inclusionAI/Ring-lite-rl-data huggingface dataset:
If integers $m$, $n$, and $k$ satisfy the equation $m^2 + n^2 + 1 = kmn$, what values can $k$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are three equal-radius, tangent circles. What is the area of the lobe between the circles if the radius of each circle is $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^{2}(\sqrt{3}-\frac{\pi}{2})}
Given vectors $\overrightarrow {m}=(a,-1)$, $\overrightarrow {n}=(2b-1,3)$ where $a > 0$ and $b > 0$. If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, determine the value of $\dfrac{2}{a}+\dfrac{1}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8+4\sqrt {3}}
30 students participated in the mathematical Olympiad. Each of them was given 8 problems to solve. The jury estimated their work with the following rule:
1) Each problem was worth \( k \) points, if it wasn't solved by exactly \( k \) students;
2) Each student received the maximum possible points in each problem or got 0 in it.

Lasha got the least number of points. What's the maximal number of points he could have?

**Remark:** Rule 1 means that if the problem was solved by exactly \( k \) students, then each of them got \( 30 - k \) points in it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$32$}
In how many ways can $n$ square envelopes of different sizes be arranged by inclusion? For instance, with $6$ envelopes $A, B, C, D, E, F$ (listed in decreasing order of their sizes), one way of arranging them would be $F \in C \in B$, $E \in B, D \in A$, where $I \in J$ means envelope $I$ is placed in envelope $J$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Given a function $f(x)=e^{x}+ax-3$ where $a\in \mathbb{R}$, if for any $x_{1}$, $x_{2}\in [1,+\infty)$ and $x_{1} < x_{2}$, the inequality $x_{2}f(x_{1})-x_{1}f(x_{2}) < a(x_{1}-x_{2})$ holds, then the range of values for $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 3]}
Exercise 6. In 5 boxes are found respectively $402, 403, 404, 405$ and 406 stones. The only allowed operation is to take 4 stones from a pile that has at least 4 stones and to put one in each of the other piles. What is the maximum number of stones that can be in a single pile?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
5. Points $M, N, P$ are the feet of the altitudes dropped from the vertices of triangle $A B C$ with angles $37.5^{0}, 67.5^{0}, 75^{0}$ to its sides. Find the ratio of the areas of triangles $M N P$ and $A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2\sqrt{3}+3\sqrt{2}-\sqrt{6}-4):8}
At the end of the term, Bob wrote his current grades in singing in a row and placed multiplication signs between some of them. The product of the resulting numbers turned out to be 2007. What grade does Bob get in singing for the term? (The music teacher does not give "fails".)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $ABC$ be a triangle with circumcenter $O$, incenter $I, \angle B=45^{\circ}$, and $OI \parallel BC$. Find $\cos \angle C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{\sqrt{2}}{2}}
In the Cartesian coordinate system $xOy$, point $P$ is a common point of the graph of the function $y=2\ln x$ and the circle $M$: $(x-3)^{2}+y^{2}=r^{2}$, and they have a common tangent line at point $P$. If the graph of the quadratic function $y=f(x)$ passes through points $O$, $P$, and $M$, then the maximum value of $y=f(x)$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {9}{8}}
The average of the numbers $3$, $4$, $5$, $6$, $a$, $10$, $11$ is $8$. What is the $60$th percentile of this set of data?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
There are 51 students taking a test with 6 questions, and the scoring rules are as follows: each person starts with 6 points, with 4 points added for each correct answer, 1 point deducted for each incorrect answer, and no change in points for unanswered questions. What is the minimum number of students that must have the same score? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Shvecov D.V.

On the hypotenuse $AC$ of the right triangle $ABC$, a point $C_{1}$ is marked such that $BC = CC_{1}$. Then, on the leg $AB$, a point $C_{2}$ is marked such that

$AC_{2} = AC_{1}$; the point $A_{2}$ is defined similarly. Find the angle $AMC$, where $M$ is the midpoint of the segment $A_{2}C_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that this number will be divisible by 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{5}$}
Problem 9.7. In front of a sweet-tooth lie five boxes of candies: the first box contains 11 candies, the second - 22 candies, the third - 33 candies, the fourth - 44 candies, the fifth - 55 candies. In one move, the sweet-tooth can take four candies from one box and distribute them, one candy to each of the remaining four boxes.

