inclusionAI/Ring-lite-rl-data huggingface dataset:
11.2. On the sides $A B$ and $A C$ of an equilateral triangle $A B C$ with side length 10, points $P$ and $Q$ are taken such that the segment $P Q$ is tangent to the inscribed circle of the triangle and its length is 4. Find the area of triangle $A P Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{\sqrt{3}}}
There are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from $1$ to $64$. At the center of the table, there are $1996$ light bulbs, all initially turned off. 

Every minute, the chips move simultaneously in a circular pattern as follows: chip $1$ moves one booth forward, chip $2$ moves two booths forward, and so on, such that multiple chips can occupy the same booth at the same time. 

A bulb is lit whenever chip $1$ shares a booth with any other chip. 

Determine the position of chip $1$ on the first minute when all $1996$ bulbs are lit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\left\lfloor\sqrt{P(x)}\right\rfloor = \sqrt{P(\left\lfloor x \right\rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{850}
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\varphi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
已知欧拉常数 $\gamma=\lim\limits _{n \rightarrow+\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\right)$ 约等于 0.5772156649 。记 $I=\frac{100}{\int_{0}^{1}\left(\frac{1}{x}-\left[\frac{1}{x}\right]\right) \mathrm{d} x}$, 则 $[I]=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236}
Given that $a \in A$, and $a-1 \notin A$ and $a+1 \notin A$, $a$ is called an isolated element of set $A$. How many four-element subsets of the set $M=\{1,2, \cdots, 9\}$ have no isolated elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given that 800 students were surveyed, and their pasta and pizza preferences were lasagna (150 students), manicotti (120 students), ravioli (180 students), spaghetti (200 students), and pizza (150 students), calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred pizza.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
In the rhombus \(ABCD\), the heights \(BP\) and \(BQ\) intersect the diagonal \(AC\) at points \(M\) and \(N\) (point \(M\) lies between \(A\) and \(N\)), \(AM = p\) and \(MN = q\). Find \(PQ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{q(2p+q)}{p+q}}
There are two concentric circles, with six non-overlapping points on the circumference of the outer circle and three non-overlapping points on the circumference of the inner circle. Determine the minimum number of straight lines that can be formed using these nine points. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
A train 280 meters long travels at a speed of 18 meters per second across a bridge. It takes 20 seconds from the moment the tail of the train gets on the bridge until the head of the train leaves the bridge. Therefore, the length of the bridge is $\qquad$ meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{640}
If $\operatorname{tg} \alpha$ and $\operatorname{tg} \beta$ are the two roots of the quadratic equation $x^{2} + p x + q = 0$, then find the value of $\frac{\sin (\alpha + \beta)}{\cos (\alpha - \beta)} =$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{p}{q+1}}
The rules of a card arrangement game are as follows: Arrange nine cards marked with $1,2, \cdots, 9$ randomly in a row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is called an operation. The game stops when no operation can be performed (i.e., the number on the first card is "1"). If an arrangement cannot be operated on and can be obtained by performing one operation on a unique other arrangement, it is called a "second stop arrangement". Find the probability of a second stop arrangement occurring among all possible arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{103}{2520}}
There are 6 people in the elevator. Their average weight is 156 lbs, not including the weight of their clothing. A group of 3 individuals, A, B, and C, enter the elevator one by one, each carrying a backpack and wearing various clothing items. The total weight of A's clothing and backpack is 18 lbs, B's is 20 lbs, and C's is 22 lbs. After each person enters, the average weight, without considering clothing and backpacks, gets updated. 

