inclusionAI/Ring-lite-rl-data huggingface dataset:
4. In an isosceles triangle $A B C$ the angle opposite the base $\overline{A B}$ is $80^{\circ}$. On the height to the base of this triangle, a point $D$ is chosen such that $|\angle D B A|=30^{\circ}$. On the segment $\overline{B D}$, a point $T$ is chosen such that $|\angle T A D|=20^{\circ}$. Determine the size of the angle $\angle T C A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
设 A_1,A_2,\cdots,A_n 是集合 \{1,2,\cdots,2018\} 的二元子集，使得 A_i+A_j,1\leq i\leq j\leq n 互不相同. 求 n 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4033}
Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67.5}
For any shape, what is the smallest positive integer \( n \) such that the total number of lattice points on and inside a convex \( n \)-gon is at least \( n+1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Determine the maximum slope of the line $OQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given that the function $f(x)$ is defined on the domain $\mathbb{R}$ and satisfies $f'(x) > f(x)$, determine the solution set of the inequality $e^{f(\ln x)} - xf(1) < 0$ (represent the answer using intervals).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, e)}
In triangle $ABC$, $\tan B= \sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\frac {3 \sqrt {3}}{2}$. Find the length of $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {7}}
There is a tunnel between locations A and B. A car leaves B for A at 08:16, and a truck leaves A for B at 09:00. It is known that the truck and the car arrive at the ends of the tunnel at the same time, but the truck leaves the tunnel 2 minutes later than the car. If the car arrives at A at 10:56 and the truck arrives at B at 12:20, at what time do they simultaneously arrive at the ends of the tunnel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:00}
8. Equilateral triangle $A B C$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $A B$ and $A C$ of $\Omega$ respectively such that $B C=D E$. Given that triangle $A B E$ has area 3 and triangle $A C D$ has area 4 , find the area of triangle $A B C$.
Proposed by: Yuan Yao
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{7}}
Five unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending from \((a,0)\) to \((3,3)\), divides the entire region into two regions of equal area. The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In triangle \(ABC\), it is given that \(|AB| = |BC|\) and \(\angle ABC = 20^\circ\). Point \(M\) is taken on \(AB\) such that \(\angle MCA = 60^\circ\), and point \(N\) is taken on \(CB\) such that \(\angle NAC = 50^\circ\). Find \(\angle NMA\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
The probability of each seed germinating is 0.9. Now, 1000 seeds have been sown. For each seed that does not germinate, 2 more seeds need to be sown. Let the number of seeds that need to be sown again be denoted as _X_. Calculate the expected value of _X_. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
A lead ball with a radius of $R$ falls into a sandpit, leaving a hole with an external diameter of 24 cm and a depth of 8 cm. The radius of the ball is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13 \text{ cm}}
15 men are equal to as many women as are equal to 12 boys. All of them together earn Rs. 432. What are the men's wages?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Six people, a mix of knights (who always tell the truth) and liars (who always lie), sat around a round table. Each of them was given a coin. Then, each person passed their coin to one of their two neighbors. After that, 3 people said, "I have one coin," and the other 3 said, "I have no coins." What is the maximum number of knights that could have been sitting at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{365}
若存在实数 $a$ 及正整数 $n$ 使得 $f(x)=\cos 2 x-a \sin x$ 在 $(0, n \pi)$ 内恰有 $2022$ 个零点, 则满足条件的正整数 $n$ 的值有 $\qquad$个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral: $$\int_0^1 \frac{x\ln x}{(1+x^2)^2} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\ln(2)}{4}}
## Task Condition

Find the derivative of the specified order.

