inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the number of elements that a set  $B$  can have, contained in  $(1, 2, ... , n)$ , according to the following property: For any elements  $a$  and  $b$  on  $B$  ( $a \ne b$ ),  $(a-b) \not| (a+b)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lceil \frac{n}{3} \right\rceil}
If a ^ 3 is divisible by some number, the least possible value of integer a is 60. What is the number that a ^ 3 is divisible by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216,000}
## Task 2 - 201242

In a fish shop, there are three water containers available for keeping live carp.

At the start of sales, there are exactly 20 carp in each of these three containers. At the end of sales, there are a total of 3 carp left. The sold carp were removed one by one.

No exchange of carp between containers took place; no new carp were added during the sales.

Calculate, rounded to 3 decimal places, the probability that at the end of sales, there is exactly one carp in each of the three containers!

Hint:

The probability $p$ to be determined is defined as follows: Let $A$ be the number of all different possibilities for the sequence of removing 57 carp from the three containers. Furthermore, let $G$ be the number of those possibilities among these where at the end of sales, there is exactly one carp in each of the three containers.

Two possible sequences of removal are considered the same if and only if they agree for each $i=1,2, \ldots, 57$ in specifying from which container the $i$-th removal of a carp took place.

With these notations, $p=\frac{G}{A}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.108}
已知 $f(x)=e^{\sin x^{2}}$, 则 $\left[-f^{(8)}(0)\right]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5040}
Given \( n, k \in \mathbb{Z}_{+} \) with \( n > k \). For real numbers \( a_{1}, a_{2}, \ldots, a_{n} \in (k-1, k) \), let positive real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) satisfy that for any set \( I \subseteq \{1,2, \ldots, n\} \), \( |I|=k \), we have \( \sum_{i \in I} x_{i} \leq \sum_{i \in I} a_{i} \). Find the maximum value of \( x_{1} x_{2} \cdots x_{n} \).

(33rd China Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_1 a_2 \cdots a_n}
If the sum of the digits of a natural number is the same as the sum of the digits of three times that number, but different from the sum of the digits of twice that number, we call such a number a "wonder number." Find the smallest "wonder number."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
A train is running at a certain speed and it crosses a post in 18 seconds. The length of the train is 200.016 meters. What is the speed of the train in km/hr?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40.0032}
How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In $\triangle ABC$, if $a= \sqrt {5}$, $b= \sqrt {15}$, $A=30^{\circ}$, then $c=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt {5}}
1 Let $w=-\frac{1}{2}+\frac{\sqrt{3}}{2} \cdot i\left(i^{2}=-1\right)$, then the number of different values that the algebraic expression $w^{m}+w^{n}+w^{l}(m, n, l$ are pairwise distinct integers) can take is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The regular octagon \( A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8} \) has a side length of 1. If we select any two points \( A_i \) and \( A_j \), what is the maximum value of \( \overrightarrow{A_i A_j} \cdot \overrightarrow{A_{1} A_{2}} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} + 1}
If $\{a_n\}$ is a sequence of real numbers for $n \ge 1$ such that
\begin{equation} a_{n+2}=a_{n+1}-a_n \tag{1} \end{equation}
\begin{equation} \sum_{n=1}^{999} a_n=1003 \tag{2} \end{equation}
\begin{equation} \sum_{n=1}^{1003}a_n=-999 \tag{3} \end{equation}
Then find the value of $$\sum_{n=1}^{2002} a_n $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{99} - 1 }
For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree. For each sapling, she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. Once a sapling is turned into a perfectly-shaped apple tree, it will remain so. 

