inclusionAI/Ring-lite-rl-data huggingface dataset:
In a competition with a total of 4 multiple-choice questions, each question has three choices: $A, B,$ and $C$. For any three contestants participating, there is at least one question for which their answers are all different. What is the maximum number of participants allowed in the competition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
8.6. What is the maximum number of sides a coin can have as a projection of a convex polyhedron with $n$ faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n-4}
A person having contracted conjunctivitis infects a total of 144 people after two rounds of infection. Determine the average number of people each infected person infects in each round.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A *short-sighted* rook is a rook that beats all squares in the same column and in the same row for which he can not go more than  $60$ -steps. 
What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a  $100\times 100$  chessboard.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178}
Given the cone $z^2=x^2+y^2$, $z\geq 0$ and the cylinder $z^2+y^2=64$, find the surface area of the section of the cone inside the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64π}
There are some positive factors of 15 and another number, and these are also the common factors in these numbers. The total number of these factors is 3. What is the second number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In a warehouse, transporting out 30 tons of goods is recorded as -30 tons. Therefore, transporting in -20 tons of goods represents ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{transporting out 20 tons}
Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^{\circ}$ and $\angle DBA = 70^{\circ}$, then find the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
A cylinder has a volume of $72\pi$ $\text{cm}^3$. Calculate the combined volume of a cone with the same height as the cylinder and a hemisphere that fits exactly on the top of the cylinder's open end. The radius of the hemisphere is equal to the radius of the cylinder and the cone. Express your answer in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72\pi \text{ cm}^3}
Two semicircles of diameters 2 and 4 are placed such that the smaller is inside the larger one and their diameters are along the same line. A square of side 1 is inscribed in the larger semicircle such that one of its sides is along the diameter of the semicircles. Calculate the area of the lune formed by the space in the smaller semicircle that is outside the larger semicircle and subtract the area covered by the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}\pi - 1}
The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Given the quadratic equation $ax^{2}+bx+c=0$ with $a > 0$ and $b, c \in \mathbb{R}$, and the roots of the equation lying in the interval $(0, 2)$, determine the minimum value of the real number $a$ given that $25a+10b+4c \geqslant 4$ for $c \geqslant 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{25}}
11 people were standing in line under the rain, each holding an umbrella. They stood so close together that the umbrellas touched each other. Once the rain stopped, people closed their umbrellas and maintained a distance of 50 cm between each other. By how many times did the length of the queue decrease? Assume people can be considered as points and umbrellas as circles with a radius of 50 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.2}
(5 points) Out of the eight playing cards which are Hearts of 2, 3, 4, 5 and Clubs of 2, 3, 4, 5, four cards are drawn and placed in a row. If the sum of the numbers on the drawn cards is 14, then the total number of different arrangements is ___ (answer in numerals).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{396}
the radius of a semi circle is 7 cm then its perimeter is ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
58 flags were planted at equal intervals on both sides of the 191.8 meters (m) long road. If flags are planted at both the beginning and end of the road, find the distance between the flags in meters (m).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.85}
a trader cheats both his supplier and customer by using faulty weights . when he buys from the supplier , he takes 10 % more than the indicated weight . when he sells to his customer , he gives the customer a weight such that 50 % of that is added to the weight , the weight claimed by the trader is obtained . if he charges the cost price of the weight that he claims , find his profit percentage .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
Mona plays a multiplayer video game on the weekends. She joins groups with four other players to complete objectives in the game. The past weekend, she joined 9 groups and grouped with 33 unique players. In one of the groups, she had grouped with a certain number of players before that weekend, and another group included one person she had grouped with before. How many players had Mona grouped with before in the first group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the sum of the interior numbers of the eighth row of Pascal's Triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, find how many integers $P(4)$ can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
3. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of collected apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned out that each of the children still had a whole percentage, but now of the remaining number of apples. What is the minimum number of apples that could have been collected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A circle with a radius of \(1 + \sqrt{2}\) is circumscribed around an isosceles right triangle. Find the radius of a circle that is tangent to the legs of this triangle and internally tangent to the circumscribed circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$, such as $\{3, 31\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
求 $\sum\limits_{i=1}^{2024}\left[\frac{19^{i}}{20}\right]$ 模 $7$ 的余数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2 + 4px - p + 1$ an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that $k \in \mathbb{Z}$, if the curve $x^{2}+y^{2}=k^{2}$ has no intersection points with the curve $xy=k$, then $k=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Given are sheets and the numbers  $00, 01, \ldots, 99$  are written on them. We must put them in boxes  $000, 001, \ldots, 999$  so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Given $x$, $y$, and $a \in R^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{3}}
In the binomial expansion of (x^2 - $\frac {3}{x}$)^n, the binomial coefficient of the fifth term is the maximum and unique. Find the value of $C_{ n }^{ 0 }$ - $\frac {1}{2}$ $C_{ n }^{ 1 }$ + $\frac {1}{4}$ $C_{ n }^{ 2 }$ - ... + (-1)^n $\frac {1}{2}$ $C_{ n }^{ n }$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{256}}
a train 600 m long can cross an over brdge of 100 m long in 70 sec and then find the speed of the train ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the range of the function
\[f(x) = \sqrt{x^2 - 10x + 34} - \sqrt{x^2 + 4}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-5,\sqrt{26}]}
Given the proposition ∀x > 1, x² + x + 1 > 0, what is the negation of this proposition? Express your answer as a logical statement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃x > 1, x² + x + 1 ≤ 0}
If the complex number \( z \) satisfies \( |z| = 2 \), then the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \) is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Find the biggest real number  $C$ , such that for every different positive real numbers  $a_1,a_2...a_{2019}$  that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
In $\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively. Angle $\angle BAC = 60^\circ$ and $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
已知奇素数 $x 、 y 、 z$ 满足
$$
x\left|\left(y^{5}+1\right), y\right|\left(z^{5}+1\right), z \mid\left(x^{5}+1\right)
$$
求乘积 $x y z$ 的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013}
$已知f(x) = (x-1)e^{x} + \frac{1}{2}ax^{2}.$
$若f(x)有2个不同零点，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(0,+\infty)$}
The trains. At Würzltown station, an old lady looked out of the window and shouted:
- Conductor! How long is the trip from here to Madville?
- All trains take 5 hours in either direction, ma'am, he replied.
- And how many trains will I encounter on my way?

