inclusionAI/Ring-lite-rl-data huggingface dataset:
Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$670,\!000$}
Find the value of \(c\) in the joint probability density function \(f_{X,Y}(x,y) = c\) for \((x,y) \in \left[\frac{1}{2}, 1\right] \times \left[-x+\frac{3}{2}, 1\right] \cup \left[0, 1\right] \times \left[-x+\frac{1}{2}, -x+1\right]\) and \(0\) otherwise, given that the total probability is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{5}}
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:

- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.

How old is this capricious lady?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the line integral \( \oint_{C_1} \left(\frac{y\hat{i} - x\hat{j}}{x^2 + y^2}\right) \cdot d\vec{r} \), where \( C_1 \) is a circle of radius 3 centered at the origin in \( \mathbb{R}^2 \). Determine the result without relying solely on parametrization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
A quadrilateral is given, where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( C \) is the curve joining the origin \((0, 0, 0)\) to the point \((1, 2, 3)\) defined by the parametric curve \( \mathbf{r}(t) = \frac{\ln t}{\ln 2} \mathbf{i} + t^{\frac{3}{2}} \mathbf{j} + t \cos(\pi t) \mathbf{k} \), \( 1 \leq t \leq 4 \), and \( \mathbf{F}(x, y, z) = 3x^2 z \mathbf{i} + z^2 \mathbf{j} + (x^3 + 2yz) \mathbf{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Calculate the integral \( \int_{0}^{\pi} e^{a\cos(\theta)} \cos(a\sin(\theta)) \, d\theta \) using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the numerator of

$$
\frac{\overbrace{1010 \, 11 \ldots 11 \, 0101}^{2011 \text{ ones}}}{\underbrace{1100 \, 11 \ldots 11 \, 0011}_{2011 \text{ ones}}}
$$

when reduced.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9091}
I was walking with my mother-in-law. We were walking slowly at a speed of 3 km/h along Saint Catherine Street in Bordeaux, which, as everyone (or maybe not everyone) knows, is straight. Suddenly, I remembered that I needed to mail a letter as soon as possible in a mailbox located a bit further down the street. Leaving my mother-in-law, who continued walking peacefully, I hurried forward, mailed the letter, and then walked back towards my mother-in-law at a speed of 5 km/h. I clearly remember that I left my mother-in-law alone for only 3 minutes.

