inclusionAI/Ring-lite-rl-data huggingface dataset:
What is the maximum number of idempotent and linearly independent matrices in $M_n(F)$ (considered as a vector space over the field $F$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Find the distance from the ellipsoid \(x^2 + y^2 + 4z^2 = 4\) to the plane \(x + y + z = 6\). Use the method of Lagrange multipliers to solve the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
A square is divided into 2022 rectangles, with no two rectangles sharing an interior point. What is the maximum number of distinct lines that can be formed by the sides of these rectangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2025}
There are 1000 toy bears in a row. The bears come in three colors: white, brown, and black. Among any three consecutive bears, there is at least one bear of each color. Iskander tries to guess the colors of the bears. He made five guesses:

- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
- The 400th bear from the left is black;
- The 600th bear from the left is brown;
- The 800th bear from the left is white.

It turns out that exactly one of his guesses is wrong. What could be the number of the bear whose color Iskander did NOT guess correctly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\frac{93}{125}$ . Find the sum of all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
Determine the infimum of the set \( \{\lvert\lvert f\rvert\rvert_{\infty} : f \in F\} \), where \( F = \{ f \in C[0,1] : f(0)=1, \int_0^1 (1-t)f(t)dt =2 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of homomorphisms from the ring of Gaussian integers \( \mathbb{Z}[i] \) onto the ring \( \mathbb{Z}_{13}(i) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the covariance between the first and third order statistics of three independent random variables with a uniform distribution on the interval $[0,1]$. The variance of the $k$th order statistic is given by $\frac{k(4-k)}{80}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{80}}
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60718293}
An empty fuel tank is filled with brand Z gasoline. When the tank is 1/4 empty, it is filled with brand Y gasoline. When the tank is half empty again, it is filled with brand Z gasoline. When the tank is half empty again, it is filled with brand Y gasoline. At this time, what percent of the gasoline in the tank is brand Z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.75\%}
In a village with $N$ people, each person owns one dog. A certain number of dogs, $K$, have contracted rabies. Every morning, villagers take their dogs for a walk and can identify which dogs, except their own, have rabies. They agree to shoot their own dog at night if they can conclude it has rabies. The disease is not contagious, and $K$ is constant with $K \geq 1$.

On the first night, no dogs are shot. On the second night, no dogs are shot. On the third night, no dogs are shot. On the fourth night, several gunshots are heard.

How many dogs were killed on the fourth night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that the graph of the power function $y=(m^{2}-2m-2)x^{m^{2}+4m}$ is symmetric with respect to the origin and does not intersect the $x$-axis or $y$-axis, determine the value of the integer $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the operator norm of $R_t$ as an element of $M_2(A)$, where $A$ is a normed real vector space and $A^2$ is equipped with a norm derived from an inner product satisfying the parallelogram identity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A quadrilateral $ABCD$ is inscribed in a circle of radius $R.$ It is known that $\measuredangle ADB = 90^\circ$ and the distance between the centers of the circles inscribed in the triangles $ADB$ and $BCD$ is $6.$ Calculate $R.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a random sample $X_1$ and $X_2$ of size $n=2$ from a uniform distribution on the interval $[0, \theta]$, determine the efficiency of the maximum likelihood estimator of $\theta$ relative to the estimator $\frac{3}{2} \bar{X}$, where $\bar{X}$ is the sample mean.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{16}}
An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{2}}
In the city of liars and knights, there are 366 inhabitants, each born on a different day of a leap year. All inhabitants of the city answered two questions. To the question "Were you born in February?" 100 people answered affirmatively, and to the question "Were you born on the 30th day?" 60 people answered affirmatively. How many knights were born in February?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given that the cross-section of a cylinder is a square with side length $2$, what is the surface area of the cylinder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
In a finite graph represented by the adjacency matrix \( B \) and transition matrix \( P \), calculate the probability of reaching state D before reaching state E, starting from state A. The matrices are given as follows:

\[ B = \begin{bmatrix} 
0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\
\frac{1}{3} & 0 & \frac{1}{3} & 0 & \frac{1}{3} \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{2} & 0 & 0 & 0 & \frac{1}{2} \\
0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\
\end{bmatrix} \]

\[ P = \begin{pmatrix} 
0 & 1 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 \\
\end{pmatrix} \]