This seemingly odd question puzzled the conductor, but he readily answered:
- Trains from Würzltown to Madville and from Madville to Würzltown depart at five minutes past the hour, every hour, with an exact interval of one hour.

The old lady prompted one of her compartment neighbors to find the answer to her question.

How many trains will she encounter on her way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:
\[
A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1
\]
Determine the value of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]

[i]Proposed by Daniel Zhu.[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167}
Suppose  $n$  is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least  $n$ . Express the  $n-$ th smallest number in this set in terms of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n + \sqrt{n} - 1)^2}
In a single-round chess tournament with 10 players, how many rounds are needed at the minimum for a clear winner to emerge ahead of time? (In each round, participants are paired up. A win awards 1 point, a draw awards 0.5 points, and a loss awards 0 points).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given $m \neq n$, $x = m^4 - m^3n$, $y = nm^3 - n^4$, compare the size of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x > y}
A sequence of positive integers is to be placed in each box. The product of any four adjacent integers is always 120. Given the sequence:

|   |   | 2 |   |   | 4 |   | x |   |   | 3 |   |   |

Find the value of x. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Inside a semicircle with a radius of 12, there is a circle with a radius of 6 and a small semicircle, all pairwise tangent to each other, as shown in the diagram. Find the radius of the small semicircle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
已知复数$z _ { 1 } 、 z _ { 2 }$满足$\vert z _ { 1 } \cdot z _ { 2 } \vert = 3 , z _ { 1 } + z _ { 2 } = 2 i$，求$\vert z _ { 1 } \vert$的最大值及最小值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left|z_1\right|$ 的最大值为 3 , 最小值为 1 .}
How many single-digit odd numbers and two-digit even numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
6. In square $A B C D$, there is a point $M$, point $P$ is on $A B$, point $Q$ is on $C D$, and $\angle A P M=150^{\circ}$, $\angle M Q C=35^{\circ}$, then $\angle P M Q=$ $\qquad$ $\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Given the vector $\overrightarrow{a}=(2,l)$ and the conditions $\overrightarrow{a} \cdot \overrightarrow{b}=10$ and $|\overrightarrow{a}+\overrightarrow{b}|=5 \sqrt {2}$, find $|\overrightarrow{b}|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
There are 2 boys and 3 girls, a total of 5 students standing in a row. If boy A does not stand at either end, and exactly 2 of the 3 girls stand next to each other, then the number of different arrangements is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33.33\%}
Let point $P$ lie on the curve $y= \frac {1}{2}e^{x}$, and point $Q$ lie on the curve $y=\ln (2x)$. Find the minimum value of $|PQ|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {2}(1-\ln 2)}
The factors of $x^4+64$ are:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(x^2-4x+8)(x^2+4x+8)$}
If the positive numbers $a$ and $b$ satisfy the equation $2+\log_{2}a=3+\log_{3}b=\log_{6}(a+b)$, find the value of $\frac{1}{a}+\frac{1}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Let $a, b$ be two real numbers such that \(\sqrt[3]{a} - \sqrt[3]{b} = 10\) and \(ab = \left( \frac{8 - a - b}{6}\right)^3\). Find \(a - b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Determine the value of $f_{1991}(2^{1990})$ where for a given positive integer $k$, the function $f_1(k)$ is defined as the square of the sum of its digits. The sequence is recursively defined as $f_{n+1}(k) = f_1(f_n(k))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
A box contains 3 black balls and 4 white balls. Balls are drawn randomly one at a time without replacement. The process continues until all balls of one color have been drawn. Calculate the probability that the last ball drawn is a black ball. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{7}}
There is a mathematical operator "$\odot$" that satisfies the following equations: $2 \odot 4=8, 4 \odot 6=14, 5 \odot 3=13, 8 \odot 7=23$. According to this rule, what is $9 \odot 3=?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