$$
y=\left(2 x^{3}+1\right) \cos x, y^{V}=?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(30x^{2}-120)\cosx-(2x^{3}-120x+1)\sinx}
Evaluate the sum $\sum\limits_{n=1}^\infty \left(\frac{1}{2}\right)^n \prod\limits_{k=0}^{n-1}\frac{j+\frac{1}{2}}{j+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{sqrt(2)-1}
Let $M=\left\{1,2, \cdots, 2^{m} n\right\}$ (where $m$ and $n$ are positive integers) be a set consisting of continuous $2^{m}n$ positive integers. Find the minimum value of the positive integer $k$ such that any $k$-element subset of $M$ contains $m+1$ numbers $a_1, a_2, \cdots, a_{m+1}$ that satisfy $a_i$ divides $a_{i+1}$ for $i=1,2, \cdots, m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2^m - 1)n + 1}
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A company had a sales revenue of 10 million yuan last year, with annual costs consisting of 5 million yuan in production costs and 2 million yuan in advertising costs. If the annual profit must be taxed at a rate of p%, and the portion of the annual advertising expense that exceeds 2% of the annual sales revenue is also taxed at p%, with no other taxes applied. It is known that the company paid a total of 1.2 million yuan in taxes last year. Calculate the tax rate p%. Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25%}
Evaluate the integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2\pi}{\sqrt{3}}\)}
A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home by bus in the afternoon $15$ times, and commuted by train (either morning or afternoon) $9$ times. Find $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$16$}
There were 12 seats prepared for the jury at a round table, each with a name indicated on it. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his designated seat but in the next one clockwise. Each jury member arriving at the table afterward would either sit in their designated seat, or, if it was already occupied, they would walk around the table clockwise and sit in the next available seat. The resulting seating arrangement of the jury members depends on the order in which they approached the table. How many different ways can the jury be seated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
11.3. The teacher is going to give the children a problem of the following type. He will inform them that he has thought of a polynomial $P(x)$ of degree 2017 with integer coefficients, the leading coefficient of which is 1. Then he will tell them $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$, and separately inform them of the value of the expression $P\left(n_{1}\right) \cdot P\left(n_{2}\right) \cdot \ldots \cdot P\left(n_{k}\right)$. Based on this data, the children are supposed to find the polynomial that the teacher could have thought of. For what smallest $k$ can the teacher formulate such a problem so that the polynomial found by the children necessarily coincides with the one thought of?

(G. Zhukov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Nine identical books cost 11 rubles and some kopecks, and thirteen identical books cost 15 rubles and some kopecks. Determine the exact cost of one book.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \text{ ruble } 23 \text{ kopeks}}
In the figure, the center of square \(ABCD\) is \(O\), and its area is \(1989 \text{ cm}^2\). Point \(P\) is a point inside the square such that \(\angle OPB = 45^\circ\) and \(PA : PB = 5 : 14\). Find \(PB\) in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this operation necessary to sort $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find
$$
\lim _{n \rightarrow \infty}\left[\frac{1}{n^{5}} \sum_{h=1}^{n} \sum_{k=1}^{n}\left(5 h^{4}-18 h^{2} k^{2}+5 k^{4}\right)\right].
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A clothing store found in its sales that a certain brand of clothing can sell an average of 20 pieces per day, with a profit of $40 per piece. In order to prepare for "Double Eleven," the mall decided to take appropriate price reduction measures to increase profits and reduce inventory. According to market research, if the price of each piece of clothing is reduced by $4, then an average of 8 more pieces can be sold per day. If you want to make a profit of $1200 per day on average by selling this type of clothing, how much should the price of each piece of clothing be reduced by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Three male students and three female students, a total of six students, stand in a row. If female students do not stand at the end of the row, and female students A and B are not adjacent to female student C, then how many different arrangements are possible? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \cdot a_5 = -15$, find the value of $a_{12}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{25}{3}}
Consider the set  $A=\{1,2,3\ldots ,2^n\}, n\ge 2$ . Find the number of subsets  $B$  of  $A$  such that for any two elements of  $A$  whose sum is a power of  $2$  exactly one of them is in  $B$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{n+1} }
Consider the set \( S = \{1, 2, 3, \ldots, 98\} \). Determine the smallest natural number \( n \) such that in any subset of \( S \) with \( n \) elements, it is possible to select 10 numbers where, no matter how these 10 numbers are divided into two groups of 5, one group will contain a number that is relatively prime to the other four numbers, and the other group will contain a number that is not relatively prime to the other four numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

2. No digit occurs more than once in the same integer.

3. The digits in each integer are in increasing order.

4. Any two integers have at least one digit in common (possibly at different positions).

5. There is no digit which appears in all the integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
The base of a right rectangular prism is a rhombus with a side length of 2 and an angle of $30^{\circ}$. The height of the prism is 1. A cross-section is made through the base edge at an angle of $60^{\circ}$ to the base. Find the area of this cross-section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{3}}{3}}
Find the largest value of the non-negative integer (a) for which:
$$ \displaystyle \lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4 $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$, and $(5,8,12)$ as shown in the figures below. He can join two pieces by sticking exactly the sides of the same length together. For example, he can join the side 10 of the first piece with the side 10 of the second piece, but he cannot join the side 10 of the first piece with the side 8 of the third piece because they do not have the same length. What is the greatest perimeter Pedrinho can obtain by joining the three pieces?

