let  $a, b, c$  be positive integers such that  $a + 2b +3c = 100$ .
Find the greatest value of  $M = abc$

Answer: $6171$


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

Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.

Determine the maximum possible number of positive integers among the 2018 integers.

Answer: $1009$


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

A relatively prime date is defined as a date where the day and the month number are coprime. Determine how many relatively prime dates are in the month with 31 days and the highest number of non-relatively prime dates?

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>.

If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$?

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>.

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

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


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

Two adjacent faces of a tetrahedron are equilateral triangles with a side length of 1 and form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that contains the given edge.

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


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

Given that the function $y = f(x)$ is an even function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = \log_2(x+2) - 3$. Find the values of $f(6)$ and $f(f(0))$.

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>.

Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$.

Answer: $\frac{17\pi}{36}$


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

Simplify the expression: $({1-\frac{1}{{x+3}}})÷\frac{{{x^2}-9}}{{{x^2}+6x+9}}$, then choose a suitable number from $-3$, $2$, $3$ to substitute and evaluate.

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>.

Let $P(x) = 3\sqrt[3]{x}$, and $Q(x) = x^3$. Determine $P(Q(P(Q(P(Q(4))))))$.

Answer: $108$


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

Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \). Additionally, find its height dropped from vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).

Vertices:
- \( A_{1}(-1, 2, 4) \)
- \( A_{2}(-1, -2, -4) \)
- \( A_{3}(3, 0, -1) \)
- \( A_{4}(7, -3, 1) \)

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>.

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

Answer: $100$


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

A three-digit number has distinct digits. By arbitrarily swapping the positions of its digits, five other three-digit numbers can be obtained. If the sum of these six three-digit numbers equals 2220, then among all the numbers that meet this condition, the smallest three-digit number is ____.

Answer: $127$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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 between 200 and 400 are divisible by 8?

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 the function $f$ mapping from set $M$ to set $N$, where $M=\{a, b, c\}$ and $N=\{-3, -2, -1, 0, 1, 2, 3\}$, how many mappings $f$ satisfy the condition $f(a) + f(b) + f(c) = 0$?

Answer: $37$


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

**p22.** Consider the series  $\{A_n\}^{\infty}_{n=0}$ , where  $A_0 = 1$  and for every  $n > 0$ ,   $$ A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]}, $$    where  $[x]$  denotes the largest integer value smaller than or equal to  $x$ . Find the  $(2023^{3^2}+20)$ -th element of the series.**p23.** The side lengths of triangle  $\vartriangle ABC$  are  $5$ ,  $7$  and  $8$ . Construct equilateral triangles  $\vartriangle A_1BC$ ,  $\vartriangle B_1CA$ , and  $\vartriangle C_1AB$  such that  $A_1$ , $B_1$ , $C_1$  lie outside of  $\vartriangle ABC$ . Let  $A_2$ , $B_2$ , and  $C_2$  be the centers of  $\vartriangle A_1BC$ ,  $\vartriangle B_1CA$ , and  $\vartriangle C_1AB$ , respectively. What is the area of  $\vartriangle A_2B_2C_2$ ?**p24.**There are  $20$  people participating in a random tag game around an  $20$ -gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the  $20$ -gon (no matter where they were at the beginning). If there are currently  $10$  taggers, let  $E$  be the expected number of untagged people at the end of the next round. If  $E$  can be written as  $\frac{a}{b}$  for  $a, b$  relatively prime positive integers, compute  $a + b$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: $653$


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

Given the parametric equation of curve $C\_1$ is $\begin{cases} x=3\cos \alpha \ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C\_2$ is $\rho \cos \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$.

(I) Find the rectangular coordinate equation of curve $C\_2$ and the maximum value of the distance $|OP|$ between the moving point $P$ on curve $C\_1$ and the coordinate origin $O$;

(II) If curve $C\_2$ intersects with curve $C\_1$ at points $A$ and $B$, and intersects with the $x$-axis at point $E$, find the value of $|EA|+|EB|$.

Answer: $\frac{6 \sqrt{3}}{5}$


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

What fraction of the area of an isosceles trapezoid $KLMN (KL \parallel MN)$ is the area of triangle $ABC$, where $A$ is the midpoint of base $KL$, $B$ is the midpoint of base $MN$, and $C$ is the midpoint of leg $KN$?

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>.

In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\sin C+\cos C=1-\sin \frac{C}{2}$.

$(1)$ Find the value of $\sin C$.

$(2)$ If $a^{2}+b^{2}=4(a+b)-8$, find the value of side $c$.

Answer: $1+ \sqrt{7}$


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

On the side \( BC \) of triangle \( ABC \), points \( A_1 \) and \( A_2 \) are marked such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On the side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) lie on the same line. Find \( B_1B_2 \).

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>.

Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money.

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>.