At any point, the sweet-tooth can take the candies from any box and leave. What is the maximum number of candies he can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159}
Let \(a, b\) be nonzero complex numbers, and \(\frac{a}{b}\) is not a real number. Define:
\[ 
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\}, 
\]
\[ 
R_{a, b}=\left\{z \mid z \text{ is a nonzero complex number, and } L_{a, b} = L_{z a, z}\right\}.
\]

Find the maximum number of elements in the set \(R_{a, b}\) as \(a, b\) change.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The alphabet contains \( n > 1 \) letters; a word is any finite sequence of letters in which any two neighboring letters are different. A word is called good if it is not possible to delete all letters except for four such that the remaining sequence is of the form \( a a b b \), where \( a \) and \( b \) are different letters. Find the maximum possible number of letters in a good word.

(D. Khramiov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n+1}
A pen and its ink refill together cost $\;\$1.10$. The pen costs $\;\$1$ more than the ink refill. What is the cost of the pen in dollars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.05}
设 A 为数集 \{1,2,3,\cdots,2024\} 的 n 元子集，且 A 中的任意两个数既不互素又不存在整除关系. 求 n 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
The pentagon \(ABCDE\) is inscribed around a circle. The angles at its vertices \(A\), \(C\), and \(E\) are \(100^{\circ}\). Find the angle \(ACE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40^	ext{circ}}
Company C sells a line of 40 products with an average retail price of $1,800. If none of these products sells for less than $500, and exactly 15 of the products sell for less than $1,400, what is the greatest possible selling price of the most expensive product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30,900}
1st Irish 1988 Problem 5 Let X = {1, 2, 3, 4, 5, 6, 7}. How many 7-tuples (X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 ) are there such that each X i is a different subset of X with three elements and the union of the X i is X?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7!\times6184400}
Eleven best football teams played one match against each other. Each team scored 1 goal in the first match, 2 goals in the second match, up to 10 goals in the tenth match. What is the maximum number of drawn matches that could have occurred?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
In the sequence $\{a_n\}$, we have $a_1=1$, and for $n \geqslant 2$, $a_n=a_{n-1}+n$. If the inequality $\frac{λ}{n} > \frac{n+1}{a_n+1}$ is satisfied for any $n \in \mathbb{N}^{*}$, then the range of the real number $λ$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2, +\infty)}
What is the volume of the region above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{7 \pi(2 - \sqrt{2})}{3}$}
There are  $20$  geese numbered  $1-20$  standing in a line. The even numbered geese are standing at the front in the order  $2,4,\dots,20,$  where  $2$  is at the front of the line. Then the odd numbered geese are standing behind them in the order,  $1,3,5,\dots ,19,$  where  $19$  is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered  $1,2,\dots,20$  (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?

*Author: Ray Li*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given the parabolic equation \( y^2 = 2px \) with \( p > 0 \) and focus \( F \), a line passing through \( F \) and having an inclination angle \( \theta \) intersects the parabola at points \( A \) and \( B \). Find the maximum area of the triangle \( \triangle ABO \) (where \( O \) is the origin).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p^2}{2}}
There are 7 clean sheets of paper on a table. Every minute, Vasya selects any 4 out of them and draws one star on each of the chosen sheets. Vasya wants each sheet to end up with a different number of stars (meaning no two sheets have the same number of stars). What is the minimum total number of stars Vasya will have to draw to achieve this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Given that $|\overrightarrow{a}|=5$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -12$, find the projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
A train of some length moving at a speed of 72 kmph takes 91.9926405887529 seconds to cross another train of length 540 m, moving at 36 kmph in the same direction. What is the length of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{379.93}
For every positive integer $n$, the function $f$ is defined as follows:
- $f(2n+1) = 2f(2n)$
- $f(2n) = f(2n-1) + 1$
- $f(1) = 0$

