inclusionAI/Ring-lite-rl-data huggingface dataset:
In a knockout tournament with $2^n$ equally skilled players, $S_1, S_2, ..., S_{2^n}$, are participating. In each round, players are divided into pairs at random, and the winner from each pair moves on to the next round. If $S_2$ reaches the semi-final, then the probability that $S_1$ wins the tournament is $\frac{1}{20}$. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A train 120 meters long running at the rate of 54 kmph takes 51.99584033277338 seconds to cross a bridge of a certain length. How long is the bridge in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{659.94}
Find a primitive element $k$ of the field $F=\mathbb{Q}(i,\sqrt{5}+i)$. Provide your answer as a single element that generates the field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5} + i}
Find the monic polynomial of least degree in \( \mathbb{R}[x] \) that is a square root of \(-4\) modulo the ideal generated by \((x^2 + 1)^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 + 3x}
Determine the number of real solutions  $a$  to the equation:
\[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \]
Here, if  $x$  is a real number, then  $[\,x\,]$  denotes the greatest integer that is less than or equal to  $x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate $\displaystyle\lim_{N \rightarrow \infty}\int_{|z|=2}\frac{1}{P^3(z) - 1}dz$, where $P_N(z) = \displaystyle\sum_{n = 0}^N\frac{z^n}{n!}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2\pi i}{3}$}
In a circle, $AB$ is the diameter. Tangents $AD$ and $BC$ are drawn such that lines $AC$ and $BD$ intersect at a point $E$ on the circle. Given that $|AD|=a$ and $|BC|=b$ with $a \neq b$, determine the diameter of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{ab}}
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers, find \( p + q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2734}
We consider an  $n \times n$  table, with  $n\ge1$ . Aya wishes to color  $k$  cells of this table so that that there is a unique way to place  $n$  tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of  $k$  for which Aya's wish is achievable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1)}{2}}
How many lines are tangent to the circle $x^2+(y-2)^2=1$ and have equal intercepts on both coordinate axes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,\,$ and $\angle B=90^\circ.\,$ Then the area of the set of all fold points of $\triangle ABC\,$ can be written in the form $q\pi-r\sqrt{s},\,$ where $q, r,\,$ and $s\,$ are positive integers and $s\,$ is not divisible by the square of any prime. What is $q+r+s\,$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{597}
Determine all values of $x$ for which the series \( \sum^\infty_{n=1}\Big(1+\frac{1}{2}+\ldots+\frac{1}{n}\big)\frac{\sin nx}{n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Let $ABCD$ be a unit square, and let $E$ be a point on segment $AC$ such that $AE = 1$. Let $DE$ meet $AB$ at $F$ and $BE$ meet $AD$ at $G$. Find the area of $CFG$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{1}{2}$}
Given the polynomial \( P(x) = 99(x^{101} - 1) - 101(x^{99} - 1) \), find the value of \( Q(1) \), where \( Q(x) \) is the quotient when \( P(x) \) is divided by \((x-1)^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{656601}
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it is occupied.One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
102
}
Given that \(4a^{2}+kab+9b^{2}\) is a perfect square, find the value of the constant \(k\). Express your answer as a numerical value, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±12}
In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that could have been made?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Calculate the surface integral \( \iint_S E \cdot \vec{n} \, d\sigma \) where \( S \) is the parametric surface given by \( X(s,t) = [st, s^2, t^2]^T \) for \( 0 \leq s \leq t \leq 1 \), and \( E \) is the vector field \( E(x,y,z) = [3yz, zx, 2xy]^T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
1. Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.
Proposed by: Reagan Choi
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{7}}
If the function $f(x) = (m-1)x^{\alpha}$ is a power function, then the graph of the function $g(x) = \log_{a}(x-m)$ (where $a > 0$, $a \neq 1$) passes through a fixed point A, whose coordinates are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 0)}
Given the implicit function defined by the equation \( y - \sin y = x \), find the value of the third derivative \( y'''(\pi) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
Given the function \( F(x) = \int_x^{\sin x} \left(\int_0^{\sin t} \sqrt{1+u^4} \,du\right)\,dt \), find \( F''(\pi) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} = \frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Calculate the integral: \[ \int_{0}^{2} \sqrt{1+x^3} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.241}
A tangent is drawn through a point P on the curve \(y=e^{-x}\). The tangent, the positive y-axis, and a horizontal line through P form a triangle. Determine the largest possible area of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e^2}}
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
Given a normal distribution of cheese weights X with mean 10 kg and standard deviation 2 kg, determine the weight threshold h such that the probability of a cheese weighing more than h is at least 0.991. Calculate the value of h.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.27}
Given the function $f(x)=x^{2}\ln x+ax$ has a decreasing interval, the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2e^{-\frac{3}{2}})}
Genesis has a wall clock in his home that rings following a specific pattern: it rings every 2 hours for the first 12 hours of the day, then every 4 hours for the remaining 12 hours. If it will first ring at 1 A.M, how many times will it ring in a day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
已知 $\sin x+\sin y=\frac{1}{3}, \cos x-\cos y=\frac{1}{5}$, 求 $17\sin (x-y)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-15}
Let $m$ be the largest real solution to the equation $\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$ There are positive integers $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$263$}
Given a set of data: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ forms an arithmetic sequence with a common difference of $d$, and the variance of this set of data equals $1$, then the common difference $d$ equals ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\frac{1}{2}}
Trent lives in a city where each block is 50 meters long. He walked 4 blocks from his house to the bus stop at a speed of 4 km/h. He then rode the bus 7 blocks to the library at a speed of 15 km/h. After spending time in the library, he picked up a bicycle from a sharing station and rode 5 blocks to the park at an average speed of 10 km/h. Later, he came home the same way, but this time he jogged 5 blocks back to the park's bike sharing station at a speed of 6 km/h, took the bus for 7 blocks and then walked the final 4 blocks to his house. In total, it took him 2 hours to complete the entire journey. What was his average speed during the entire trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21840}
A vessel with a capacity of 100 liters is filled with a brine solution containing 10 kg of dissolved salt. Every minute, 3 liters of water flows into it, and the same amount of the resulting mixture is pumped into another vessel of the same capacity, initially filled with water, from which the excess liquid overflows. Determine the time at which the amount of salt in both vessels is equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{100}{3}}
The solution to the inequality
\[\frac{x + c}{x^2 + ax + b} \le 0\]is $x \in (-\infty,-1) \cup [1,2).$  Find $a + b + c.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Tony made a sandwich with two slices of bread for lunch every day this week. On Saturday, he was extra hungry from doing yard work and made two sandwiches. Tony started with a loaf of bread with a certain number of slices and now has 6 slices left. How many slices were in the loaf Tony started with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $f(x)$ be a function defined as:

$$f(x)=\int^{x}_{\gamma}\frac{\delta}{t(t^{\delta}+2+t^{-\delta})}$$

where $\gamma, \delta \in \mathbb{R^+}$. Calculate the limit of $f(x)$ as $x$ approaches infinity, expressed in terms of $\gamma$ and $\delta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{\gamma^{\delta}+1}\)}
Find the value of \( a \) such that the equation \( a^x = |x - 4| - |2x - 8| \) has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
1. In a tennis tournament, 1152 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by drawing lots among participants with the same number of points (those who do not find a pair are awarded a point without playing). A player is eliminated after the second loss. The tournament continues as long as at least one pair of opponents can be formed. How many rounds will have to be played?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the function $f(x)=x^3-3x-a$, find the value of $(M-N)$, where $M$ and $-N$ are the maximum and minimum values of $f(x)$ on the interval $[0, 3]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
If \( P \) is the circumcenter of \( \triangle ABC \), and
\[
\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, \quad \text{with} \quad \angle C=120^{\circ},
\]
determine the value of the real number \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages starting from page two have a red mark. How many pages of the encyclopedia have both colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
The main line train starts at 5:00 am and the harbor line train starts at 5:02 am. Each train has a frequency of 10 minutes. If a guy goes in the morning at a random time, what is the probability of him getting the main line train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Tom and Jerry are running on a figure-eight track in the same direction at constant speeds. Initially, Jerry is directly above Tom. After 20 minutes, Tom is directly above Jerry, and neither has completed a full lap. Another 15 minutes later, Tom returns to his starting point. Determine the time from the start when Tom will catch up with Jerry.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given that the radius of the incircle of an equilateral triangle \(ABC\) is 2 and the center of this circle is \(I\). If point \(P\) satisfies \(P I = 1\), what is the maximum value of the ratio of the areas of \(\triangle APB\) to \(\triangle APC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 + \sqrt{5}}{2}}
What range of values must the expression \(\lambda = \frac{\sum_{i=1}^{4} S_{i}}{S}\) satisfy, where \(S_{1}, S_{2}, S_{3}, S_{4}\) are the face areas of a tetrahedron, with the largest of these areas being \(S\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 < λ ≤ 4}
What is the Gromov-Witten invariant \(\langle p, p, \ell\rangle_{0, 1}\) that counts degree 1, genus 0 curves in \(\mathbb{CP}^2\), where \(p\) and \(\ell\) are the cohomology classes Poincaré dual to a point and a line, respectively?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive? Express your answer as a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2148}
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \frac{(2n)!}{2^n \cdot n!} \right)^{\frac{1}{n}} \left( \tan{\left( \frac{\pi\sqrt[n+1]{(n+1)!}}{4\sqrt[n]{n!}} \right)} - 1 \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{e}}
Determine the codimension of the commutator ideal of \( \mathfrak{gl}(V) \) in \( \mathfrak{gl}(V) \) when \( V \) is an infinite-dimensional vector space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( N \) be the set of natural numbers, \( R \) be the set of real numbers, and \( S \) be the set of functions \( f: N \rightarrow R \) that satisfy the following two conditions:
1. \( f(1) = 2 \);
2. \( f(n+1) \geq f(n) \geq \frac{n}{n+1} f(2n) \) for \( n = 1, 2, \ldots \)