The maximum value of the function \( y = \tan x - \frac{2}{|\cos x|} \) is to be determined.

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 that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation when one piece of each product is sold.

Answer: $5.92$


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

Given that the sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$ by $255$.
$(1)$ Find all the rational terms of $x$ in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$;
$(2)$ If $(2x-1)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots+a_{n}x^{n}$, find the value of $(a_{0}+a_{2}+a_{4})^{2}-(a_{1}+a_{3})^{2}$.

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>.

Using the digits $1, 2, 3, 4$, 24 unique four-digit numbers can be formed without repeating any digit. If these 24 four-digit numbers are arranged in ascending order, find the sum of the two middle numbers.

Answer: $4844$


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

In triangle \( \triangle ABC \), \( AB = BC = 2 \) and \( AC = 3 \). Let \( O \) be the incenter of \( \triangle ABC \). If \( \overrightarrow{AO} = p \overrightarrow{AB} + q \overrightarrow{AC} \), find the value of \( \frac{p}{q} \).

Answer: $2/3$


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>.

In an isosceles triangle $\triangle ABC$ with vertex angle $A = \frac{2\pi}{3}$ and base $BC = 2\sqrt{3}$, find the dot product $\vec{BA} \cdot \vec{AC}$.

Answer: $-2$


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

Let $ABCD$ be an isosceles trapezoid with $AB$ and $CD$ as parallel bases, and $AB > CD$. A point $P$ inside the trapezoid connects to the vertices $A$, $B$, $C$, $D$, creating four triangles. The areas of these triangles, starting from the triangle with base $\overline{CD}$ moving clockwise, are $3$, $4$, $6$, and $7$. Determine the ratio $\frac{AB}{CD}$.
- **A**: $2$
- **B**: $\frac{5}{2}$
- **C**: $\frac{7}{3}$
- **D**: $3$
- **E**: $\sqrt{5}$

Answer: $\frac{7}{3}$


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>.

Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$.

Answer: $-7$


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

From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______.

Answer: $\frac{6}{35}$


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

In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is

Answer: $7 \sqrt{3}$


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 \( \frac{1}{3} \leqslant a \leqslant 1 \), if \( f(x)=a x^{2}-2 x+1 \) attains its maximum value \( M(a) \) and minimum value \( N(a) \) on the interval \([1,3]\), and let \( g(a)=M(a)-N(a) \), then the minimum value of \( g(a) \) is \(\quad\) .

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


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

Given that $4^{-1} \equiv 57 \pmod{119}$, find $64^{-1} \pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.)

Answer: $29$


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

Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
What is the value of \( f(90) \)?

Answer: $999$


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

Suppose six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Calculate the value of $b$.

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


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>.

The volume of a sphere is $72\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bottom) of the cylinder.

Answer: $90\pi \sqrt[3]{4}$


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>.

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $2\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$

Answer: $16$


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

The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number?

Answer: $78$


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

The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,

$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is _______.

$(2)$ If three seeds are sown per hole, the probability of having seedlings in this hole is _______; the probability of having strong seedlings in this hole is _______.

Answer: $0.936$


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

Given that $α∈(0, \dfrac{π}{2})$ and $β∈(\dfrac{π}{2},π)$, with $\cos β=-\dfrac{1}{3}$ and $\sin (α+β)=\dfrac{7}{9}$.

(1) Find the value of $\sin α$;

(2) Find the value of $\sin (2α+β)$.

Answer: $\dfrac{10\sqrt{2}}{27}$


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

Given the function $f(x)=2\sqrt{3}\sin x\cos x+2\cos^{2}x-1$.

(I) Find the axis of symmetry and the center of symmetry of $f(x)$;

(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi }{6}, \frac{\pi }{4}\right]$.

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>.

We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:
1. $g(b)=3b+1$ if $b$ is odd
2. $g(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $g$?

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>.

Reading material: In class, the teacher explained the following two problems on the blackboard:

Problem 1: Calculate: $77.7\times 11-77.7\times 5+77.7\times 4$.
Solution: $77.7\times 11-77.7\times 5+77.7\times 4=77.7\times \left(11-5+4\right)=777$.
This problem uses the distributive property of multiplication.
Problem 2: Calculate $(\frac{1}{2}×\frac{3}{2})×(\frac{2}{3}×\frac{4}{3})×(\frac{3}{4}×\frac{5}{4})\times …×(\frac{18}{19}×\frac{20}{19})$.
Solution: The original expression $=\frac{1}{2}×(\frac{3}{2}×\frac{2}{3})×(\frac{4}{3}×\frac{3}{4})×…×(\frac{19}{18}×\frac{18}{19})×\frac{20}{19}=\frac{1}{2}×\frac{20}{19}=\frac{10}{19}$.
This problem uses the commutative and associative properties of multiplication.