Determine the remainder when $f(2005)$ is divided by $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
All the vertices of a regular pyramid $P A B C D$ lie on the lateral surface of a cylinder, the axis of which is perpendicular to the plane $P A B$. Find the radius of the base of the cylinder if $A B = a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{\sqrt{3}}}
Evaluate the line integral $\displaystyle \int _C\dfrac{1+2z}{1+z}dz$ where $C$ is the smooth curve lying in $\Omega=\{z\in \mathbb C:Im z>0\}$ with initial point $-1+2i$ and final point $1+2i$, given that the Residue Theorem cannot be applied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 - 0.5\ln 2 + i\frac{\pi}{4}}
Let $a, b, c$ be distinct positive integers such that $b+c-a$, $c+a-b$, and $a+b-c$ are all perfect squares. What is the largest possible value of $a+b+c$ smaller than 100?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
$7 \cdot 32$ Find the smallest natural number $n$, such that every convex 100-gon can be represented as the intersection of $n$ triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the number of irreducible submodules isomorphic to $V_4$ in an $\mathfrak{sl}(2,\mathbb{C})$-module $V$ given that the eigenvalues of the action $h$ are 6, 4, 2, 0, -2, -4, and -6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations 
\[
2x + y = 2022 \quad \text{and} \quad y = |x-a| + |x-b| + |x-c|
\]
Determine the minimum value of $c$ such that the system has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11475}
设 $n(n \geqslant 3)$ 是正整数，集合 $M=$ $\{1,2, \cdots, 2 n\}$ 。求最小的正整数 $k$, 使得对于 $M$ 的任何一个 $k$ 元子集，其中必有 $4$ 个互不相同的元素之和等于 $4 n+1$ 。请提供当 n = 48 时的 k 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
According to relevant regulations, the behavior of a motor vehicle driver with a blood alcohol content of 20 mg/100 ml or more is considered drunk driving. Suppose that after drinking, the blood alcohol content is $p_0$ mg/100 ml, and after $x$ hours, the alcohol content drops to $p$ mg/100 ml, satisfying the relationship $$p=p_{0}\cdot e^{rx}$$ (where $r$ is a constant). If a person's blood alcohol content is 89 mg/100 ml after drinking, and 61 mg/100 ml 2 hours later, then this person can drive after \_\_\_\_\_\_ hours. (Round up to the nearest hour)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that $$32+ \frac {2}{7} = 3 \frac {2}{7}$$, $$33+ \frac {3}{26} = 3 \frac {3}{26}$$, $$34+ \frac {4}{63} = 3 \frac {4}{63}$$, ..., $$32014+ \frac {m}{n} = 2014 \cdot 3 \frac {m}{n}$$, ..., $$32016+ \frac {a}{b} = 2016 \cdot 3 \frac {a}{b}$$, then find the value of $$\frac {b+1}{a^{2}}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
There are prime numbers $a$, $b$, and $c$ such that the system of equations
$$a \cdot x - 3  \cdot y + 6  \cdot z = 8$$
$$b  \cdot x + 3\frac12  \cdot y + 2\frac13  \cdot z = -28$$
$$c  \cdot x - 5\frac12  \cdot y + 18\frac13  \cdot z = 0$$
has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{154}
Ali and Veli go hunting. The probability that each will successfully hit a duck is 1/2 on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. Calculate the probability that Veli hits more ducks than Ali. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $f: \mathbb{N} \to \mathbb{R}$ be the function \[ f(n) = \sum_{k=1}^{\infty} \frac{1}{\operatorname{lcm}(k,n)^2}. \] It is well-known that $f(1) = \frac{\pi^2}{6}$. What is the smallest positive integer $m$ such that $m \cdot f(10)$ is the square of a rational multiple of $\pi$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C=60^\circ$, $b=\sqrt{6}$, and $c=3$, find the measure of angle $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^\circ}
Consider a circle with a radius of 5 cm. If we draw a rectangle of maximum size inside the circle and a parallelogram of maximum size tangent to the circle from outside such that the longer sides of the parallelogram are tangent to the circle and the vertices of the rectangles are touching the shorter sides of the parallelogram, what is the sum of the areas of the rectangle and the parallelogram?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses each of the numbers from 1 to 1993 exactly once. At the end of the game, the second player receives a reward equal to the absolute value of the algebraic sum written on the board. What is the maximum reward the second player can guarantee for themselves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1993}
When $x=$____, the expressions $\frac{x-1}{2}$ and $\frac{x-2}{3}$ are opposite in sign.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{7}{5}"]}
If 6 women take an hour to dig a ditch, then how long should 18 women take to dig a ditch of the same type?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
