inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider the functions $f, g: [0,1] \rightarrow [0,1]$ defined by $f(x) = \frac{1}{2} x(x+1)$ and $g(x) = \frac{1}{2} x^2(x+1)$. What is the area enclosed between the graphs of the inverses $f^{-1}$ and $g^{-1}$ on the interval $x \in [0,1]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
Let the function $f(x) = x\ln(\sqrt{x^{2}+1}+x) + x^{2} - x\sin x$, then the range of $x$ for which $f(x) > f(2x-1)$ holds is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \dfrac{1}{3},1\right)}
Determine the value of $c \in \mathbb{R}$ for which there exists a sequence of irrational numbers $\{x_n\}_{n\in\mathbb{N}}$ such that $x_n \to c$ and $\lim_{n\to\infty} f(x_n) = 1-c$, where $f$ is a given function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In the triangular prism $P-ABC$, the three edges $PA$, $PB$, and $PC$ are mutually perpendicular, with $PA=1$, $PB=2$, and $PC=2$. If $Q$ is any point on the circumsphere of the triangular prism $P-ABC$, what is the maximum distance from $Q$ to the plane $ABC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2} + \frac{\sqrt{6}}{6}}
If the function $f(x)=\sin x+a\cos x$ is a strictly monotonic function on $(\frac{2\pi}{3},\frac{7\pi}{6})$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\frac{\sqrt{3}}{3}, \sqrt{3}]}
Let \( p = 10007 \), a prime number. Determine the number of \( 2 \times 2 \) matrices \( X \) with elements from \( \mathbb{Z}_p \) such that \( X^2 \equiv I \pmod{p} \), where \( I \) is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100150058}
Identify the probability distribution that resembles the function \( \frac{1}{\sqrt{2\pi x}}\exp\left(-\frac{x^2}{2x}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(1)}
Donggyu wants to complete the diagram so that each box contains a positive integer and each box in the top two rows contains the product of the integers in the two boxes below it. He wants the integer in the top box to be 720. How many different values can the integer n take? Express your answer as a single integer representing the count of possible n values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given two integers \( x \) and \( y \), compute the lowest 8 bits of the integer division \( \left\lfloor \frac{x}{y} \right\rfloor \). Specifically, find \( r = \left( \frac{x}{y} \right) \mod 256 \). For example, if \( x = 18423 \) and \( y = 29 \), then \( r = 123 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123}
Determine the total number of constraints provided by a 2D rigid body transformation matrix in homogeneous representation, given by:
\[
\begin{bmatrix} 
\textbf{R} & \textbf{t} \\ 
0^T & 1
\end{bmatrix}
\]
where \(\textbf{R}\) is a rotation matrix with the orthonormal constraint \(R^TR=1\). Consider any additional constraints from the homogeneous representation and the translation vector \(\textbf{t}\). What is the total number of constraints in the 2D case?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
There were 20 different non-integer numbers written on the board. For each of these numbers \( x \), Vasya wrote down separately \( [x] \) and \( \frac{1}{\{x\}} \). What is the minimum number of different numbers that Vasya could have obtained?

Here, \( [x] \) and \( \{x\} \) denote the integer and fractional parts of the number \( x \), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A train starts from Amritsar to Bombay at 9 am. It reaches the destination after some days at 9:30 am. Every day a train starts. The train comes across 4 trains on the way. How many days does it take for the train to reach its destination?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
There are 26 positive integers d between 200 and 300 (both inclusive) that are not divisible by a certain number, 3, or 5. What is the first number that these integers are not divisible by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the maximum integer \( n \) such that the Taylor series expansion of \( \sinh(x^4) \) is given by:
\[ \sinh(x^4) = x^4 + \frac{1}{6}x^{12} + \mathcal{o}(x^n) \quad \text{as} \ x \to 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the
six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \in \mathbb{R}$
with $a + b + c + d = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Knowing that segment \( CD \) has a length of 6 and its midpoint is \( M \), two triangles \( \triangle ACD \) and \( \triangle BCD \) are constructed on the same side with \( CD \) as a common side and both having a perimeter of 16, such that \( \angle AMB = 90^{\circ} \). Find the minimum area of \( \triangle AMB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{400}{41}}
Let $p_1, p_2, p_3, p_4$ be four distinct prime numbers, and let $1 = d_1 < d_2 < \ldots < d_{16} = n$ be the divisors of $n = p_1 p_2 p_3 p_4$. Determine all $n < 2001$ such that $d_9 - d_8 = 22$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1995}
Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2144}
Olya wrote fractions of the form $\frac{1}{n}$ on cards, where $n$ represents all possible divisors of the number $6^{100}$ (including the unit and the number itself). She arranged these cards in some order. 

