inclusionAI/Ring-lite-rl-data huggingface dataset:
Given a regular polygon with $2n+1$ sides. How many triangles can be formed whose vertices are the vertices of the polygon and contain the center of the polygon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1)(2n+1)}{6}}
As shown in the diagram, in parallelogram \(ABCD\), \(EF \parallel AD\), \(GH \parallel AB\), the area of \(\triangle EBH\) is 6, the area of \(\triangle DGF\) is 8, and the area of \(\triangle ECG\) is 18. Find the area of parallelogram \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In an abandoned chemistry lab, Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant rate of $3\tfrac{1}{4}$ miles per hour for the first hour, at $3\tfrac{3}{4}$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4$}
On the edges \(AB\), \(BC\), and \(AD\) of the tetrahedron \(ABCD\), points \(K\), \(N\), and \(M\) are chosen, respectively, such that \(AK:KB = BN:NC = 2:1\) and \(AM:MD = 3:1\). Construct the section of the tetrahedron by the plane passing through points \(K\), \(M\), and \(N\). In what ratio does this plane divide edge \(CD\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:3}
双曲线$\frac{{{x}^{2}}}{{{a2}}}-\frac{{{y}^{2}}}{{{b2}}}=1(a>0,b>0)$的左焦点为$F(-2,0)$，点$A(0,\sqrt{5})$，点$P$为双曲线右支上的动点，且$\vartriangle APF$周长的最小值为8，则双曲线的离心率为______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The base of a rectangular prism is an equilateral triangle with sides of unit length. A plane intersects the lateral surface of the prism in an isosceles right triangle. What are the side lengths of this intersecting triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{3}{2}}, \sqrt{3}}
Let $N=\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9, inclusive, and $N$ is a 7-digit positive integer. If $N$ is divisible by 792, determine all possible ordered triples $(A, B, C)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(0,5,5),(4,5,1),(6,4,9)$}
In an institute, there are truth-tellers, who always tell the truth, and liars, who always lie. One day, each of the employees made two statements:

1) There are fewer than ten people in the institute who work more than I do.
2) In the institute, at least one hundred people have a salary greater than mine.

It is known that the workload of all employees is different, and their salaries are also different. How many people work in the institute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Given that $x > 2$, $y > 0$ and they satisfy $2^{x \cdot 2^{y}} = 16$, calculate the minimum value of $\frac {2}{x-2}$ + $\frac {2}{y}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the areas of the regions bounded by the curves given in polar coordinates.

$$
r=\cos 2 \phi
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{127}{924}}
In a chess tournament, 12 participants played. After the tournament, each participant compiled 12 lists. The first list includes only the participant himself, the second list includes himself and those he won against, the third list includes everyone from the second list and those they won against, and so on. The twelfth list includes everyone from the eleventh list and those they won against. It is known that for any participant, there is a person in their twelfth list who was not in their eleventh list. How many drawn games were played in the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Find the coefficient of the $x^{3}$ term in the expansion of $(x^{2}+1)^{2}(x-1)^{6}$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-32}
The graph of the linear function $y=kx+2$ intersects the $x$-axis at a point that is $2$ units away from the origin. The value of $k$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm1}
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50^\circ}
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Given the equation of the ellipse $x^{2}+2y^{2}=4$, calculate the coordinates of the foci. Express your answer as a pair of coordinates $(x, y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{2}, 0), (-\sqrt{2}, 0)}
In the figure, there are graphs of two linear functions \( y = f(x) \) and \( y = g(x) \), with the coordinates of their intersections with the axes indicated. Find the maximum point of the function \( y = f(x) \cdot g(x) \) (i.e., the value of \( x \) at which the function \( f(x) \cdot g(x) \) is maximized). If the answer is a fractional number, it must be recorded using a decimal point, e.g., \( 0.15 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given 2023 identical-looking coins, where two of them are counterfeit coins with the same weight and the remaining 2021 coins are genuine coins with the same weight, the counterfeit coins have a different weight than the genuine coins. Using a balance scale, what is the minimum number of weighings needed to determine whether a counterfeit coin or a genuine coin is heavier?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
M4. The squares on each side of a right-angled scalene triangle are constructed and three further line segments drawn from the corners of the squares to create a hexagon, as shown. The squares on these three further line segments are then constructed (outside the hexagon). The combined area of the two equal-sized squares is \(2018 \mathrm{~cm}^{2}\). What is the total area of the six squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8072}
Find the area of the region bounded by a function  $y=-x^4+16x^3-78x^2+50x-2$  and the tangent line which is tangent to the curve at exactly two distinct points.