The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{789953}
7.191. $\log _{10} x+\log _{\sqrt{10}} x+\log _{\sqrt[3]{10}} x+\ldots+\log _{\sqrt[1]{10}} x=5.5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[10]{10}}
Given \( z \in \mathbf{C} \). If the equation \( x^{2} - 2 z x + \frac{3}{4} + \mathrm{i} = 0 \) (where \(\mathrm{i}\) is the imaginary unit) has real roots, then find the minimum value of \( |z| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the cubic function f(x) = mx^3 - x on the interval (-∞, +∞), find the range of values for m such that the function is decreasing. Express your answer as an inequality or interval, using any necessary mathematical notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m < 0}
Given that the two foci of the ellipse $C$ are $F\_1$ and $F\_2$, a line passing through point $F\_1$ intersects the ellipse $C$ at points $M$ and $N$. If $|MF\_2|=|F\_1F\_2|$, and $|MF\_1|=2$, $|NF\_1|=1$, determine the eccentricity of the ellipse $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Example 12 Find the minimum value of $\frac{3}{\cos x}+\frac{2}{\sin x}\left(0<x<\frac{\pi}{2}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{(\sqrt[3]{9}+\sqrt[3]{4})^{3}}}
Let \( A, B, C \) be points in that order along a line, such that \( AB = 20 \) and \( BC = 18 \). Let \( \omega \) be a circle of nonzero radius centered at \( B \), and let \( \ell_{1} \) and \( \ell_{2} \) be tangents to \( \omega \) through \( A \) and \( C \), respectively. Let \( K \) be the intersection of \( \ell_{1} \) and \( \ell_{2} \). Let \( X \) lie on segment \(\overline{KA}\) and \( Y \) lie on segment \(\overline{KC}\) such that \( XY \parallel BC \) and \( XY \) is tangent to \( \omega \). What is the largest possible integer length for \( XY \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Sandy has 10 books, Benny has 24 books, and Tim has 33 books. Each of their friends, Rachel, Alex, and Jordan, also have books. Rachel has twice as many books as Benny, Alex has 3 less than half of Tim's books, and Jordan has the same number of books as the sum of Sandy's and Benny's books. How many books do they all have together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Determine the relationship (less than, equal to, or greater than) between $2\sqrt{2}-\sqrt{7}$ and $\sqrt{6}-\sqrt{5}$. (Please fill in "<", "=", or ">" on the line)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}-\sqrt{7} < \sqrt{6}-\sqrt{5}}
Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so that $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. If the area of triangle $CD'X'$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime, find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1369}
Among the 27 points consisting of the 8 vertices of a cube, the midpoints of its 12 edges, the centers of its 6 faces, and the center of the cube, how many groups of three collinear points are there? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
The price of a stock decreased by 5% last year and changed by some percentage this year. The net percentage change in the price of the stock is 4.499999999999993%. What was the percentage increase in the price of the stock this year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1}
Regular polygons $I C A O, V E N T I$, and $A L B E D O$ lie on a plane. Given that $I N=1$, compute the number of possible values of $O N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If $a$, $b$ are integers and $s = a^3 + b^3 - 60ab(a + b) \geq 2012$, find the least possible value of $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
In the isosceles trapezoid $ABCD$, $AD \parallel BC$, $\angle B = 45^\circ$. Point $P$ is on the side $BC$. The area of $\triangle PAD$ is $\frac{1}{2}$, and $\angle APD = 90^\circ$. Find the minimum value of $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. Find, $\mathbb{E}[max(2.5A+B, A + 2.5B)]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{2\sqrt{\pi}}\)}
Compute

$$\log_2\left(\prod\limits_{a=1}^{2015}\prod\limits_{b=1}^{2015}(1+e^{2\pi iab/2015})\right)$$

Here $i$ is the imaginary unit (that is, $i^2=-1)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13725}
Consider the following diagram showing a rectangular grid of dots consisting of 3 rows and 4 columns. How many rectangles can be formed in this grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
With three different digits, all greater than  $0$ , six different three-digit numbers are formed. If we add these six numbers together the result is  $4.218$ . The sum of the three largest numbers minus the sum of the three smallest numbers equals  $792$ . Find the three digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8, 7, 4}
In a certain boys camp, 20% of the boys are from School A, 25% are from School B and the rest are from School C. Of those from School A, 30% study Science and 40% study Mathematics. Of the boys from School B, 50% study Science and 10% study English. No boys from School C study English. The rest have not specified their subject of interest. If there are 56 boys in the camp that are from School A but do not study Science, and 35 boys from School B who study English, how many total boys are there in the camp?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
In the expansion of \((x+y+z)^{8}\), find the sum of the coefficients for all terms of the form \(x^{2} y^{a} z^{b}\) where \(a, b \in \mathbf{N}\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1792}
Two circles are given: the first has center A(5,3) and radius 12, and the second has center B(2,-1) and radius 6. Find the shortest distance between the two circles. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
The two faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are possible positions for a beam.)  No two beams have intersecting interiors.  The interiors of each of the four faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
3030
\]}
There are two inlets and one outlet to a cistern. One of the inlets takes some time to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern. Both of the inlets are turned on at 9:00 am with the cistern completely empty, and at 10:00 am, the outlet is turned on and it takes 1 more hour to fill the cistern completely. The outlet working alone takes 2 hours to empty the cistern when the cistern is full. How much time does the first inlet take to fill up the cistern?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Znayka cut a semicircle from paper. Neznaika marked a point \( C \) on the diameter \( AB \) of this semicircle and cut two semicircles with diameters \( AC \) and \( CB \) from Znayka's semicircle. The area of the remaining figure turned out to be \( 10\pi \). Find the length of the part of the chord lying inside, which passes through point \( C \) perpendicularly to \( AB \). If necessary, round the answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.32}
Jenna runs a wholesale business. She pays $3 for each widget and resells it for a certain amount. Every month she has to pay $10,000 in rent, 20% of her total profit in taxes, and $2,500 each to four workers. If Jenna sells 5000 widgets one month and her total profit is $4000, how much does she sell each widget for?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$7.96}
Find the difference between the second largest and the third smallest three-digit numbers, each of which has 1, 6, and 8 on each digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
If set $A=\{x\in N\left|\right.-1 \lt x\leqslant 2\}$, $B=\{x\left|\right.x=ab,a,b\in A\}$, then the number of non-empty proper subsets of set $B$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3375}
There are 120 people that own pets. 25 people own only dogs, 18 people own only cats, 12 people own only birds, 15 people own only snakes, 7 people own only hamsters, 8 people own both cats and dogs, 5 people own both dogs and birds, 6 people own both cats and birds, 7 people own both cats and snakes, 10 people own both dogs and snakes, 4 people own both dogs and hamsters, 3 people own both cats and hamsters, 5 people own both birds and hamsters, 2 people own both birds and snakes, 3 people own both snakes and hamsters, 3 people own cats, dogs, and birds, 4 people own cats, dogs, and snakes, 2 people own cats, snakes, and hamsters, and 1 person owns all the categories: cats, dogs, birds, snakes, and hamsters. How many total snakes are there among these pet owners?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Evaluate the double sum and integral: $$ \sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) \, dx. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3}}{2}}
What is the closed form for the following combinatorial sum involving binomial coefficients?
$$\sum_{i=a+1}^n {i-1 \choose a}{n-i \choose k-a}$$

