inclusionAI/Ring-lite-rl-data huggingface dataset:
An equilateral triangle \( \triangle ABC \) has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides of the triangle are drawn. The 15 intersection points of these lines, including the vertices of \( \triangle ABC \), are called lattice points. Out of these 15 lattice points, find the minimum number \( n \) such that among any \( n \) chosen points, there are always three points that form an isosceles triangle (including an equilateral triangle).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
How many solutions are there to the equation $x^2(x+1)^2\equiv 0\pmod{1000}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Mike, Ted, and Sarah planted tomatoes and peppers. In the morning, Mike planted 50 tomato seeds and 30 pepper seeds, while Ted planted twice as many tomato seeds as Mike and half as many pepper seeds. Sarah planted 30 seeds more than Mike for both tomatoes and peppers. In the afternoon, Mike planted 60 tomato seeds and 40 pepper seeds, while Ted planted 20 fewer tomato seeds than Mike and the same number of pepper seeds. Sarah planted 20 more tomato seeds and 10 more pepper seeds in the afternoon than she did in the morning. How many tomato and pepper seeds did they plant altogether?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{685}
From the vertex of the obtuse angle $A$ of triangle $ABC$, a perpendicular $AD$ is drawn. A circle with center $D$ and radius $DA$ is drawn, which intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Find $AC$ if $AB = c$, $AM = m$, and $AN = n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{mc}{n}}
On a table lie two spheres that touch each other externally. A cone touches the table with its lateral surface as well as both spheres (externally). The vertex of the cone is on the segment connecting the points where the spheres touch the table. It is known that the rays connecting the vertex of the cone with the centers of the spheres form equal angles with the table. Find the maximum possible angle at the vertex of the cone. (The angle at the vertex of the cone is defined as the angle between its generatrices in the axial cross-section.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \operatorname{arcctg} 2}
How many 9-digit numbers are there in the decimal system that are divisible by 11 and in which every digit except zero occurs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31680}
Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2043231}
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x(x+2)(x+5)}{(x+1)(x+3)}}
6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is 2, and

the lateral edge is 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15\sqrt{2}}{4}}
A person moved along a slope with a ratio of 1:2 and traveled 100√5 meters. Calculate the increase in the person's position. Express your answer in meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
An outlet pipe A empties a tank which is full in 10 hours. There is also another outlet pipe B with a different emptying rate. If the inlet pipe is kept open, which lets water in at the rate of 16 litres/min, and both outlet pipes A and B are working together, the tank would take 8 hours longer to be emptied. Find the capacity of the tank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21600}
Somewhere in the universe, $n$ students are taking a 10-question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest $n$ such that the performance is necessarily laughable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{253}
We multiply the number $1234567 . . . 1000$ (juxtaposed are all natural numbers 1 to 1000) by a number from 1 to 9, and strike out all 1’s in the product. We multiply the number obtained by a nonzero one-digit number once again, and strike out the 1’s, and so on, many times over. What is the least number one can obtain in this manner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A TV station is broadcasting 6 advertisements in a row, including 3 different commercial ads, 2 different Olympic promotional ads, and 1 public service ad. The last ad cannot be a commercial ad, and the Olympic promotional ads cannot be played consecutively with each other or with the public service ad. How many different broadcasting methods are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 1 }
The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{16}}
Given \(a, b, c \in (0, 1]\) and \(\lambda \) is a real number such that
\[ \frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c), \]
find the maximum value of \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{64}{27}}
Let \( A, B, C, D \) be four points in space that are not coplanar. Each pair of points is connected by an edge with a probability of \( \frac{1}{2} \), and whether or not there is an edge between any two pairs of points is independent of the others. Determine the probability that \( A \) and \( B \) can be connected by a path (composed of one or multiple edges).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Find all pairs  $(m, n)$  of positive integers, for which number  $2^n - 13^m$  is a cube of a positive integer.