Proposed by Kunihiko Chikaya
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1296/5}
It is known that point $M$ is a point on the parabola $y^2 = 4x$, and $F$ is the focus of the parabola. Point $A$ is on the circle $C$: $(x-4)^2 + (y-1)^2 = 1$. Find the minimum value of $|MA| + |MF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. If the measure of $\widehat{ABC}$ is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
The equation of a circle is given as $x^2+y^2-2x=0$. Find the coordinates of the center and the radius of the circle. Express your answer in the format: Center: (x, y), Radius: r.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Center: (1, 0), Radius: 1}
Using the digits 2, 3, 4, 5, and 6, form two three-digit numbers that are squares. What is the digit that is placed in the center of the grid? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle \(ABC\), the height \(AM\) is not less than \(BC\), and the height \(BH\) is not less than \(AC\). Find the angles of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^
\circ, 45^
\circ, 90^
\circ}
4 设X＝｛00,01，··，98,99｝是100个二位数码的集合，A是X的子集，满足：对任何一个由0到9中的数字构成的无穷序列中，都有两个相邻的数字组成的二位数码属于A，求｜A｜的最小值．（第52届莫斯科数学奥林匹克试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in  $3$ , Anne and Carl in  $5$ .  How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{59}{20}}
Five boys and six girls are to be seated in a row of eleven chairs so that they sit one at a time from one end to the other. The probability that there are no more boys than girls seated at any point during the process is  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Evaluate  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let  $p$  be a prime number. A rational number  $x$ , with  $0 < x < 1$ , is written in lowest terms. The rational number obtained from  $x$  by adding  $p$  to both the numerator and the denominator differs from  $x$  by  $1/p^2$ . Determine all rational numbers  $x$  with this property.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{p^2 - p - 1}{p^2}}
Two cars of length 120 m and some length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. They will be clear of each other in 19.99840012798976 seconds from the moment they meet. What is the length of the second car?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Find the largest real number $k$, such that for any positive real numbers $a,b$,
$$(a+b)(ab+1)(b+1)\geq kab^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{4}}
Given that \(a, b, c\) are positive integers satisfying
$$
a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120 \text {, }
$$
determine the maximum possible value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Two of the vertices of a cube are to be chosen at random. What is the probability that they will be the endpoints of a space diagonal of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.

At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them away!" Find the largest possible value of $N$, given that no one lied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A square is divided into  $25$  unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the equation $\left\lfloor \left\lfloor \frac{91}{n} \right\rfloor \div n \right\rfloor = 1$, where $n$ is a positive integer, find all possible values of $n$. Express your answer as a list of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7, 8, 9}
8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the second smallest number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Evaluate the expression \( \frac{\sum_{k=0}^{6}\csc^2\left(x+\frac{k\pi}{7}\right)}{7\csc^2(7x)} \) when \( x=\frac{\pi}{8} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
We have a 99x99 checkerboard. There is a fly on each square. In one step, each fly must move exactly one square diagonally (multiple flies can end up on the same square). After one step, what is the minimum number of empty squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $120^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:\sqrt{3}}
In an isosceles triangle, the base is twice the height. Find the angles of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^{\circ}, 45^{\circ}, 90^{\circ}}
Given a convex quadrilateral \(ABCD\) with side lengths \(AB = 7\), \(BC = 4\), \(CD = 5\), and \(DA = 6\), determine the range \((a, b]\) of its area \(S\). Then, find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{210}}
$A$ is located 60 kilometers west of $B$. Individuals A and B depart from location $A$ while individuals C and D depart from location $B$ at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After $n$ hours, B and C meet, and $n$ hours later, A catches up to D at location $C$. What is the distance between locations $B$ and $C$ in kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let  $2000 < N < 2100$  be an integer. Suppose the last day of year  $N$  is a Tuesday while the first day of year  $N+2$  is a Friday. The fourth Sunday of year  $N+3$  is the  $m$ th day of January. What is  $m$ ?