Attempt to solve: (1) Calculate: $99\times 118\frac{4}{5}+99×(-\frac{1}{5})-99×18\frac{3}{5}$.
Application of solution: (2) Subtract $\frac{1}{2}$ from $24$, then subtract $\frac{1}{3}$ from the remaining, then subtract $\frac{1}{4}$ from the remaining, and so on, until subtracting the remaining $\frac{1}{24}$. What is the final result?
(3) Given a rational number $a\neq 1$, calling $\frac{1}{1-a}$ the reciprocal difference of $a$, such as the reciprocal difference of $a=2$ is $\frac{1}{1-2}=-1$, the reciprocal difference of $-1$ is $\frac{1}{1-(-1)}=\frac{1}{2}$. If $a_{1}=-2$, $a_{1}$'s reciprocal difference is $a_{2}$, $a_{2}$'s reciprocal difference is $a_{3}$, $a_{3}$'s reciprocal difference is $a_{4}$, and so on. Find the value of $a_{1}+a_{2}+a_{3}-2a_{4}-2a_{5}-2a_{6}+3a_{7}+3a_{8}+3a_{9}$.

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


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

The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence to be number 0, determine after what time interval (in minutes) they will coincide the 19th time. If necessary, round the answer to two decimal places following the rounding rules.

Answer: $1243.64$


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>.

Evaluate $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{49}\rceil$$Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$

Answer: $245$


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

Given the function $$f(x)=4\sin(x- \frac {π}{6})\cos x+1$$.
(Ⅰ) Find the smallest positive period of f(x);
(Ⅱ) Find the maximum and minimum values of f(x) in the interval $$\[-\frac {π}{4}, \frac {π}{4}\]$$ .

Answer: $-2$


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

How many positive 3-digit numbers are multiples of 30, but not of 75?

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 right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid).

Answer: $34$


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

If the maximum and minimum values of the exponential function $f(x) = a^x$ on the interval $[1, 2]$ differ by $\frac{a}{2}$, then find the value of $a$.

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


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

The value of x that satisfies the equation \( x^{x^x} = 2 \) is calculated.

Answer: $\sqrt{2}$


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

If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functions: (1) $f(x) = \frac{1}{x}$; (2) $f(x) = x + 1$; (3) $f(x) = \begin{cases} -x^2 & \text{if}\ x \geq 0 \\ x^2 & \text{if}\ x < 0 \end{cases}$; identify which can be called an "ideal function" by their respective sequence numbers.

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>.

A steak initially at a temperature of 5°C is put into an oven. After 15 minutes, its temperature reaches 45°C. After another 15 minutes, its temperature is 77°C. The oven maintains a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

Answer: $205$


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

Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$.

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


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

Compute the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6}\left(\frac{x}{2}\right) \cos ^{2}\left(\frac{x}{2}\right) d x
$$

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


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

At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 10 minutes, and when she came back, she found Ivan and Foma in front of the door, with Foma having apparently arrived after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the shop to reopen.

Answer: $0.75$


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

$(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:
① When $n$ is a positive integer, $f(f(n))=4n+9$;
② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.
$(2)$ The function $f$ is defined on the set of ordered pairs of positive integers, and satisfies the following properties:
① $f(x,x)=x$;
② $f(x,y)=f(y,x)$;
③ $(x+y)f(x,y)=yf(x,x+y)$. Find $f(14,52)$.

Answer: $364$


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

Let \( n = 2^{31} \times 3^{19} \times 5^7 \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)?

Answer: $13307$


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

A triangle has sides of lengths 30, 70, and 80. When an altitude is drawn to the side of length 80, the longer segment of this side that is intercepted by the altitude is:

Answer: $65$


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

Let  $p = 2027$  be the smallest prime greater than  $2018$ , and let  $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$ . Let  $\mathrm{GF}(p)$  be the integers modulo  $p$ , and let  $\mathrm{GF}(p)(X)$  be the set of rational functions with coefficients in  $\mathrm{GF}(p)$  (so that all coefficients are taken modulo  $p$ ). That is,  $\mathrm{GF}(p)(X)$  is the set of fractions  $\frac{P(X)}{Q(X)}$  of polynomials with coefficients in  $\mathrm{GF}(p)$ , where  $Q(X)$  is not the zero polynomial. Let  $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$  be a function satisfying \[
    D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2}
\]for any  $f,g\in \mathrm{GF}(p)(X)$  with  $g\neq 0$ , and such that for any nonconstant polynomial  $f$ ,  $D(f)$  is a polynomial with degree less than that of  $f$ . If the number of possible values of  $D(P(X))$  can be written as  $a^b$ , where  $a$ ,  $b$  are positive integers with  $a$  minimized, compute  $ab$ .

*Proposed by Brandon Wang*

Answer: $4114810$


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

Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle.

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>.