*Proposed by Oleksiy Masalitin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 9)}
The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that these two people are not adjacent in the front row. Calculate the number of different ways to adjust their positions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{560}
Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+i$，另两根的积为 $5+6i$，求 $b$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $m \in \mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and a line passing through a fixed point $B$ is given by $mx-y-m+3=0$. These two lines intersect at point $P(x, y)$. Find the maximum value of $|PA|+|PB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{5}}
Given the conditions $p$: {$x ||x-a| < 3 $}, and $q$: {$x |x^{2}-2x-3 < 0 $}, and $\neg p$ is a necessary but not sufficient condition for $\neg q$, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,2]}
From 1812 identical squares with a side length of 1 mm, a rectangular frame was made for a group photo. The boundaries of the photo coincide with the inner boundaries of the frame. Then the photo was cut along the millimeter grid line into two rectangular parts. Now, two frames were needed, which used up 2018 of the same squares. Find the dimensions of the original photo.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101 \times 803 \text{ mm}}
Naeun, Leejun, Yoonseo, Seohyun, and Soyul are interviewed in order. When Yoonseo and Seohyeon switched, who will be the 4th person being interviewed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Yoonseo}
Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{204}}
Generalize the cube-octahedron-cube problem to an n-dimensional hypercube. What is the ratio of the side lengths of the innermost and outermost hypercubes in terms of n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{n}$}
Calculate: $\frac{145}{273} \times 2 \frac{173}{245} \div 21 \frac{13}{15}=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7395}{112504}}
In a plane, there is a square \(ABCD\) with side length 1 and a point \(X\). It is known that \(XA = \sqrt{5}\) and \(XC = \sqrt{7}\). What is the length of \(XB\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6 - \sqrt{10}}}
Let \( N \) be the set of all positive integers, and let \( f \) be a function from \( N \) to \( N \) such that for any \( s \) and \( t \) in \( N \), the following condition is satisfied:
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2}.
$$

Determine the minimum possible value of \( f(1998) \) among all such functions \( f \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1998}
Xiaoming has 4 identical coins, each with a head and a tail. He wants to stack the 4 coins in such a way that the heads of adjacent coins do not face each other. Find the number of different stacking methods. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the largest integer $k$ such that:
\[
k \leq \sqrt{2} + \sqrt[3]{\frac{3}{2}} + \sqrt[4]{\frac{4}{3}} + \sqrt[5]{\frac{5}{4}} + \cdots + \sqrt[2015]{\frac{2015}{2014}}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Two trains of different lengths run at the speed of 60 kmph and 48 kmph in opposite directions in parallel tracks. The time they take to cross each other is 9.99920006399488 seconds. If the other train is 160 m long, how long is the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{139.98}
袋子 $A$ 中装有 2 张 10 元纸币和 3 张 1 元纸币，袋子 $B$ 中装有 4 张 5 元纸币和 3 张 1元纸币。现随机从两个袋子中各取出两张纸币，则 $A$ 中剩下的纸币面值之和大于 $B$ 中剩下的纸币面值之和的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let \(f(x)\) be an odd function defined on \(\mathbb{R}\), and \(f(2)=0\). When \(x > 0\), it always holds that \(xf''(x)-f(x) < 0\). Find the solution set of the inequality \(x^{2}f(x) > 0\). Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2) \cup (0, 2)}
3. What is the minimum number of 3-cell corners that need to be painted in a $5 \times 5$ square so that no more corners can be painted? (Painted corners should not overlap.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
11. Among the positive integers $1,2, \cdots, 20210418$, how many numbers have a digit sum of 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["5100"]}
On a circular railway, there are \( n \) stations. At times, the station attendants communicate with each other via radio. At any given moment, the communication session is held by only two people. Over the course of a day, there was exactly one radio session between each pair of stations. For each station (considering only the sessions it engaged in), it turned out that it communicated with other stations alternately in the order of their arrangement on the railway (either clockwise or counterclockwise, with different directions for different stations), starting with one neighboring station and ending with the other. What could \( n \) be equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $a, b, c, d, e$ be nonnegative integers such that $625 a+250 b+100 c+40 d+16 e=15^{3}$. What is the maximum possible value of $a+b+c+d+e$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
Given the ellipse $\frac{x^2}{3} + y^2 = 1$, let line $l$ intersect the ellipse at points $A$ and $B$. The distance from the coordinate origin $O$ to line $l$ is $\frac{\sqrt{3}}{2}$. Determine the maximum area of $\triangle AOB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
How many polynomials of degree exactly $5$ with real coefficients map the set \( \{1, 2, 3, 4, 5, 6\} \) to a permutation of itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{718}
When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.5}