Find the smallest natural number \( M \) such that for any \( f \in S \) and any \( n \in N \), \( f(n) < M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1944}
Using the vertices of a single rectangular solid (cuboid), how many different pyramids can be formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Three numbers 1, $a$, 2 form a geometric sequence, then the real number $a$ = ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{2}}
A chess tournament was held among the actors of the Karabas Barabas theater. Each participant played with each of the others exactly once. For a win, one received one soldo, for a draw - half a soldo, and for a loss - nothing. It turned out that among any three participants, there is a chess player who earned exactly 1.5 soldos in games with the other two. What is the maximum number of actors that could have participated in this tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}\frac{n}{n+1}\left(\frac{2x+1}{x}\right)^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
若复数 $z_{1}, z_{2}$ 分别满足 $\left|z_{1}-3 \mathrm{i}\right|=2,\left|z_{2}-8\right|=1$ ，则在复平面上所有由复数 $z_{1}-z_{2}$ 对应的点围成的面积是 $x \pi$。求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How can I evaluate the integral $$\int_0^\frac{\pi}{2} x\ln(\sin x) \, dx$$ in terms of the Riemann zeta function $\zeta(3)$ and the natural logarithm of 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{7}{16}\zeta(3) - \frac{\pi^2}{8}\ln(2)\)}
There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are isosceles right triangles. If the area of square $ABCD$ is 25, find the area of $KLMN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
As a seasoned executive at a mainstream headphone company, Alex is skeptical about the potential sales of a new line of niche gaming headphones. The company currently sells 100,000 units of their mainstream headphones every month. They are considering launching the niche gaming headphones and project that they could sell 5% of the mainstream units in the first month.

If the company decides to launch the gaming headphones, they plan to increase production by 2% each month, based on the initial projected sales of the gaming headphones. Calculate how many units of the gaming headphones the company expects to produce in the third month after the launch.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5200}
若正整数 $n$ 使得方程 $x^{3}+y^{3}=z^{n}$ 有正整数解 $(x, y, z)$ ，称 $n$ 为 "好数"。则不超过 2019 的 "好数"的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1346}
A fly and $\mathrm{k}$ spiders are moving on a $2019 \times 2019$ grid. On its turn, the fly can move by 1 square and the $k$ spiders can each move by 1 square. What is the minimal $k$ for which the spiders are guaranteed to catch the fly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the maximum value for \(a, b \geq 1\) of the expression

$$
\frac{|7a + 8b - ab| + |2a + 8b - 6ab|}{a \sqrt{1 + b^2}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{2}}
Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$. What is the minimum possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?\Based on a proposal by Ahaan Rungta
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
210
}
There are 34 kindergartners, 48 first graders, and some second graders in Ms. Watson's class. Additionally, there are 5 third graders and 6 absent students who are a mix of first and second graders. Overall, Ms. Watson has 120 students in her class. Four new students of different grade levels are also joining the class this week. How many second graders are currently in Ms. Watson's class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
On each non-boundary unit segment of an  $8\times 8$  chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{7}}