After arranging, she noted the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, and so on, until finally, she recorded the sum of the numbers on all the cards. Each sum was recorded on the board as an irreducible fraction. 

What is the least number of different denominators that could appear on the numbers on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the integer $n > 2$ that satisfies the inequality $(\lg n)^{2^{100}} < \sqrt{n}$, where $\lg$ denotes the binary logarithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^{108}}}
Given two independent standard normal variables $X$ and $Y$, determine the correlation of the samples $(x, y)$ that satisfy the condition $x + y > 0$. Assume $n$ samples are generated. What is the correlation of these samples?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{\pi - 1}}
In $\triangle DEF$ with sides $5$, $12$, and $13$, a circle with center $Q$ and radius $2$ rolls around inside the triangle, always keeping tangency to at least one side of the triangle. When $Q$ first returns to its original position, through what distance has $Q$ traveled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m$, $n$, and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{346}
Find the function \( f: \mathbb{R} \to \mathbb{R}_0^+ \) that satisfies the equation \( f(x+2y) + f(x) = 2(f(x+y) + f(y)) \) for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax^2}
Two circles with diameters \( d \) and \( D \) (\( d < D \)) are placed inside a semicircle such that each circle touches both the arc and the diameter of the semicircle, as well as the other circle. A line is drawn through the centers of the circles, intersecting the extension of the diameter of the semicircle at point \( M \). From point \( M \), a tangent is drawn to the arc of the semicircle, touching it at point \( N \). Find \( MN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{D d}{D - d}}
Using the digits 1, 2, 3, 4, and 5, form distinct five-digit numbers without repeating any digit and ensuring that the digit 4 doesn't appear in either the first or last position. Additionally, among the digits 1, 3, and 5, exactly two must be adjacent to each other. Determine the total number of different five-digit numbers that meet these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Isosceles triangles $A B C(A B=B C)$ and $A_{1} B_{1} C_{1}\left(A_{1} B_{1}=B_{1} C_{1}\right)$ are equal. The vertices $A_{1}, B_{1}$ and $C_{1}$ are located respectively on the extensions of side $B C$ beyond point $C$, side $B A$ beyond point $A$, side $A C$ beyond point $C$, and $B_{1} C_{1} \perp B C$. Find the angle $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arcsin3/5}
Find the number of different numbers of the form  $\left\lfloor\frac{i^2}{2015} \right\rfloor$ , with  $i = 1,2, ..., 2015$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval (0,1). Determine the probability density function (PDF) of the random variable $V = XY$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln v}
Let $m$ and $n$ be positive integers such that $\gcd(m, n) = 1$ and
$$\sum_{k=0}^{2020} (-1)^k \binom{2020}{k} \cos(2020 \cos^{-1}(\frac{k}{2020})) = \frac{m}{n}.$$

Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n = 12 = 2 \times 2 \times 3$, then the answer would be $2 + 2 + 3 = 7$.)

