inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the expression with the pattern of alternating subtraction and addition of squared numbers, starting with 100 and ending with 1. What is the result of this expression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5050}
What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1304}
A frustum of a cone with a height of 8 units contains a sphere O1 with a radius of 2 units. The center of sphere O1 lies on the axis of the frustum, and it is tangent to the upper base and the side surface of the frustum. Another sphere O2 with a radius of 3 units can be placed inside the frustum such that it is tangent to sphere O1, the lower base of the frustum, and the side surface of the frustum at one point each. Excluding sphere O2, what is the maximum number of additional spheres with a radius of 3 units that can be placed inside the frustum? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26597}
Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Find the coefficient of ${x}^{2}{y}^{7}$ in the expansion of $((x+y)(x-y{)}^{8})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A porter needs to transport 200 buns from the kitchen to the construction site (he is currently in the kitchen). Each time, he can carry 40 buns. However, he eats 1 bun each way, whether traveling from the kitchen to the construction site or from the construction site to the kitchen. How many buns can the porter transport to the construction site at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{191}
给定正整数 $n$, 非负实数 $a_{1}$, $a_{2}, \cdots, a_{n}$ 满足 $a_{1}+a_{2}+\cdots+a_{n}=4$ 。记
$$
S=54(a_{1}+a_{1} a_{2}+\cdots+a_{1} a_{2} \cdots a_{n})-13\sqrt{13}.
$$
求 $S$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{305}
Given 15 points in space, 5 of which are collinear, what is the maximum possible number of unique planes that can be determined by these points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{445}
In Anchuria, there is a checkers championship that takes place over several rounds. The days and cities hosting the rounds are determined by a draw. According to the rules of the championship, no two rounds can be held in the same city, and no two rounds can be held on the same day. A lottery is conducted among the fans: the grand prize goes to the person who correctly guesses which cities and on which days all rounds will be held before the championship begins. If no one guesses correctly, the grand prize goes to the championship organizing committee. There are a total of eight cities in Anchuria, and the championship spans a total of eight days. How many rounds should be in the championship so that the organizing committee has the highest probability of winning the grand prize?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Lindsey saved $50 in September, $37 in October, and $11 in November. Lindsey's mom said that since Lindsey had saved more than a certain amount, she would give Lindsey $25. After spending $87 on a video game, Lindsey had $36 left. What was the minimum amount Lindsey needed to save for her mom to give her $25?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$98}
Circles $\mathcal{C}_1, \mathcal{C}_2,$ and $\mathcal{C}_3$ have their centers at $(0,0), (12,0)$, and $(24,0)$, and have radii $1, 2$, and $4$, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y),$ and that $x=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given variables x, y, and z, if x is less than y by 45% and y exceeds z by 30%, determine the percentage by which z exceeds x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39.86\%}
In triangle $ABC$, angle $A$ is $40^{\circ}$. The triangle is randomly thrown onto a table.

Find the probability that vertex $A$ will be east of the other two vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. Find the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The set  $M= \{1;2;3;\ldots ; 29;30\}$  is  divided in  $k$ subsets such that if  $a+b=n^2, (a,b \in M, a\neq b, n$  is an
integer number  $)$ , then  $a$  and  $b$  belong different subsets.
Determine the minimum value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A frog is placed at the origin on the number line , and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$, or to the closest point with a greater integer coordinate that is a multiple of $13$. A move sequence is a sequence of coordinates that correspond to valid moves, beginning with $0$ and ending with $39$. For example, $0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39$ is a move sequence. How many move sequences are possible for the frog?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{2 x^{3}-40 x-8}{x(x+4)(x-2)} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x+\ln|x|+\ln|x+4|-6\cdot\ln|x-2|+C}
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条, 则由这 3030 条线段能构成不全等的三角形的个数为  (用数字作答).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510555}
In a regular pentagonal prism, a diagonal is defined as a line segment connecting two vertices that are neither on the same lateral face nor on the same base. How many diagonals does a regular pentagonal prism have? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Question 13: Given that $A$ and $B$ are two subsets of $\{1,2, \ldots, 100\}$, satisfying: $|A|=|B|, A \cap B=\emptyset$, and for any $x \in A, 2 x+2 \in B$. Try to find the maximum value of $|A \cup B|$.