(Note: The figures provided in the original problem have been omitted for text-based translation purposes.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
There are  $ 2010 $  people sitting around a round table. First, we give one person  $ x $  a candy. Next, we give candies to  $1$  st person,  $1+2$  th person,  $ 1+2+3$  th person,  $\cdots$  , and  $1+2+\cdots + 2009 $  th person clockwise from  $ x $ . Find the number of people who get at least one candy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{408}
Given two lines $l_{1}$: $mx+y+2m-3=0$ and $l_{2}$: $mx+y-m+1=0$, the maximum value of the distance between line $l_{1}$ and $l_{2}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
在 $\triangle A B C$ 中, $A B=A C=2, B C$ 边有 100 个不同的点 $P_1, P_2, \cdots, P_{100}$, 记 $m_i=A P_2^2+$ $B P_i \cdot P_i C(i=1,2, \cdots, 100)$, 求 $m_1+m_2+\cdots+m_{10}$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^\circ$, $\angle B = 105^\circ$, $\angle C = 90^\circ$ and $AB = 2$, $BC = CD = DE = \sqrt{2}$. If the length of $AE$ is $\sqrt{a} - b$ where $a$ and $b$ are integers, what is the value of $a + b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ $\mathbf{d}$ be four distinct unit vectors in space such that 
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{d} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{7}.\]
Find $\mathbf{a} \cdot \mathbf{d}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{19}{21}}
An n-pointed star is formed by a series of equal-length segments, with each segment measuring 2n. The angles A1, A2,..., An are equal, and the angles B1, B2,..., Bn are equal. If angle A1 = angle B1 - 10 degrees, find the value of n. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can be assembled into $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are identical after rotation are considered the same).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find all natural numbers n that satisfy the equation 1*2 + 2*3 + 3*4 +... + n*(n+1) = 3n^2 - 3n + 2. Express your answer as a list of natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2, 3}
Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each subsequent placement must share at least one edge with a piece placed in the previous placement, and apart from sharing edges, there should be no other overlaps (see diagram). After the 20th placement, the total number of equilateral triangle pieces used is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{571}
In the sequence below, each △ represents a number, and starting from the third term, each number is the sum of the two preceding numbers. Calculate the sum of all 9 numbers listed: △, △, △, △, 7, △, △, △, 47. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{122}
The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
在由实数构成的 $n \times n$ ($n \geq 2$) 数表中，第 $i$ 行 $n$ 个数之和为 $s_i$，第 $j$ 列 $n$ 个数之和为 $t_j$，记 $a_{ij} = s_i - t_j$，$i,j = 1, 2, \dots, n$。求 $n^2$ 个数 $a_{ij}$ ($i,j = 1, 2, \dots, n$) 中正数个数的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n(n-1)$}
A conical flask contains some water. When the flask is oriented so that its base is horizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep. When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2 inches deep. What is the height of the cone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{2}+\\frac{\\sqrt{93}}{6}"]}
Assign each letter a digit from 1, 2, 3, 4, 5 such that all the inequalities are satisfied.

$$
3>\mathrm{A}>\mathrm{M}<\mathrm{E}<\mathrm{H}<\mathrm{A}
$$

Different letters must correspond to different digits. Write the number ZAMENA as the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{541234}
5. Ms. Olga Ivanovna, the class teacher of 5B, is staging a "Mathematical Ballet." She wants to arrange the boys and girls so that there are exactly 2 boys 5 meters away from each girl. What is the maximum number of girls that can participate in the ballet, given that 5 boys are participating?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\ 3\sqrt{2}$}
设实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 5 } \in [ - 1 , 1 ]$，求下列表达式的最小值

$P = x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 } + \cdots + x _ { 2 0 0 4 } x _ { 2 0 0 5 } + x _ { 2 0 0 5 } x _ { 1 }$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{−2003}
3. Encipher the message SURRENDER IMMEDIATELY using the affine transformation $C \equiv 11 P+18(\bmod 26)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{IEXXK FZKXC UUKZC STKJW}
Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9) > f(16)$. The probability that $f(1) > f(16) > f(25)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1, 2, \ldots, 100\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{730}
Let \(X\), \(Y\), and \(Z\) be points on the sides \(BC\), \(AC\), and \(AB\) of \(\triangle ABC\), respectively, such that \(AX\), \(BY\), and \(CZ\) concur at point \(O\). The area of \(\triangle BOC\) is \(a\). If \(BX : XC = 2 : 3\) and \(CY : YA = 1 : 2\), what is the area of \(\triangle AOC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3a}
Find the sum of all values of  $a$  such that there are positive integers  $a$  and  $b$  satisfying  $(a - b)\sqrt{ab} = 2016$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
In a geometric sequence $\{a_n\}$, $a_1=1$, and the sum of the first $n$ terms is denoted by $S_n$. It is given that $S_7 - 4S_6 + 3S_5 = 0$. Find the value of $S_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Two sides of a regular $n$-gon are extended to meet at a $28^{\circ}$ angle. What is the smallest possible value for $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}