inclusionAI/Ring-lite-rl-data huggingface dataset:
On weekdays, a scatterbrained scientist takes the Moscow metro ring line from Taganskaya station to Kievskaya station to get to work, and makes the return trip in the evening. The scientist boards the first available train that arrives. It is known that trains run at approximately equal intervals in both directions. The northern route (via Belorusskaya) takes 17 minutes to travel from Kievskaya to Taganskaya or vice versa, while the southern route (via Paveletskaya) takes 11 minutes.

The scientist, as usual, counts everything. One day, he calculated that, based on years of observations:

- A train traveling counterclockwise arrives at Kievskaya on average 1 minute and 15 seconds after a train traveling clockwise. The same applies to Taganskaya;
- The scientist's average commute from home to work takes 1 minute less than the commute from work to home.

Find the expected value of the interval between trains traveling in one direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \text{ minutes}}
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ of $\triangle ABC$ respectively, with $b=6$, and $$accosB=a^{2}-b^{2}+ \frac { \sqrt {7}}{4}bc$$, $O$ is a point inside $\triangle ABC$, and satisfies $$\overrightarrow {OA}+ \overrightarrow {OB}+ \overrightarrow {OC}= \overrightarrow {0}，∠BAO=30 °$$ , then $$| \overrightarrow {OA}|$$=____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1330}
Four pieces of lumber are placed in parallel positions, perpendicular to line M:

- Piece W is 5m long
- Piece X is 3m long and its left end is 3m from line M
- Piece Y is 5m long and is 2m from line M
- Piece Z is 4m long and is 1.5m from line M

A single cut, perpendicular to the pieces of lumber, is made along the dotted line L. The total length of lumber on each side of L is the same. What is the length, in meters, of the part of piece W to the left of the cut? Express your answer as a decimal number with two places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.75}
A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The line passing through the points (3,9) and (-1,1) has an x-intercept of ( ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{2}}
Determine the number of  $ 8$ -tuples of nonnegative integers  $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$  satisfying  $ 0\le a_k\le k$ , for each  $ k \equal{} 1,2,3,4$ , and  $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1540}
If some positive integers are 2 and 3, then their product plus the first integer is 33. What is the second integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
For each positive integer $n$, let $a(n)$ be the number of zeros in the base $3$ representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^{3}} $$ converge? Please provide the sum of the bounds of the interval where x satisfies the condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Calculate the value of the integral $\oint_{\gamma} \frac{z}{\cos z}$, where $\gamma$ is the circle $|z|= 5$ oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi^2 i}
Define binary operations $\diamondsuit$ and $\heartsuit$ by \[a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}\]for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsuit\, 2$ and \[a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}\]for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$11$}
An urn contains $5$ green balls and $5$ blue balls. A second urn contains $20$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.6$. Find $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N = 20}
在数列  $\left[\frac{1^{2}}{2019}\right],\left[\frac{2^{2}}{2019}\right], \cdots,\left[\frac{2019^{2}}{2019}\right]$  中，共含有_____个互不相等的整数（[x]表示不超过实数 $x$ 的最大整数)。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1515}
Plot points \(A, B, C\) at coordinates \((0,0), (0,1)\), and \((1,1)\) in the plane, respectively. Let \(S\) denote the union of the two line segments \(AB\) and \(BC\). Let \(X_1\) be the area swept out when Bobby rotates \(S\) counterclockwise 45 degrees about point \(A\). Let \(X_2\) be the area swept out when Calvin rotates \(S\) clockwise 45 degrees about point \(A\). Find \(\frac{X_1 + X_2}{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10! ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3960}
Given a sequence $\{a_n\}$ that satisfies $a_{n+1}-a_n=3$ and $a_3=-2$. If for all $k \in [4,+\infty)$, $\frac{a_{k+1}a_{k+2}}{a_k} \geqslant a_m$ always holds, where $m,k \in \mathbb{N}^*$, then the maximum value of the real number $m$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In a class of 25 students, 3 students did not borrow any books from the library, 10 students each borrowed 1 book, 4 students each borrowed 2 books, and the rest borrowed at least 3 books. If the average number of books per student was 2.5, what is the maximum number of books any single student could have borrowed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In a square piece of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if, after cutting out the hole, 209 cells remained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
As shown in Figure 1, a solid metal sphere with a radius of 20 has a circular hole with a radius of 12 drilled through it along one of its diameters. The remaining part is then melted down and cast into a new solid sphere. What is the radius of the new sphere?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
On a 6 by 6 grid of points, what fraction of the larger square's area is inside the shaded square if the shaded square is rotated 45 degrees with vertices at points (2,2), (3,3), (2,4), and (1,3)? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{18}}
设 $a_{1}, a_{2}, a_{3}, a_{4}$ 是 4 个有理数, 使得
$$
\left\{a_{i} a_{j} \mid 1 \leqslant i<j \leqslant 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}
$$
求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的所有可能值之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Using the four digits 1, 9, 8, 5 to form a number in the form \(a^{b^{c^{d}}}\), if you want to maximize its value, what is the value of \(c\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A company sells a brand of cars in two locations, A and B, with profits (in ten thousand yuan) of $L_1 = -x^2 + 21x$ and $L_2 = 2x$, respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 units in both locations, calculate the maximum profit it can achieve. Express your answer in ten thousand yuan, rounded to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is it possible that $P$ wins the first four rounds that he plays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Inside circle \(\omega\), there are two circles \(\omega_1\) and \(\omega_2\) that intersect at points \(K\) and \(L\), and touch circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(M\), and \(N\) are collinear. Find the radius of circle \(\omega\), given that the radii of circles \(\omega_1\) and \(\omega_2\) are 3 and 5, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
9000 plus a certain percentage of 9032 equals 10500. What is the percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.61\%}
For each nonnegative integer $k$, let $d(k)$ denote the number of I's in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)-2$). Let $m$ be a positive integer. Express
$$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$
in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.Let \(m = 2\). Find the value of \(a+f(2)+g(2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the least positive integer of the form <u>a</u> <u>b</u> <u>a</u> <u>a</u> <u>b</u> <u>a</u>, where a and b are distinct digits, such that the integer can be written as a product of six distinct primes
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{282282}
If x is less than y by a certain percentage, then y exceeds x by 81.81818181818181%. What is the percentage by which x is less than y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45\%}
How many permutations $p(n)$ of the set $\{1, 2, \dots, 35\}$ satisfy the condition that if $a \mid b$ (where $a$ divides $b$), then $p(a) \mid p(b)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given that \(a\), \(b\), and \(c\) are three distinct real numbers, and in the quadratic equations