---

The translation maintains the original format and line breaks as requested.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
设a、b为正整数，$a ^ { 2 } + b ^ { 2 }$除以a+b的商为q，余数为r，且$q ^ { 2 } + r = 2010$．求ab 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1643}
Let  $a,x,y$  be positive integer such that  $a>100,x>100,y>100$  and  $y^2-1=a^2(x^2-1)$   . Find the minimum value of  $\frac{a}{x}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expected number of rounds played until the game ends?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16/3}
From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught up with the cyclist at point $D$, which is $\frac{2}{3}$ km from point $C$. Find the speed of the bus (in km/h), given that the distance between points $A$ and $B$ is 4 km and the speeds of the bus and the cyclist are constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
If \( a^3 + b^3 + c^3 = 3abc = 6 \) and \( a^2 + b^2 + c^2 = 8 \), find the value of \( \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Calculate the definite integral:

$$
\int_{1}^{64} \frac{6-\sqrt{x}+\sqrt[4]{x}}{\sqrt{x^{3}}-7 x-6 \sqrt[4]{x^{3}}} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \ln \left( \frac{2}{2 \sqrt{2} + 1} \right)}
The cards in a stack are numbered consecutively from 1 to $2n$ from top to bottom. The top $n$ cards are removed to form pile $A$ and the remaining cards form pile $B$. The cards are restacked by alternating cards from pile $B$ and $A$, starting with a card from $B$. Given this process, find the total number of cards ($2n$) in the stack if card number 201 retains its original position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{402}
On the Saturday of a weekend softball tournament, Team A plays Team D, Team B plays Team E, and Team C gets a bye (no match). The winner of Team A vs. Team D plays against Team C in the afternoon, while the winner of Team B vs. Team E has no further matches on Saturday. On Sunday, the winners of Saturday's afternoon matches play for first and second places, and the remaining teams play based on their win-loss status for third, fourth, and fifth places. There are no ties. Determine the total number of possible five-team ranking sequences at the end of the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Problem 7.5. In each room of the hotel, no more than 3 people can be accommodated. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. In one room, only men or only women can be accommodated; also, fans of different teams cannot be accommodated together. How many rooms need to be reserved to definitely accommodate all the fans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
At a certain university, 69% of the professors are women, and 70% of the professors are tenured. If a certain percentage of the professors are women, tenured, or both, and 52% of the men are tenured, what percent of the professors are women, tenured, or both?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85.12\%}
A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \geq 0$. Find $\operatorname{gcd}(a_{999}, a_{2004})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{677}
Determine the smallest positive integer $n$ such that there exist positive integers $x_1, x_2, \ldots, x_n$ so that each natural number from 1001 to 2021 inclusive can be expressed as the sum of one or more distinct terms $x_i$ (where $i = 1, 2, \ldots, n$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Example 5 Given that the function $f(n)$ is defined on the set of positive integers and takes positive integer values, is strictly monotonically increasing, and satisfies $f[f(n)]=3n$, find the value of $f(2004)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3825}
Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Let \( O \) be the center of the circumcircle of \( \triangle ABC \). \( P \) and \( Q \) are the midpoints of \( AO \) and \( BC \), respectively. Suppose \( \angle CBA = 4 \angle OPQ \) and \( \angle ACB = 6 \angle OPQ \). Find the size of \( \angle OPQ \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12^\circ}
15 ladies and 30 gentlemen attend a luxurious party. At the start of the party, each lady shakes hands with a random gentleman. At the end of the party, each lady shakes hands with another random gentleman. A lady may shake hands with the same gentleman twice (first at the start and then at the end of the party), and no two ladies shake hands with the same gentleman at the same time.

Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that the collection of ladies and gentlemen that shook hands at least once can be arranged in a single circle such that each lady is directly adjacent to someone if and only if she shook hands with that person. Find the remainder when $m$ is divided by 10000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a scalene triangle \( ABC \), one of the angles is equal to the difference of the other two angles, and one of the angles is twice as large as another. The angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circumcircle of the triangle at points \( L \), \( O \), and \( M \) respectively. Find the area of triangle \( LOM \) if the area of triangle \( ABC \) is 2. If the answer is not an integer, round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Scatterbrained Scientist walks to work every morning and back home every evening. He has two umbrellas and takes one with him under two conditions: it is raining outside or there is no umbrella at his destination. Over time, he calculates that he carries an umbrella 20% of the time. Find the probability of rain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
A cargo ship carrying 6 kinds of items - doohickies, geegaws, widgets, yamyams, thingamajigs, and whatchamacallits - arrives at the port. Each item weighs 3, 11, 5, 7, 13, and 17 pounds, respectively, and each item is weighed as it is unloaded. Let a, b, c, d, e, and f represent the number of doohickies, geegaws, widgets, yamyams, thingamajigs, and whatchamacallits, respectively.

If the product of the individual weights of the unloaded items equals 351,125,648,000 pounds and the sum of the individual weights equals 3255 pounds, how many widgets (c) have been unloaded?