After A enters, the average weight becomes 159 lbs. When B enters, the average weight increases to 162 lbs. After C enters, the new average weight is reduced to 161 lbs. What is the true weight (excluding clothing and backpack) of the third person, C, who entered the elevator?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131}
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+2)a_n$, and $a_nb_n=\frac{1}{2}$. If $\lambda > T_n$ holds true for any $n\in\mathbb{N}^*$, find the minimum value of the real number $\lambda$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
An arithmetic sequence alternates between even and odd terms. The sums of the odd terms and the even terms are 24 and 30, respectively. The last term exceeds the first term by 10.5. Find the total number of terms in the arithmetic sequence. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
10. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ with edges $A B=4, A D=2$, $A A_{1}=3 \sqrt{2}$, a plane is drawn through the diagonal $B D_{1}$, intersecting the edge $A A_{1}$ such that the section of the parallelepiped by this plane has the smallest perimeter. Find the area of this section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{2}}
The angle bisectors of triangle \( A B C \) intersect at point \( I \), and the external angle bisectors of angles \( B \) and \( C \) intersect at point \( J \). The circle \( \omega_{b} \) with center at point \( O_{b} \) passes through point \( B \) and is tangent to line \( C I \) at point \( I \). The circle \( \omega_{c} \) with center at point \( O_{c} \) passes through point \( C \) and is tangent to line \( B I \) at point \( I \). The segments \( O_{b} O_{c} \) and \( I J \) intersect at point \( K \). Find the ratio \( I K / K J \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
8. Let $A, B, C, D$ be four non-coplanar points in space, and connect an edge between each pair of points with a probability of $\frac{1}{2}$. Whether any two pairs of points are connected is independent of each other. Then the probability that $A$ and $B$ can be connected by (a single edge or a sequence of edges forming a) spatial polyline is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Let \( M \) be a subset of the set \(\{1, 2, 3, \cdots, 15\}\), and suppose that the product of any three different elements in \( M \) is not a perfect square. Let \( |M| \) denote the number of elements in the set \( M \). Find the maximum value of \( |M| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
From two pieces of alloys (with different lead contents) weighing 6 kg and 12 kg, respectively, equal-mass pieces were cut off. Each of the cut pieces was fused with the remainder of the other piece, after which the percentage of lead in both resulting alloys became identical. What are the masses of each of the cut pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \text{ kg}}
Given the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4045}
Let the set \( M = \{1, 2, \cdots, 19\} \) and \( A = \{a_1, a_2, \cdots, a_k\} \subseteq M \). Determine the smallest \( k \), such that for any \( b \in M \), there exist \( a_i, a_j \in A \), satisfying \( a_i = b \) or \( a_i \pm a_j = b\) (where \(a_i\) and \(a_j\) can be the same).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $f(x)$ be a polynomial of degree 3 with real coefficients, satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Let $P(x) \in \mathbb{Z}[x]$ be a polynomial with degree $t-1$, and suppose it satisfies the equation $$xP(x) = x^t + P(x-1)$$ for all real $x$. Determine $t \equiv ? \pmod{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}ax^2 + bx$ has an extreme value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{36}}
Sally has a gold credit card with a certain spending limit, and a platinum card with twice the spending limit of the gold card. She has a balance on her gold card that is a certain fraction of the spending limit on that card, and she has a balance on her platinum card that is 1/9 of the spending limit on that card. If Sally transfers the entire balance on her gold card to her platinum card, 0.7222222222222222 portion of her limit on the platinum card will remain unspent. What fraction of the spending limit on the gold card is her current balance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
A person crosses a 1080 m long street in some time. His speed is 5.4 km per hour. How much time did he take to cross the street?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\frac{[H_PH_QH_RH_S]}{[PQRS]}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
The coordinates of the fixed point that the function $f(x)=\log_{x}(3x-2)+2$ (where $a > 0$ and $a \neq 1$) always passes through are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
A railway freight station decided to organize and dispatch 8 coal freight trains into two groups, each containing 4 trains, with the conditions that trains A and B cannot be in the same group, train A departs first, and train B departs last. Find the total number of different departure sequences for these 8 trains. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0, 1, 2, \ldots, 12$ such that the value of the resulting expression is divisible by 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{816}
There are 10 different electronic products, among which 2 products are unstable. Technicians test them one by one until both unstable products are identified, and the testing ends. Calculate the number of methods to exactly finish the testing in 3 tries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A positive integer $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \overline{aa}$, which is a multiple of $a^2$. Find all possible values of $\frac{b}{a^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the sum of the alternating series: $$\sum_{n=-\infty}^{\infty} (-1)^n \frac{2 n-1}{(2 n-1)^2+y^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4019}
Select 5 different letters from the word "equation" to arrange in a row, ensuring that "qu" are together and in the same order. Calculate the total number of different arrangements. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base?  Give your response in base 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10001_2}
A 360 m long train is running at a speed of 55 km/hr. It crossed a platform of certain length in 57.59539236861051 seconds. What is the length of the platform?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{520}
The length of a cuboid is increased by 40%, its width is halved, and its height is increased by 25%. What is the percentage change in the volume of the cuboid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5\%}
In the context of group theory, determine the largest prime number $p$ for which the multiplicative group $(U(p^2), \cdot \mod p)$, where $U(p^2)$ represents the group of units modulo $p^2$, is isomorphic to a subgroup of the symmetric group $S_{50}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
某班有 47 个学生，所用教室有 6 排，每排有 8 个座位，用 (i,j) 表示位于第 i 排第 j 列的座位. 新学期准备调整座位，设某学生原来的座位为 (i,j)，如果调整后的座位为 (m,n)，则称该生作了移动 [a,b]=[i-m,j-n]，并称 a+b 为该生的位置数. 所有学生的位置数之和记为 S，求 S 的最大值与最小值之差.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{de} f$ 是偶数的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing each watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on a piece of paper the weights of the scales he used. What is the smallest number of different weights that could appear in his records if the mass of each weight is an integer kilogram?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Given the function $f(x)={2^x}+\frac{x}{3}$, find the interval in which the zero point $x_0$ of the function lies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, -1)}
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$, and let $x^{*2}$ be defined as $x^{*2} = x*x$. Then, the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let \( n \) be the 200th smallest positive real solution to the equation \( x - \frac{\pi}{2} = \tan x \). Find the greatest integer that does not exceed \( \frac{n}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{314}
Suppose a Markov chain is given as follows. $p_{ii}=1-3a$ and $p_{ij}=a\ \forall i\ne j$ where $P=(p_{ij}), 1\le i,j\le 4$. Find $P_{1,1}^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{4}(1+3(1-4a)^n)\)}
An isosceles triangle \( ABC \) \((AB = BC)\) is inscribed in a circle. The diameter \( AD \) intersects the side \( BC \) at point \( E \), with \( DE : EA = k \). Find the ratio \( \frac{CE}{BC} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2k}{1+k}}
The words "PUNK CD FOR SALE" are painted on a clear window. Count how many of the letters in the sign look the same from both sides of the window due to being symmetrical either vertically or horizontally. Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Using Qin Jiushao's algorithm, calculate the value of v_3 when evaluating the polynomial f(x) = x^5 + 3x^4 - x^3 + 2x - 1 at x = 2. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find the smallest positive integer $x$ such that $10^x\equiv 1 \pmod{1997}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{998}
If John makes a contribution to a charity fund at school, the average contribution size will increase by 75% reaching $100 per person. If there were 10 other contributions made before John's, what is the size of his donation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$528.57}
If the function $f(x)=\frac{1}{3}x^3-ax^2+x-5$ has no extreme points, then the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1,1]}
There were 3 candidates in an election and they received some votes. The winning candidate got 65.21739130434783% of the total votes in that election. If the other two candidates received 5000 and 15000 votes respectively, how many votes did the first candidate receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37500}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\sin \frac{A}{2},\cos \frac{A}{2})$ and $n=(\cos \frac{A}{2},-\cos \frac{A}{2})$, and $2m\cdot n+|m|=\frac{ \sqrt{2}}{2}$, find $\angle A=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi }{12}}
## Problem Statement