At what distance from the mailbox were we when we parted ways?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200 \text{ meters}}
The sequence 12, 15, 18, 21, 51, 81, $\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\mathrm{th}}$ term of the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{318}
Find the total number of times the digit ‘ $2$ ’ appears in the set of integers  $\{1,2,..,1000\}$ . For example, the digit ’ $2$ ’ appears twice in the integer  $229$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Find the largest integer $K$ such that for each $k∈$ {$2,3,4,...,K$}, the number $2^{30}-k$ has a prime factor less than $30$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
What fraction of the entire wall is painted red if Matilda paints half of her section red and Ellie paints one third of her section red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{12}}
Randomly select seven numbers from the set $\{1, 2, \cdots, 63\}$. Let the smallest of these numbers be denoted as $x$. What is the expected value $\mathrm{E}(x)$ of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
On their way driving to Los Angeles, Benjamin and James see lots of billboard ads. Curious, Benjamin starts to count how many they see within specific periods each hour. In the first hour, Benjamin notes the billboards during a 25-minute period and counts 15 billboards. In the second hour, he only counts during a 45-minute period and sees 31 billboards. In the third hour, he counts during a 20-minute period and sees 12 billboards. James asks Benjamin the average number of billboards they've seen per hour considering the varying periods he counted. What should Benjamin tell him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.78}
Sean originally has 55 blocks. He decided to share them with his 5 friends, giving each friend an equal number of blocks. Before they get a chance to play with the blocks, a hippopotamus eats 29 of the remaining blocks. How many blocks does Sean have left now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the smallest possible value of n such that there are n distinct lines in the plane, one of which intersects exactly 5 of the n lines, another intersects exactly 9 of the n lines, and yet another intersects exactly 11 of the n lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A five-digit number $\overline{ABCDE}$ is a multiple of 2014, and $\overline{CDE}$ has exactly 16 factors. What is the smallest value of $\overline{ABCDE}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24168}
The apothem of a regular hexagonal pyramid is $h$, and the dihedral angle at the base is $60^{\circ}$. Find the total surface area of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\sqrt{3}}{2} h^2}
Given the polynomial equation
$$
x^{5}+3 x^{4}-4044118 x^{3}-12132362 x^{2}-12132363 x-2011^{2}=0
$$
where all roots are integers, find the number of distinct roots of the equation. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of ways to choose a set of $n$ distinct non-negative integers $i_1, i_2, \ldots, i_n$ from the set $\{0, 1, \ldots, n-1\}$ such that their sum is a multiple of $n$. Each $i_j$ must be unique.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 - (-1)^n}{2}}
Evaluate the following limit: $$\lim_{n\to\infty}\left(1-\frac{1}{2}\right)\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{2^4}\right)\cdots\left(1-\frac{1}{2^{2^n}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set \( \prod_{n\in\mathbb{N}} (n+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Two sisters were picking currants: the older one was using a 4-liter bucket, and the younger one was using a 3.5-liter bucket. The older sister was always working faster than the younger one. When the older sister had collected three-quarters of her bucket, and the younger sister had collected more than half of hers, the girls exchanged buckets. Continuing to work at the same speeds as before exchanging buckets, they finished at the same time. How many times faster was the older sister picking berries compared to the younger sister?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
Find a bijective function that maps the real numbers to the interval $(-1, 1)$, different from the standard arctangent function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{x}{1 + |x|}}
The function $y=(m^{2}-m-1){x}^{m^{2}-2m-1}$ is a power function, and it is a decreasing function for $x\in (0,+\infty)$. Find the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of ways to distribute 15 distinct balls into 5 indistinguishable boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255514355}
Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
${\dfrac{8}{15}}$}
In how many ways can a random number of identical men be arranged in a line of 15 chairs such that no two men sit next to each other, and each empty chair has at least one man sitting next to it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Given a point $P(\frac{1}{2},0)$, determine the number of lines passing through point $P$ that intersect the hyperbola $4x^{2}-y^{2}=1$ at only one point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the sum of $n=14$ iid random variables $X_1, \dots, X_{14}$ with PDF \( f_{X_i}(x) = \frac{1}{\theta} e^{-\frac{x-\delta}{\theta}}, \quad x > \delta \) and 0 otherwise, where \( \theta = 2 \) and \( \delta = 0 \), calculate the probability \( P(Y < 2.467) \) for \( Y = \frac{\sqrt{28}Z}{\sqrt{S_{14}}} \) and \( Z \sim \mathcal{N}(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.99}
In order to assess the number of students in various classes at a school participating in an extracurricular group, 5 classes are randomly selected. The number of students participating in the group from each class is taken as sample data. It is known that the sample mean is 7, the sample variance is 4, and the sample data are all distinct. Find the maximum value in the sample data.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_{N}$ denote the probability that the product of these two integers has a units digit of 0. The maximum possible value of $p_{N}$ over all possible choices of $N$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2800}
In the right trapezoid \( ABCD \), it is known that \(\angle A = \angle D = 90^\circ\), \(DE \perp AC\) at point \(E\), \(\angle ACD = \angle EBC = 30^\circ\), and \(AD = \sqrt{3}\). Find \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( A = \left(a_1, a_2, \ldots, a_{2001}\right) \) be a sequence of positive integers. Let \( m \) be the number of 3-element subsequences \( (a_i, a_j, a_k) \) with \( 1 \leq i < j < k \leq 2001 \) such that \( a_j = a_i + 1 \) and \( a_k = a_j + 1 \). Considering all such sequences \( A \), find the greatest value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{667^3}
A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, what is the value of $n$ in the form of an improper fraction \frac{k}{m}? Please provide the sum of k and m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{359}
For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$. 
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19999999999999999999999}
Given that $7^{-1} \equiv 55 \pmod{78}$, find $49^{-1} \pmod{78}$, as a residue modulo 78. (Give an answer between 0 and 77, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Given that $P$ is a point on the circle $(x-3)^{2}+(y-4)^{2}=4$, then the maximum distance from point $Q(\cos \alpha ,\sin \alpha )$ to point $P$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The first train is 100 m long. They will be clear of each other in 10.999120070394369 seconds from the moment they meet. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119.98}
Determine how many roots of the equation

$$
4 \sin 2x + 3 \cos 2x - 2 \sin x - 4 \cos x + 1 = 0
$$

are located in the interval \(\left[10^{2014!} \pi, 10^{2014!+2015} \pi\right]\). In the answer, write down the sum of all the digits of the found number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18135}
Find all positive integers $ m,n $  that satisfy the equation  \[ 3.2^m +1 = n^2 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(m, n) \in \{(3, 5), (4, 7)\}}
Dylan has a \( 100 \times 100 \) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
Given two circles with radii $R$ and $r(R>r)$, having internal tangency. Find the radius of the third circle that is tangent to the first two circles and their common diameter.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4rR(R-r)}{(R+r)^2}}
8. Given the set $A=\{1,2, \cdots, 6\}$, the mapping $f: A \rightarrow A$ is a one-to-one mapping,
$$
\begin{array}{l}
f^{(2)}(x)=f(f(x)), \\
f^{(n+1)}(x)=f\left(f^{(n)}(x)\right)(x \geqslant 1) .
\end{array}
$$