Proposed by Ankit Bisain
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
209601}
Find the smallest positive integer $n$ that satisfies the following two properties:

1. $n$ has exactly 144 distinct positive divisors.
2. Among the positive divisors of $n$, there are ten consecutive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110880}
A car's wheels are 28 inches in diameter. How far (in miles) will the car travel if its wheels revolve 10,000 times without slipping?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.88\text{ miles}}
If 5% more is gained by selling an article for Rs. 350 than by selling it for another amount, the cost of the article is Rs. 40. What is the lower selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{348}
If the equation $x^{2}+(k^{2}-4)x+k-1=0$ has two roots that are opposite numbers, solve for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Brent received 5 Kit-Kat bars, some amount in Hershey kisses, 8 boxes of Nerds, and 11 lollipops. He also had 10 Baby Ruths and half that amount in Reese Peanut butter cups. After giving his little sister 5 lollipops, he had 49 pieces of candy left. What is the ratio of the number of Hershey kisses to the number of Kit-Kat bars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
Let \( N \) be the number of 2015-tuples of (not necessarily distinct) subsets
 \( \left( {{S}_{1},{S}_{2},\ldots ,{S}_{2015}}\right) \) of \( \{ 1,2,\ldots ,{2015}\} \) such that the number o
 permutations \( \sigma \) of \( \{ 1,2,\ldots ,{2015}\} \) satisfying \( \sigma \left( i\right)  \in  {S}_{i} \) for all
 \( 1 \leq  i \leq  {2015} \) is odd. Let \( {k}_{2},{k}_{3} \) be the largest integers such that \( {2}^{{k}_{2}}|N \)
 and \( {3}^{{k}_{3}} \mid  N \) respectively. Find \( {k}_{2} + {k}_{3} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2030612
}
3. One of the angles of the trapezoid is $60^{\circ}$. Find the ratio of its bases, if it is known that a circle can be inscribed in this trapezoid and a circle can be circumscribed around this trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
Consider two geometric sequences $16$, $a_1$, $a_2$, $\ldots$ and $56$, $b_1$, $b_2$, $\ldots$ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
Let  $S$  be the increasing sequence of positive integers whose binary representation has exactly  $8$  ones. Let  $N$  be the  $1000^{th}$  number in  $S$ . Find the remainder when  $N$  is divided by  $1000$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $A$ be a $7\times 7$ matrix with rank $3$, and let $a$ be a $7\times 1$ column vector. Determine the least possible rank of the matrix $A + (a a^T)$. ($a^T$ denotes the transpose of the vector $a$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of unique real solutions for the equation \((x-2)^2 (x+2)^2 = 4 + \log(x+4)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, $\ldots$). What is the largest integer $j$ such that $P$ is divisible by $101^j$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1980
}
All integers from 1 to 100 are written in a string in an unknown order. With one question about any 50 numbers, you can find out the order of these 50 numbers relative to each other. What is the minimum number of questions needed to definitely find out the order of all 100 numbers?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the contour integral \( \oint_{\gamma} \frac{1}{\sin z} \, dz \) along a simple, closed path \( \gamma \) that encloses the origin and the set \( \{\pi n : n \in \mathbb{Z} \setminus \{0\}\} \). Use Cauchy's residue theorem to determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Kanul spent 15% of his cash on buying raw materials, 10% on buying machinery, and 25% on employee wages. He also invested 35% of his cash into a fixed deposit account. After all these expenses, he still had 15% of his total amount left as cash with him. He also received a loan of $2,000, which formed 20% of his total available amount. What was Kanul's total initial amount before spending and receiving the loan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$53,333.33}
Find a tight bound for the recurrence relation \(T(n) = T\left(\frac{n}{3}\right) + T\left(\frac{n}{6}\right) + n\) using the substitution method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
A worker's wage is reduced by 30%. To restore their original salary, what percentage raise is required on their new reduced wage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.86\%}
Determine the cardinality of the set of all infinite sequences composed of the digits $0, 1, -1$, such that each sequence contains each digit at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Given 10 points on a plane. We draw the perpendicular bisectors of the segments determined by them. In how many points can these lines intersect at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{750}
Determine the sign of the permutation \( \sigma \) given by:
\[ \sigma = \left[ \begin{array}{cccccc} 1 & \cdots & k & k+1 & \cdots & k+l \\ l+1 & \cdots & l+k & 1 & \cdots & l \end{array} \right] .\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{kl}}
Given that
$$\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}=2\omega^2\;\;\;\;\;(1)$$