2. (15 points) The bisectors of the obtuse angles at the base of a trapezoid intersect on the other base of the trapezoid. Find the area of the trapezoid if its height is 12 cm, and the lengths of the bisectors are 15 cm and $13 \mathrm{~cm}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{260.4}
Compute the exact value of the following limit: $$\lim_{n \to \infty} \frac{{n^2 \choose n}}{{n^2+n-1 \choose n}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1}}
Let  $\Omega$  be a circle with radius  $18$  and let  $\mathcal{S}$  be the region inside  $\Omega$  that the centroid of  $\triangle XYZ$  sweeps through as  $X$  varies along all possible points lying outside of  $\Omega$ ,  $Y$  varies along all possible points lying on  $\Omega$  and  $XZ$  is tangent to the circle. Compute the greatest integer less than or equal to the area of  $\mathcal{S}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{904}
A square with a side of 100 was cut into two equal rectangles. They were joined together as shown in the picture. Find the perimeter of the resulting figure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Bag A initially contains 3 white balls and 5 black balls, while bag B initially contains 4 white balls and 6 black balls. A ball is randomly taken from bag A and put into bag B. After thoroughly mixing, a ball is then randomly taken from bag B and put back into bag A. Find the probability that the number of white balls in bag A does not decrease. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {35}{44}}
$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. Determine the largest possible value of $n$ if the last number in the row is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying \[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\] is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$929$}
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Define the annoyingness of a permutation of the first \( n \) integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence \( 1, 2, \ldots, n \) appears. For instance, the annoyingness of \( 3, 2, 1 \) is 3, and the annoyingness of \( 1, 3, 4, 2 \) is 2. A random permutation of \( 1, 2, \ldots, 2022 \) is selected. Compute the expected value of the annoyingness of this permutation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011.5}
Origami. Mitya plans to fold a square sheet of paper \(ABCD\). Mitya calls a fold beautiful if side \(AB\) intersects side \(CD\) and the four resulting right triangles are congruent.

Before this, Vanya chooses a random point \(F\) on the sheet. Find the probability that Mitya can make a beautiful fold passing through point \(F\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
The weight of cows of the Red Gorbatov breed is a random variable \(X\) distributed according to the normal law with a mean \(a = 470 \, \text{kg}\) and a standard deviation \(\sigma = 30 \, \text{kg}\). What is the probability that two out of three randomly selected cows will have a weight more than 470 kg and less than 530 kg?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.357}
A function f(n) has the property that f(f(n)) + f(n) = 2n + 3, and f(0) = 1. If f(2015) = 2016, what is the value of n for which we are finding f(n)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
A primary school conducted a height survey. For students with heights not exceeding 130 cm, there are 99 students with an average height of 122 cm. For students with heights not less than 160 cm, there are 72 students with an average height of 163 cm. The average height of students with heights exceeding 130 cm is 155 cm. The average height of students with heights below 160 cm is 148 cm. How many students are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{621}
Suppose that the number $\sqrt{7200} - 61$ can be expressed in the form $(\sqrt a - b)^3,$ where $a$ and $b$ are positive integers. Find $a+b.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
On the board, we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold:

1. The sum of every $40$ consecutive numbers is equal to $0$.
2. The sum of every $42$ consecutive numbers is not equal to $0$.

We denote by $S_n$ the sum of the $n$ numbers on the board. Find the maximum possible value of $S_n$ for all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}