Then the number of mappings $f$ that satisfy $f^{(12)}(x)=x$ for any $x \in A$ is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1011}}
Positive integers  $a,b,c$  exist such that  $a+b+c+1$ ,  $a^2+b^2+c^2 +1$ ,  $a^3+b^3+c^3+1,$  and  $a^4+b^4+c^4+7459$  are all multiples of  $p$  for some prime  $p$ . Find the sum of all possible values of  $p$  less than  $1000$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in the case of a tie, the order is arbitrary). It is noted that each participant won half of their points against the last ten ranked players. How many people participated in the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given a convex pentagon  $ABCDE$  with  $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$ . Find the area of  $ABCDE$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number $i \in \{0, 1, . . . , 5\}$, some coin has exactly $i$ dots on it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79200}
The bases of three identical cones are located in the same plane and touch each other. The axial cross-section of each cone is an equilateral triangle with side length $a$. Find the radius of a sphere that touches the lateral surface of each cone and the plane in which their bases are located.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a(2 - \sqrt{3})}{2}}
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 100 residents of this island lined up, and each of them said one of the following phrases:

- "To the left of me there are as many liars as knights."
- "To the left of me there is 1 more liar than knights."
- "To the left of me there are 2 more liars than knights."
- "To the left of me there are 99 more liars than knights."

It is known that each phrase was said by exactly one person. What is the minimum number of liars that can be among these 100 residents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
The product $20! \cdot 21! \cdot 22! \cdot \ldots \cdot 28!$ can be expressed in the form $m \cdot n^3$, where $m$ and $n$ are positive integers, and $m$ is not divisible by the cube of any prime. Find $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{825}
Four cars $A, B, C$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counter-clockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. When will all the cars meet for the first time after the race starts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{371}
6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. For the sake of secrecy, he will not visit the same village twice, and he will not visit villages in a row that are directly connected by a road. How many villages can he guarantee to fly over? He can start from any village.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
The real number sequence \( x_0, x_1, x_2, \cdots \), is defined as \( x_0 = 1989 \) and \( x_n = -\frac{1989}{n} \sum_{k=0}^{n-1} x_k \) for \( n \geq 1 \). Calculate the value of \( \sum_{n=0}^{1988} 2^n x_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1989}
Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian Ji's bottom-tier horse being worse than King Qi's bottom-tier horse, determine the probability that Tian Ji's selected horse wins the race.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Find all positive integer pairs \((a, b)\) such that the equation \(5^b + 2 = 7^c\) is satisfied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1)}
The terminal side of angle $α$ passes through point $P(x,4)$, and $\sin α= \frac {4}{5}$, then $x=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±3}
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Brandon wants to maximize $\frac{\Box}{\Box} +\Box$ by placing the numbers $1$, $2$, and $3$ in the boxes. If each number may only be used once, what is the maximum value attainable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5
}
Find the number of tuples \((k_1, k_2, \ldots, k_{26})\) where each \(k_i\) is either 0, 1, or 3, such that the sum \(k_1 + k_2 + \cdots + k_{26} = 15\). Express this as finding the coefficient of \(x^{15}\) in the expansion of \((1 + x + x^3)^{26}\), and calculate this coefficient directly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{853423740}
Evaluate the limit: \[ \lim_{x\rightarrow a}\frac{a^{a^x}-a^{x^a}}{a^x-x^a} \] without using l'Hospital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{a^a} \ln a}
Evaluate the limit: $$ \lim_{n\to\infty} n\int_{1}^{2} \frac{1}{x^2(1+x^n)}dx $$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Find the maximum value of the expression \( T \) for real numbers \( x, y, z \geq 0 \):
\[ T = \frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3} + \frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3} + \frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{16}\)}
Evaluate the integral \( \int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Calculate $$\lim_{n\rightarrow \infty}\sum_{k=1}^{n}\cos\left(\frac{2\pi k}{2n+1}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1/2}
Find the locus of the center of a circle that cuts the family of circles given by \(x^2+y^2+4x(\lambda-\frac{3}{2})+3y(\lambda-\frac{4}{3})-6(\lambda+2)=0\) orthogonally.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4x + 3y = 6}
The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}