Calculate the definite integral:

$$
\int_{2 \operatorname{arctan} 2}^{2 \operatorname{arctan} 3} \frac{d x}{\cos x(1-\cos x)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}+\ln2-\ln3}
Sara has 47 red flowers, 63 yellow flowers, 54 blue flowers, 29 orange flowers, and 36 pink flowers. She wants to make bouquets with the same number of each color flower in each bouquet. What is the greatest number of bouquets she can make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the smallest positive integer that satisfies the congruence $5x \equiv 17 \pmod{31}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Let $A, B, C$ be points in that order along a line, such that $AB = 20$ and $BC = 18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY \parallel BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{nice}
if its center is at the origin $(0,0)$ and it passes through at least one lattice point
(i.e. a point with integer coordinates).
Define the points $A = (20,15)$ and $B = (20,16)$. How many nice circles intersect the open segment $AB$?

For reference, the numbers $601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $691$ are the only prime numbers between $600$ and $700$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the Cartesian coordinate system $xOy$, the hyperbola $C$: $x^{2}- \frac{y^{2}}{b^{2}}=1$ ($b > 0$) has two asymptotes that intersect with the circle $O$: $x^{2}+y^{2}=2$ at four points sequentially named $A$, $B$, $C$, and $D$. If the area of rectangle $ABCD$ is $b$, then the value of $b$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}}
What is the maximal number of checkers that can be placed on an $8 \times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Two types of shapes composed of unit squares, each with an area of 3, are placed in an $8 \times 14$ rectangular grid. It is required that there are no common points between any two shapes. What is the maximum number of these two types of shapes that can be placed in the $8 \times 14$ rectangular grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}