Determine the probability \( p_A \) of reaching state D before state E from state A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{8}}
A right circular cylindrical tank lies on its side with a length of 12 feet and a radius of 4 feet. The water level in the tank is 18 inches above the lowest part. Calculate the volume of water that must be drained from the tank. Round your answer to the nearest tenth of a cubic foot.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78.3}
Rosencrantz and Guildenstern are playing a game where they repeatedly flip coins. Rosencrantz wins if 1 heads followed by 2009 tails appears. Guildenstern wins if 2010 heads come in a row. They will flip coins until someone wins. What is the probability that Rosencrantz wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{\frac{2^{2009}-1}{3 \cdot 2^{2008}-1}}$}
Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1956}
For which values of \( n \) does the cyclotomic extension \( \mathbb{Q}(\zeta_n) \) over \( \mathbb{Q} \) contain the imaginary unit \( i \)? Determine the condition on \( n \) such that \( i \in \mathbb{Q}(\zeta_n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \mid n}
Given a twice differentiable function $k: \Bbb R \to \Bbb R$ such that $k(a) = k(e) = 0$, $k(b) = k(d) = 2$, and $k(c) = 1$, with $a < b < c < d < e$, determine the minimum number of solutions for the equation $(k'(x))^2 + k(x)k''(x) = 0$ within the interval $[a, e]$. Find the minimum number of solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the determinant of the matrix $A = I + yx^H$, where $x, y \in \mathbb{C}^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + x^H y}
Find the general form of all $4 \times 4$ matrices that commute with every other $4 \times 4$ matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda I_4}
Determine whether the set \( X = \{ f \in \mathbb{L}_1(\mathbb{R}^n) \mid m(f^{-1}((0,\infty))) = 0 \} \) is open, closed, or neither in \( \mathbb{L}_1(\mathbb{R}^n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{closed}
For a 3x3 matrix \( A \) with the characteristic polynomial \( c^3 - 3c^2 + 4 = 0 \), find the trace of the adjoint of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq a_{k} \leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1540}
Given natural numbers \( m \) and \( n \). There are two piles on the table: the first pile contains \( n \) stones, and the second pile contains \( m \) stones. Petya and Vasya play the following game. Petya starts. In one move, a player can break one of the available piles on the table into several smaller piles. The player who cannot make a move loses. For which values of \( m \) and \( n \) can Petya ensure victory regardless of Vasya's play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \neq n}
Find the surface area of the part of the sphere \(x^2+y^2+z^2=a^2\) that lies inside the cylinder \(x^2+y^2=ax\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a^2(\pi - 2)}
Find a non-zero solution to the equation \(3\sin(\sqrt{x})=x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.966}
Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, and the year 2013 is the first time in his life he would experience a year with no repeated digits, how old is he in 2013?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
$\begin{array} {l}\text{In the convex quadrilateral}ABCD,\mathrm{point}X\text{is selected on side}AD,\text{and the diagonals intersect at point}E.\text{It is known that} \\ AC=BD,\angle ABX=\angle AXB=50^o,\angle CAD=51^o,\angle AED=80^o.\text{Find the value of angle}\angle AXC. \end{array}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
80°}
Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is $2018 \mathrm{~cm}$. The sum of all the distances between the second point and the other points, including the first one, is $2000 \mathrm{~cm}$. What is the distance between the first and second points? Express your answer in centimeters (cm) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the iterated integral \( \int_{0}^{2}\left (\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2} \, dy\right ) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{9}}
What is the expected value of a random variable $X$ that follows the negative binomial distribution with parameters $r$ and $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{r}{p}}
Given a twice differentiable function $f(x)$ with $f(a) = 0, f(b) = 2, f(c) = -1, f(d) = 2, f(e) = 0$, where $a < b < c < d < e$, determine the minimum number of zeros of the function $g(x) = (f'(x))^2 + f(x)f''(x)$ in the interval $[a, e]$. What is this minimum number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
You and a friend are playing a game with a fair coin. You go first, and you each take turns flipping the coin, recording the result each time. A player wins the game when they get two consecutive heads. What is the probability that you win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Friends Vasya, Petya, and Kolya live in the same house. One day, Vasya and Petya set off to go fishing at the lake on foot. Kolya stayed home, promising his friends to meet them on the way back with a bicycle. Vasya was the first to head back home, and at the same time, Kolya set off to meet him on the bicycle. Petya, traveling at the same speed as Vasya, left the lake for home at the moment Vasya and Kolya met. Kolya, after meeting Vasya, immediately turned around and brought Vasya home on the bicycle, then immediately set off again on the bicycle towards the lake. Upon meeting Petya, Kolya once again turned around and brought Petya home. As a result, the time Petya spent traveling from the lake to home was $5 / 4$ of the time Vasya took for the same journey. How much slower would Vasya have been if he walked the entire way home by himself? (8 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the area of the region where the polar curves \( r = 2 - \cos(\theta) \) and \( r = 3 \cos(\theta) \) intersect, given that the intersection angles are \( \theta = \pm \frac{\pi}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
Find the value of $x$ such that \( \sum_{k=1}^{2014} k^k \equiv x \pmod{10} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( A \) be the sum of the decimal digits of \( 4444 \, 4444 \), and \( B \) be the sum of the decimal digits of \( A \). Find the sum of the decimal digits of \( B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Compute the Lebesgue measure of the set \( S = \{(x, y, z) \in \mathbb{R}^3 : x \geq 1, \ y^2 + z^2 \leq \frac{1}{x^2} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Given a right triangular prism ABC-A1B1C1 with volume V, points P and Q lie on the edges AA1 and OC1 respectively, such that AP = C1Q. Calculate the volume of the quadrangular pyramid B-APQC as a fraction of the original volume V. Express your answer in the form of V/n, where n is the denominator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{V}{3}}
Evaluate the limit \( L = \lim\limits_{n\to\infty} \frac{2^{n^{1.1}}}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{203}
A positive integer \( b \) is called good if there exist positive integers \( 1 = {a}_{1},{a}_{2},\ldots ,{a}_{2023} = b \) such that \( \left| {{a}_{i + 1} - {a}_{i}}\right|  = {2}^{i} \) . Find
 the number of the good integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2^{2021}$}
There are 6 movie tickets numbered 1, 2, 3, 4, 5, and 6. They need to be distributed among four people: A, B, C, and D. Each person should receive at least one ticket and a maximum of two tickets. If a person gets two tickets, they must be consecutive. Calculate the number of different ways the tickets can be distributed. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Find the value of \( \eta = \int_0^1 f(x) \, \mathrm{d}x \) given the integral equation \( f(x) = x + \int_0^1 t(x+t)f(t) \, \mathrm{d}t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{42}{23}}
Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which  