\[
x^{2} + ax + b = 0
\]

\[
x^{2} + bx + c = 0
\]

\[
x^{2} + cx + a = 0
\]

any two of the equations have exactly one common root, find the value of \(a^{2} + b^{2} + c^{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a circular truncated cone with upper and lower base radii of $2$ and $4$ respectively, and the central angle of the sector ring on its lateral surface is $90^{\circ}$, then the surface area of the circular truncated cone is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68\pi}
Mike's Age. "Pat O'Connor," said Colonel Crackham, "is now \(1 \frac{1}{3}\) times as old as he was when he built the pigsty under the window of his living room. Little Mike, who was 3 years and 4 months old when Pat built the pigsty, is now 2 years older than half the age Biddy, Pat's wife, was when Pat built his pigsty. So, when little Mike reaches the age Pat was when he built the pigsty, the combined age of all three will be one hundred years. How old is little Mike now?"
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 \frac{16}{21}}
方程 $19 x+93 y=4 x y$ 的整数解的组数是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the real numbers \( a \), \( b \), and \( c \) such that \( f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \) holds for any real number \( x \), find the maximum value of \( a + b + c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the value of $\int_{\vert z \vert =r} f(z) dz $ where $f(z) = (1-{1 \over {z^2}}) e^{z+ {1 \over z}} $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $g(t)$ be a continuous function that satisfies $\int_{-1}^{1}g(t)dt = 3$. Let $f(x,y) = g(3x-y)$. Calculate the double integral of $f(x,y)$ over the area $D=\{(x,y) : |3x|+|y|\le 1\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A $5 \times 5$ grid of squares is filled with integers. Define a rectangle as "corner-odd" if its sides are grid lines and the sum of the integers at its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?

Note: A rectangle must have four distinct corners to be considered "corner-odd"; i.e., no $1 \times k$ rectangle can be "corner-odd" for any positive integer $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
There are some kids in a summer camp. Half of the kids are going to soccer camp, and 1/3 of the kids going to soccer camp are going in the morning. 750 kids are going to soccer camp in the afternoon. 1/5 of the other half of kids in the camp are going to basketball camp. One third of the kids going to basketball camp are going in the morning and the rest in the afternoon. If 200 kids are going to the morning basketball camp, how many kids are in the summer camp in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6000}
## Problem Statement

Calculate the definite integral:

$$
\int_{\sqrt{3}}^{\sqrt{8}} \frac{x-\frac{1}{x}}{\sqrt{x^{2}+1}} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\ln\sqrt{\frac{2}{3}}}
Suppose $a, b, c$, and $d$ are pairwise distinct positive perfect squares such that $a^{b}=c^{d}$. Compute the smallest possible value of $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{305}
Let $S = \{1, 2, \dots, n\}$ for some integer $n > 1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if \begin{enumerate} \item[(i)] $\pi(k) > \pi(k+1)$ for $k=1$; \item[(ii)] $\pi(k-1) < \pi(k)$ and $\pi(k) > \pi(k+1)$ for $1 < k < n$; \item[(iii)] $\pi(k-1) < \pi(k)$ for $k=n$. \end{enumerate} (For example, if $n=5$ and $\pi$ takes values at $1, 2, 3, 4, 5$ of $2, 1, 4, 5, 3$, then $\pi$ has a local maximum of 2 at $k=1$, and a local maximum of 5 at $k=4$.) What is the average number of local maxima of a permutation of $S$, averaging over all permutations of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n+1}{3}}
A finite set \( S \) of points in the coordinate plane is called overdetermined if \( |S| \geq 2 \) and there exists a nonzero polynomial \( P(t) \), with real coefficients and of degree at most \( |S| - 2 \), satisfying \( P(x) = y \) for every point \( (x, y) \in S \).