Given that $k \geq a$ and $n \geq a+1$, find a simplified expression for this sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n}{k+1}}
Suppose that \[ \prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}} \] where $a_{1}$, $a_{2}$,..., $a_{m}$ are nonzero and $k_{1}< k_{2}< \cdots < k_{m}$. Find $a_{1996}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665280}
If the real part and the imaginary part of the complex number $(a-i)(1-i)i$ (where $i$ is the imaginary unit) are equal, then find the value of the real number $a$. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Andrew has seven pieces of wire of lengths 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, and 7 cm. He bends some of the pieces to form a wire frame in the shape of a cube with edges of length 1 cm without any overlaps. What is the smallest number of these pieces that he can use? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the integral $$\int_{0}^{2\pi} \frac{\sin^4(4x)}{\sin^2(x)} \, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{207}
Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^{n}$ such that from any three of them, at least two are orthogonal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Square \(ABCD\) with side length 2 is inscribed in a circle. Using each side of the square as a diameter, semicircular arcs are drawn. Calculate the area of the shaded region outside the circle and inside the semicircles. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Form a five-digit number using the digits 1, 2, 3, 4, 5 without repeating any digit, satisfying the condition that among the odd numbers 1, 3, 5, exactly two odd numbers are adjacent. Calculate the total number of such five-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
A circle is circumscribed around an isosceles triangle with a $45^\circ$ angle at the vertex. Another circle is inscribed within this triangle, touching the first circle internally and also touching the two equal sides of the triangle. The distance from the center of the second circle to this vertex of the triangle is 4 cm. Find the distance from the center of this second circle to the center of the circle inscribed in the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \text{ cm}}
Lev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ integers per minute and Alex runs at $7$ integers per minute (in the same direction). How many minutes from the start of the race will it take Alex to catch up to Lev (who is still running after Alex wakes up)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
$记S_n为等差数列{a_n}的前n项和.已知S_9=-a_5.$
$若a_{1}>0，求使得S_{n} \geq a_{n}的n的取值范围.默认(n\in N)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1,10]}
Let $\triangle ABC$ be a triangle with $BC = 4$, $CA= 5$, $AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
将 3 个 12\times12 的正方形沿邻边的中点剪开分成两部分；将这 6 部分接于一个边长为 6\sqrt{2} 的正六边形边上，若拼接后的图形是一个多面体的表面展开图，则该多面体的体积是__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
Find the sum of the series: $\sum_{k=0}^{15} (-1)^k (k+1) \binom{15}{k}^{-1}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If j is the least positive integer that is divisible by every integer from 10 to 15 inclusive, then j divided by a certain number is equal to 2310. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Given that $a$, $b$, and $c$ form a geometric sequence, determine the number of roots for the equation $ax^3+bx^2+cx=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
When the product of $\underbrace{\frac{2}{3} \times \frac{2}{3} \times \cdots \times \frac{2}{3}}_{10}$ is written as a decimal, what are the first two decimal places?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.01}
Suppose the set $\{1, a^2\} = \{1, a\}$, then $a =$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Cyclic quadrilateral  $ABCD$  satisfies  $\angle ABD = 70^\circ$ ,  $\angle ADB=50^\circ$ , and  $BC=CD$ .  Suppose  $AB$  intersects  $CD$  at point  $P$ , while  $AD$  intersects  $BC$  at point  $Q$ .  Compute  $\angle APQ-\angle AQP$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20^\circ}
The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{5}}