The base of a triangle is  $80$ , and one side of the base angle is  $60^\circ$ .  The sum of the lengths of the other two sides is  $90$ .  The shortest side is:

Answer: $17$


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

There are a total of $400$ machine parts. If person A works alone for $1$ day, and then person A and person B work together for $2$ days, there will still be $60$ parts unfinished. If both work together for $3$ days, they can produce $20$ parts more than needed. How many parts can each person make per day?

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>.

Given that point $M$ represents the number $9$ on the number line.
$(1)$ If point $N$ is first moved $4$ units to the left and then $6$ units to the right to reach point $M$, then the number represented by point $N$ is ______.
$(2)$ If point $M$ is moved $4$ units on the number line, then the number represented by point $M$ is ______.

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>.

Point  $M$  lies on the diagonal  $BD$  of parallelogram  $ABCD$  such that  $MD = 3BM$ . Lines  $AM$  and  $BC$  intersect in point  $N$ . What is the ratio of the area of triangle  $MND$  to the area of parallelogram  $ABCD$ ?

Answer: $3/8$


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

Two balls, one blue and one orange, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is $3^{-k}$ for $k = 1, 2, 3,...$. What is the probability that the blue ball is tossed into a higher-numbered bin than the orange ball?
A) $\frac{1}{8}$
B) $\frac{1}{9}$
C) $\frac{1}{16}$
D) $\frac{7}{16}$
E) $\frac{3}{8}$

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


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

Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sum of the first $n$ terms of each sequence is denoted as $S_n$ and $T_n$ respectively. If $$\frac {S_{n}}{T_{n}}= \frac {7n+45}{n+3}$$, and $$\frac {a_{n}}{b_{2n}}$$ is an integer, then the value of $n$ is \_\_\_\_\_\_.

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>.

A three-digit  $\overline{abc}$  number is called *Ecuadorian* if it meets the following conditions: $\bullet$   $\overline{abc}$  does not end in  $0$ . $\bullet$   $\overline{abc}$  is a multiple of  $36$ . $\bullet$   $\overline{abc} - \overline{cba}$  is positive and a multiple of  $36$ .
Determine all the Ecuadorian numbers.

Answer: $864$


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

Simplify first, then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a$ and $b$ satisfy $a^2+b^2-2a+4b=-5$.

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>.

For $x > 0$, find the maximum value of $f(x) = 1 - 2x - \frac{3}{x}$ and the value of $x$ at which it occurs.

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


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

Let  $f : Q \to  Q$  be a function satisfying the equation  $f(x + y) = f(x) + f(y) + 2547$  for all rational numbers  $x, y$ . If  $f(2004) = 2547$ , find  $f(2547)$ .

Answer: $2547$


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

Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.

Answer: $0.298$


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

Add $256_{7} + 463_{7} + 132_7$. Express your answer in base $7$.

Answer: $1214_{7}$


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

What is the sum of the fractions of the form $\frac{2}{n(n+2)}$, where $n$ takes on odd positive integers from 1 to 2011? Express your answer as a decimal to the nearest thousandth.

Answer: $0.999$


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

Suppose that $x_1+1 = x_2+2 = x_3+3 = \cdots = x_{1000}+1000 = x_1 + x_2 + x_3 + \cdots + x_{1000} + 1001$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \sum_{n=1}^{1000} x_n$.

Answer: $501$


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

A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock.

Answer: $12.925$


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

Given $\overrightarrow{a}=( \sqrt {2},m)(m > 0)$, $\overrightarrow{b}=(\sin x,\cos x)$, and the maximum value of the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$ is $2$.
1. Find $m$ and the smallest positive period of the function $f(x)$;
2. In $\triangle ABC$, $f(A- \frac {\pi}{4})+f(B- \frac {\pi}{4})=12 \sqrt {2}\sin A\sin B$, where $A$, $B$, $C$ are the angles opposite to sides $a$, $b$, $c$ respectively, and $C= \frac {\pi}{3}$, $c= \sqrt {6}$, find the area of $\triangle ABC$.

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


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

Determine the value of the sum \[ \sum_{n=0}^{332} (-1)^{n} {1008 \choose 3n} \] and find the remainder when the sum is divided by $500$.

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>.

Given the functions $f(x)=\log_{a}x$ and $g(x)=\log_{a}(2x+t-2)$, where $a > 0$ and $a\neq 1$, $t\in R$.

(1) If $0 < a < 1$, and $x\in[\frac{1}{4},2]$ such that $2f(x)\geqslant g(x)$ always holds, find the range of values for the real number $t$;

(2) If $t=4$, and $x\in[\frac{1}{4},2]$ such that the minimum value of $F(x)=2g(x)-f(x)$ is $-2$, find the value of the real number $a$.

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


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>.

$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$.

Answer: $\frac{11+6\sqrt{2}}{8}$