Given four points in space that are not coplanar, how many planes are there from which all four points are equidistant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A number is supposed to be multiplied by 9 and then squared, but instead it is divided by 9 and then the square root is taken. What is the percentage of error obtained in the final result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99.59\%}
We divided a chord $AB$ of a circle, which has the length of the radius, and the shorter arc $AB$ into three equal parts each. The line connecting the division points closer to $A$ intersects the line connecting the division points closer to $B$ at point $P$. What is the measure of angle $APB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a regular tetrahedron $\mathrm{ABCD}$ with a point $\mathrm{P}$ inside it such that $P A=P B=\sqrt{11}$ and $P C=P D=\sqrt{17}$. What is the side length of the tetrahedron $\mathrm{ABCD}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let x1, x2, x3,..., x9 be distinct positive integers such that x1 < x2 <... < x9 and their sum is 220. When the value of x1 + x2 +... + x5 is maximized, find the minimum value of x9 - x1. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Compute the norm of the functional $f \in (W^{2,1}[0,1])^*$ defined by
$$f(u) = \int_0^{1/2} u(t)\,dt.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is the value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
John and Ingrid pay 30% and 40% tax annually, respectively. However, John is eligible for a $2000 tax reduction and Ingrid has an exemption of 15% on her total income before taxation. If John's annual income is $58000 and Ingrid's is $72000, what is their combined tax rate after considering these deductions and exemptions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30.68\%}
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
The sequence $\left(z_{n}\right)$ of complex numbers satisfies the following properties: $z_{1}$ and $z_{2}$ are not real. $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \geq 1$. $\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \geq 1$. $\left|\frac{z_{3}}{z_{4}}\right|=\left|\frac{z_{4}}{z_{5}}\right|=2$ Find the product of all possible values of $z_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65536}
Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions:
1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square.
2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square.
Find the maximum value of the number of the “good square”.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n^2 - 2004n }
A store has buttons in six different colors. What is the minimum number of buttons that need to be bought in order to sew them in a row such that for any two different colors in the row, there are two adjacent buttons of those colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
If the equation $\left(a-3\right)x^{2}-4x+1=0$ with respect to $x$ is a quadratic equation, then the range of values for $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq 3}
In how many different ways can the faces of a cube be painted using six given colors (each face must be painted entirely with one color), considering that only those colorings are different which cannot be made identical by rotating the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$810$}
Two hexagons are attached to form a new polygon \( P \). Compute the minimum number of sides that \( P \) can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the inequality $(x-a+2)(x^2-ax-2) \geq 0$ for $x > 0$, find the exact value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a city, there are $2019$ metro stations. Some pairs of stations are connected by tunnels, and from any station, you can reach any other station through these tunnels. The mayor ordered the organization of several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and each station must be on at least one line. To save money, no more than $k$ lines should be made. It turned out that the mayor's order is not feasible. What is the largest value of $k$ for which this could happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
Evaluate the infinite geometric series:
$$\frac{5}{3} - \frac{5}{4} + \frac{25}{48} - \frac{125}{384} + \dots$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{21}}
In a right square prism $A B C D A_{1} B_{1} C_{1} D_{1}$, the height is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. If the answer is $x$ inches, what is the value of $100x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Compute the expected value of the number of maximal decrescent blocks in a random permutation of $\{1, \dots, n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{n+1}{2}$}
At 7:00, five sheep, designated as A, B, C, D, and E, have distances to Wolf Castle forming an arithmetic sequence with a common difference of 20 meters. At 8:00, these same five sheep have distances to Wolf Castle forming another arithmetic sequence, but with a common difference of 30 meters, and their order has changed to B, E, C, A, D. Find how many more meters the fastest sheep can run per hour compared to the slowest sheep.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(2) = 0$. If for $x > 0$, $xf'(x) + f(x) > 0$, then the solution set of the inequality $xf(x) > 0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2) \cup (2, +\infty)}
A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have independent lifetimes. What is the smallest number of bulbs to be purchased so that the succession of light bulbs lasts for at least 40 months with probability at least 0.9772?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}