*Based on a proposal by Neelabh Deka*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6$}
Find the volume of the region given by the inequality
\[
|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 6.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{49}{216}$}
Solve the system of equations: \[ \begin{aligned} x + y + z &= 1 \\ x^3 + y^3 + z^3 &= 1 \\ xyz &= -16. \end{aligned} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 4, -4)}
Determine the minimum number of times you need to spin a bottle to have at least a 50% chance of pointing to each of the 10 people in a circle at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
On the Island of Misfortune, there are knights who always tell the truth and liars who always lie. One day, $n$ islanders gathered in a room.

The first one said: "Exactly every second person in this room is a liar."

The second one said: "Exactly every third person in this room is a liar."

and so on.

The person with number $n$ said: "Exactly every $(n - 1)$-th person in this room is a liar."

How many people could be in the room, knowing that not all of them are liars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the expected value of the absolute value of the second pivot in a 4x4 matrix after performing Gauss-Jordan elimination, given that the matrix elements are drawn from a uniform distribution over [0, 1] and the first pivot is fixed at 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{36}}
各位数字之和等于 12 的四位数共有多少个?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{342}
The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{555}
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter first. One morning, $M$ groups consisting of a total of $N$ people enter and sit down. Then, a single person walks in, and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of $M+N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given \( x_{0} > 0 \), \( x_{0} \neq \sqrt{3} \), a point \( Q\left( x_{0}, 0 \right) \), and a point \( P(0, 4) \), the line \( PQ \) intersects the hyperbola \( x^{2} - \frac{y^{2}}{3} = 1 \) at points \( A \) and \( B \). If \( \overrightarrow{PQ} = t \overrightarrow{QA} = (2-t) \overrightarrow{QB} \), then \( x_{0} = \) _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Determine the maximum dimension of the annihilator \( W^\circ \) of the subspace \( W = \{ A^m ; m \geq 0 \} \) within the vector space \( V = M_n(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - 1}
Find the longest finite arithmetic progression with a common difference of 6 consisting of prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 11, 17, 23, 29}
In a doctor's clinic, patients arrive at an average rate of 10 per hour, following a Poisson distribution. The time taken for each patient's consultancy, which is exponentially distributed, averages 6 minutes (or 0.1 hour). The doctor doesn't attend to any patient if there are already 10 patients waiting. The system can be modeled as an M/M/1/11 queue (1 doctor, 10 waiting). Calculate the expected number of patients waiting in the queue at steady state.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{55}{12}}
Given the sets $S=\{x\mid (x-2)^{2} > 9\}$ and $T=\{x\mid a < x < a+8\}$, with $S\cup T=\mathbb{R}$, find the range of values for the real number $a$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3,-1)}
Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
p3. A multi-set of 11 positive integers has a median of 10, a unique mode of 11, and a mean of 12. What is the largest possible
 number that can be in this multi-set? (A multi-set is a set that allows repeated elements.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Determine the computational complexity, in terms of steps, for solving a linear system $Ax = b$ where $A \in \mathbb{R}^{n \times n}$ is a rank-deficient matrix with rank $r < n$. Express the complexity in terms of $r$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2 r)}
6.33 8 people participate in a chess round-robin tournament, where the winner gets 1 point, a draw gets 0.5 points, and the loser gets 0 points. It is known that the 8 people's scores are all different, and the second-place player's score is the sum of the scores of the last 4 players. What was the result of the match between the third-place player and the seventh-place player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-0}
Let  $1 = d_1 < d_2 < ...< d_k = n$  be all natural divisors of the natural number  $n$ . Find all possible values ​​of the number  $k$  if   $n=d_2d_3 + d_2d_5+d_3d_5$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k \in \{8, 9\} }
5. (COL) A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.8\cdot10^{-37}}
Three circles with radius one pass through a common point. Their second points of intersection are \( A \), \( B \), and \( C \). What is the radius of the circle \( ABC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A plane cuts a sphere of radius  $ 1$  into two pieces, one of which has three times the surface area of the other. What is the area of the disk that the sphere cuts out of the plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
What is the largest 4-digit integer congruent to $7 \pmod{19}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9982}