\(\log_{10}(x+y) = z\) and \(\log_{10}(x^{2}+y^{2}) = z+1.\)  
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have \(x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.\) The original answer is in \(\frac{m}{n}\) format, please find the value of \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
When Minhyuk asked Kihyun's math test score, Kihyun answered it would be 90 if he rounded it off to 10 digits. What is the biggest number possible with Kihyun's math score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
The greatest common divisor of natural numbers \( m \) and \( n \) is 1. What is the greatest possible value of \(\text{GCD}(m + 2000n, n + 2000m) ?\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3999999}
For every integers $ a,b,c$ whose greatest common divisor is $n$, if
\[ \begin{array}{l} {x + 2y + 3z = a} \\
{2x + y - 2z = b} \\
{3x + y + 5z = c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
三角形的三边长分别是整数l,m,n，且l&gt;m&gt;n. 已知$\{ \frac { 3 ^ { l } } { 1 0 ^ { 4 } } \} = \{ \frac { 3 ^ { m } } { 1 0 ^ { 4 } } \} = \{ \frac { 3 ^ { n } } { 1 0 ^ { 4 } } \}$，其中{x}=x-[x]，而［[x]表示不超过x的最大整数.求这种三角形周长的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3003}
Calculate the variance of a random variable X that follows the hyperbolic secant distribution, given that the expected value E[X] = 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Sonya the frog chooses a point uniformly at random lying within the square \([0, 6] \times [0, 6]\) in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from \([0, 1]\) and a direction uniformly at random from \{north, south, east, west\}. All of her choices are independent. Find the probability that she lands outside the square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^3 + ax^2 + bx \). Suppose there exist three real numbers \( x_1, x_2, x_3 \) such that \( x_1 + 1 \leq x_2 \leq x_3 - 1 \) and \( f(x_1) = f(x_2) = f(x_3) \). Find the minimum value of \( |a| + 2|b| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
The shopkeeper increased the price of a product by 25%, making it harder for customers to purchase as much as they needed. However, a particular customer, with a budget of $150, managed to buy only 64% of the quantity they originally intended to purchase. Assuming there was also a maximum limit of 40 units of the product available, what is the net difference in the expenditure on that product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30}
Determine the set of all cluster points of subsequences of the sequence \( \frac{n}{e} - \left[\frac{n}{e}\right] \), where \( e \) is Euler's number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
16. Check Digit (8th grade. 3 points). The Simple-Dairy Telephone Company “Simple-Telecom” uses three-digit telephone numbers. The equipment is old, so during connection, errors in individual digits of the transmitted subscriber number are possible - each digit, independently of the others, has a probability of $p=0.02$ of being replaced by some other random digit. To reduce the probability of incorrect connection, the company uses the following rule: the first digit of the number is always equal to the remainder of the sum of the two other digits divided by 10. For example, the numbers 000 and 156 are possible, while the number 234 is not. If the check digit is incorrect during connection, an error message is displayed. Find the probability that, despite the measures taken, an incorrect connection will occur.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.000131}
Evaluate the improper integral: $$\int_{-\infty}^{\infty} \frac{\sin^{4}(x)}{x^2}dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members so that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be [i]happy[\/i] if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1518
}
Find the angle between the tangents drawn to the curve \(3x^2+4xy+5y^2-4=0\) at the points of intersection with the lines \(y=-\frac{3}{2}x\) and \(y=-\frac{2}{5}x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
The area of a rectangular grassy park is $4000 \mathrm{~m}^{2}$. Two concrete paths parallel to the sides intersect in the park. The area of one path is $400 \mathrm{~m}^{2}$, and the area of the other path is $250 \mathrm{~m}^{2}$. What percentage of the park's area is covered with grass?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84.375\%}
Given the joint density function \( f_{XY}(x, y) = kxy^2 \) for \( 0 \leq x, y \leq 1 \) and 0 otherwise, where \( k = 6 \), compute the probability \( \Pr(|X - Y| < 0.5) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
Evaluate the limit of the integral as \( n \to \infty \):
\[
\lim_{n\to\infty} n\int_{-\infty}^{\infty} e^{-2n\vert x\vert}\frac{x+3}{x^4+3x^2+1} \mathrm{d}x
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}