For each integer \( n \geq 2 \), find the largest integer \( k \) (in terms of \( n \) ) such that there exists a set of \( n \) distinct points that is not overdetermined, but has \( k \) overdetermined subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}-n}
A truncated cone is circumscribed around a sphere, with the area of one base being 4 times larger than the area of the other base. Find the angle between the generatrix of the cone and the plane of its base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \left( \frac{1}{3} \right)}
Given that the three views of a certain solid are all squares with a side length of 2, if this solid is carved into a sphere, then what is the maximum surface area of the sphere? Express your answer as a multiple of pi (π).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters, given that the area of the cross is $810 \mathrm{~cm}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Brad answers 10 math problems, one after another. He answers the first problem correctly and answers the second problem incorrectly. For each of the remaining 8 problems, the probability that he answers the problem correctly equals the ratio of the number of problems that he has already answered correctly to the total number of problems that he has already answered. What is the probability that he answers exactly 5 out of the 10 problems correctly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(3) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the gcd of $a + b$ and $c$ is greater than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given the parametric equations:
$$x = t + \frac{1}{t}$$
$$y = t - \frac{1}{t}$$
Find the eccentricity of the curve. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Find the least positive integer $N>1$ satisfying the following two properties: There exists a positive integer $a$ such that $N=a(2 a-1)$. The sum $1+2+\cdots+(N-1)$ is divisible by $k$ for every integer $1 \leq k \leq 10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is at least half of the number of beetles in the tenth jar. How many beetles are in the sixth jar?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
已知函数$f ( x ) = \lg ( a x ^ { 2 } + 2 x + 1 )$的值域是一切实数，则实数a的取值范围是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$0 \leq a \leq 1$}
Two people, Jia and Yi, start simultaneously from points A and B and walk towards each other. When Jia reaches the midpoint of AB, they are 5 kilometers apart. When Yi reaches the midpoint of AB, they are $\frac{45}{8}$ kilometers apart. Find the distance between points A and B in kilometers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Construct a string by using only n characters, the string should not contain any consecutive repeated pattern, in other words, the string must not have any substring matching /(.+)\1/, what is the length l(n) of the longest string S(n)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
The volume of a regular tetrahedron \( ABCD \) is 1, and \( O \) is its center. The regular tetrahedron \( A'B'C'D' \) is symmetrical to \( ABCD \) with respect to the point \( O \). Determine the volume of the common part of these two regular tetrahedrons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
All the milk in container A, which was filled to its brim, was poured into two containers B and C. The quantity of milk in container B was some percentage less than the capacity of container A. When 150 liters were transferred from C to B, both the containers had equal quantities of milk. The initial quantity of milk in container A was 1200 liters. What was the percentage of milk in container B compared to container A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.5\%}
Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. Please provide the value of $k$ when $n = 6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine which digit is in the 1000th position after the decimal point in the decimal expansion of the number $\frac{9}{28}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The ratio of boys to girls in a math class is 5:8, the ratio of math to science students is 7:4, and the ratio of science to literature students is 3:5. If the total number of students across all three subjects (math, science, and literature) is 720, how many girls are in the math class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
How to calculate the following limit:
$$\lim_{n\to\infty}n\int_0^1 nx^{n-1}\left(\frac{1}{1+x}-\frac{1}{2}\right)\mathrm dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Two positive integers differ by 8 and their product is 168. What is the larger integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A finite set of polygons on a plane is considered properly placed if, for any two polygons in the set, there exists a line passing through the origin that intersects both. 

Find the smallest natural number \( m \) such that for any properly placed set of polygons, one can draw \( m \) lines passing through the origin such that each polygon in the set intersects at least one of these \( m \) lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many functions $f: \{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ are there such that $f(f(x)) = f(f(f(x)))$ for all $x \in \{1,2,3,4,5\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{756}
Out of 1500 people surveyed, 25% do not like television, and 15% of those who do not like television also do not like games. How many people surveyed do not like both television and games?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Joe went on a diet 5 months ago when he weighed 222 pounds. He initially lost 12 pounds in the first month, followed by a loss of 6 pounds in the second month, and then gained 2 pounds in the third month. In the fourth and fifth months, he lost a total of 8 pounds. If Joe's weight loss continues to fluctuate like this pattern, in approximately how many months will he weigh 170 pounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}