$$\frac{1}{a+\omega^2}+\frac{1}{b+\omega^2}+\frac{1}{c+\omega^2}=2\omega\;\;\;\;\;(2)$$
Find
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}=?\;\;\;\;\;\;(3)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the area enclosed by the curve defined in polar coordinates by the equation \( r = \frac{\sin 3\theta}{\sin \theta} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
3. At a basketball tournament, 16 teams participate, playing in a double round-robin format, meaning each team plays every other team twice. The top 8 teams advance to the next tournament. The ranking of the teams is determined based on the number of wins, and if multiple teams have the same number of wins, their relative ranking is determined by a draw. How many wins are minimally necessary for a team to secure advancement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A COVID vaccine has a probability \( p \) of causing harmful side effects, where \( p \) is uniformly distributed between 0 and 0.5. Researchers administered the vaccine to 10,000 volunteers and observed no adverse reactions. Determine the smallest real number \( \lambda \) such that the probability that \( p < \lambda \) is at least 90%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.00023}
A broken clock moves the minute hand each minute randomly one minute ahead and one minute behind with equal probabilities. Each minute it does so independently of all other times it moved the minute hand. The hour hand is on the other synchronized with the minute hand. The clock starts running at noon and let T denotes the first time that the clock shows either 1 o'clock or 11 o'clock, whatever happens first. Compute E(T).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3600}
Given a transitive set $\{x\}$, determine the nature of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Find the pedal equation of the curve \(y^2 = 4a(x+a)\) with respect to the origin \((0,0)\). Express your answer in terms of the distance \(p\) from the origin to the tangent, and the distance \(r\) from the origin to a point on the curve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 = a r}
Using the digits 2, 3, 5, 7, and 8 exactly once, form a five-digit integer that is as close as possible to 30000. What is the tens digit of this integer? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
\( AB \) is a diameter; \( BC \) and \( AC \) are chords, with \( \angle B C = 60^\circ \). \( D \) is the point where the extension of the diameter \( AB \) meets the tangent \( CD \). Find the ratio of the areas of triangles \( DCB \) and \( DCA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Determine all points in \(\mathbb{R}^3\) where the function \[ f(x,y,z) = \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}} \] reaches its maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(t, 2t, 3t)}
Evaluate the limit: $$\lim_{x\to\infty} x^{7/2} \left[\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\left(1-\frac1{8x^2}\right)\right] $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{64}}
A flight from Toronto to Gander takes 2 hours and 50 minutes. When the time in Toronto is 1:00 p.m., the time in Gander is 2:30 p.m. If the flight departs from Toronto at 3:00 p.m. (Toronto time), what time will the flight land in Gander (Gander time)? Express your answer in 12-hour clock format with a.m./p.m. designation (e.g., 7:20 p.m.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7:20 p.m.}
We have one $10 \mathrm{Ft}$ coin and several (at least four) $20 \mathrm{Ft}$ coins in our pocket. We successively draw one coin at a time at random until the sum of the drawn coins becomes a positive integer multiple of $30 \mathrm{Ft}$. On average, how many draws are needed to achieve this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the volume of a torus using a double integral in Cartesian coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi^2 R r^2}
Tina decides to fill a jar with coins. In the first hour she puts in 20 coins. During the next two hours she puts in some coins each time. During the fourth hour she puts in 40 coins. During the fifth hour her mother asks to borrow some money so she takes 20 coins out. There are 100 coins left after the fifth hour. How many coins did Tina put in during each of the next two hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Call a positive integer *monotonous* if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. Find the total number of monotonous positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1524}
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{n}}{n+k^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the definite integral:

$$
\int_{-\pi / 2}^{0} 2^{8} \cdot \sin^2(x) \cos^6(x) \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \pi}
What are the last 8 digits of
$$
11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19754321}
Three different 6th grade classes are combining for a square dancing unit. The first class has 17 males and 13 females, while the second class has 14 males and 18 females, and the third class has 15 males and some females. When the classes are combined, 2 students will be unable to partner with a student of the opposite gender. How many female students are in the third class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}