What is the $33$-rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the function f(x) = ax^2 + bx + 3a + b, which is an even function with a domain of [2a, a-1]. Find the value of a+b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Find the last non-zero digit of the product of all natural numbers from 1 to 40.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the degree of the field extension $K=\mathbb{Q}(\alpha\zeta)$ over $\mathbb{Q}$, where $\alpha =2^{1/5}$ and $\zeta=e^{2i\pi/5}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If \( x + k \) is a factor of \( 3x^2 + 14x + a \), find \( k \). (\( k \) is an integer.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A rectangle, a square, and an equilateral triangle all have the same perimeter. The length of the side of the square is 9 cm. Find the length of the shorter sides of the rectangle. Express your answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \text{ cm}}
In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \leq z \leq 1$. Let $S_{1}, S_{2}, \ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin ( $0,0,0$). The expected volume of the region $S_{1} \cap S_{2} \cap \cdots \cap S_{2022}$ can be expressed as $\frac{a \pi}{b}$, for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271619}
A classroom consists of a $5 \times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{962}
A motorist traveled from point A to point $\mathrm{D}$, which are 100 km apart. The road from A to D passes through points B and C. At point B, the navigator showed that 30 minutes remained to reach the destination, and the motorist immediately reduced his speed by 10 km/h. At point C, the navigator showed that 20 km remained, and the motorist reduced his speed by another 10 km/h. (The navigator determines the remaining time based on the current speed of travel.) Determine the initial speed of the car, given that the travel time from B to C was 5 minutes longer than the travel time from C to D.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \text{ km/h}}
6.1. We will call a natural number interesting if all its digits, except the first and last, are less than the arithmetic mean of the two adjacent digits. Find the largest interesting number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96433469}
In a race, A covers the distance in 20 seconds and B in 25 seconds. A beats B by 20 meters. What is the total distance of the race?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given the hyperbola $C:\frac{{{x^2}}}{{{a^2}}}-\frac{{{y^2}}}{{27}}=1({a＞0})$ with left and right foci $F_{1}$ and $F_{2}$ respectively, and one of its asymptotes has a slope of $\frac{π}{3}$. If point $P$ lies on the hyperbola and $|PF_{1}|=7$, then $|PF_{2}|=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
What is the inverse of the polynomial $2x^2+2$ in the finite field $\mathbb{Z}_3[x]/(x^3+2x^2+2)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + 2x + 2}
Let $\alpha$ be a real number, and consider the sequence $(a_n)_n$ defined by $a_0 = 1$ and $a_n = \alpha(\alpha-1)\cdots(\alpha - (n-1))$ for $n \geq 1$. Determine the exponential generating function (EGF) for this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 + x)^\alpha}
If the range of \( y = \log_{2016} (x^2 - ax + 65) \) is \( \mathbb{R}^{+} \), then the range of the value of \( a \) is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-16 < a < 16}
Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The Greene family, composed of Mr. and Mrs. Greene and their three children, went to the amusement park. They spent a total of $45 on discounted admission tickets, which originally cost $50, and spent $13 less than the original ticket cost on food and beverages. Mr. Greene spent an additional $15 on a souvenir for himself, while Mrs. Greene spent double that amount on souvenirs for their children. Each child played games costing $9 each, and the family paid $25 for round-trip transportation. The tax applied on all their purchases, except transportation and admission tickets, was 8%. Calculate the total amount the Greene family spent on their trip to the amusement park after taking into account all expenditures, discounts, and taxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$187.72}
A cube with side length 2 is cut three times to form various polyhedral pieces. It is first cut through the diagonal planes intersecting at vertex W, then an additional cut is made that passes through the midpoint of one side of the cube and perpendicular to its adjacent face, effectively creating 16 pieces. Consider the piece containing vertex W, which now forms a pyramid with a triangular base. What is the volume of this pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}