Consider the following simultaneous equations:

3a + 11b + 5c + 7d + 13e + 17f = 3255
3^a * 11^b * 5^c * 7^d * 13^e * 17^f = 351,125,648,000
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A baker bakes 5 loaves of bread an hour in one oven. He has 4 ovens. From Monday to Friday, he bakes for 5 hours, but on Saturday and Sunday, he only bakes for some hours. The baker bakes 1740 loaves of bread in 3 weeks. How many hours does the baker bake on Saturday and Sunday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The function $g(x)$ satisfies the equation
\[xg(y) = 2yg(x)\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Triangle $A B C$ has side lengths $A B=15, B C=18, C A=20$. Extend $C A$ and $C B$ to points $D$ and $E$ respectively such that $D A=A B=B E$. Line $A B$ intersects the circumcircle of $C D E$ at $P$ and $Q$. Find the length of $P Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
If $x + \frac{1}{x} = 3$, what is $x^6 + \frac{1}{x^6}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{322}
已知椭圆$\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1\left( a>b>0 \right)$的离心率为$\frac{\sqrt{2}}{2}$，右焦点为$F\left( 1,0 \right)$，三角形$ABC$的三个顶点都在椭圆上，设它的三条边$AB$、$BC$、$AC$的中点分别为$D$、$E$、$F$，且三条边所在直线的斜率分别为${{k}_{1}},{{k2}},{{k}_{3}}\left( {{k1}}{{k}_{2}}{{k3}}\ne 0 \right)$.若直线$OD$、$OE$、$OF$的斜率之和为-1（$O$为坐标原点），则$\frac{1}{{{k}_{1}}}+\frac{1}{{{k2}}}+\frac{1}{{{k}_{3}}}=$______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the limit of the following expression as $n$ approaches infinity, assuming $n$ is a positive integer: $$\lim_{n\to\infty}\frac{2-4+\cdots+(2n-2)-2n}{\sqrt{ (n^2+4)}+ \sqrt{ (n^2-4)}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given \( x \in [0, 2\pi] \) and \( 2 \cos x \leqslant |\sqrt{1+\sin 2x} - \sqrt{1-\sin 2x}| \leqslant \sqrt{2} \), find the range of values for \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{\pi}{4}, \frac{7 \pi}{4} \right]}
In a $10 \times 10$ grid square with the centers of all unit squares marked (a total of 100 points), what is the minimum number of lines, not parallel to the sides of the square, that need to be drawn to cross out all the marked points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $MATH$ be a trapezoid with $MA = AT = TH = 5$ and $MH = 11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
On one particular Wednesday, Jack worked \( t-2 \) hours and earned \( 3t-2 \) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \( 2t-7 \) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same amount of earnings. Determine the value of \( t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{3}}
4. Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:

- First, Sofia covers the entire board with non-overlapping dominoes that do not extend outside the board,
- Then, Viktor writes down a natural number $n$ on a piece of paper without seeing the arrangement of the dominoes on the board, and then examines the board with the dominoes, selects $n$ dominoes, and fixes (glues) them to the board,
- After that, Sofia removes the remaining (unfixed) dominoes from the board and tries to achieve a new complete covering of the board, which will differ from the initial one.

If Sofia manages to find a new different covering, she wins; otherwise, Viktor wins.

Determine the smallest possible value of $n$ for which Viktor can always win, regardless of the initial covering.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011^2}
The king of the Lazy Kingdom issued a decree on Sunday, April 1, 2007, eliminating Fridays from all subsequent weeks. Since then, in his kingdom, Saturday always follows Thursday and the week has only six days. What day of the week will April 9, 2008, fall on in the Lazy Kingdom? (Don't forget that the year 2008 is a leap year!)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Tuesday}}
Given a square with an area of $1$, if the square revolves around one of its sides for one full turn, the lateral surface area of the resulting geometric body is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
There are five students taking a graduation photo in a row, among them, student A cannot stand next to student B, and students B and C must stand together. The number of different arrangements is \_\_\_\_\_\_ (answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A sequence of real numbers $x_0, x_1, x_2, \ldots$ is defined as follows: $x_0 = 1989$ and for each $n \geq 1$,
\[ x_n = -\frac{1989}{n} \sum^{n-1}_{k=0} x_k. \]
Calculate the value of $\sum^{1989}_{n=0} 2^n x_n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1989}
For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The product of the digits of \( \frac{x}{k} \) is a perfect cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1998}
The rectangle $3 \times 100$ consists of 300 squares $1 \times 1$. What is the largest number of diagonals that can be drawn in the squares so that no two diagonals share endpoints? (In one square, two diagonals can be drawn without sharing endpoints. Shared internal points are allowed.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given the function f(x) = 2^x - log_{1/2} x and that a is a root of f(x), if x_0 > a, determine the relationship between f(x_0) and 0. Express your answer as a comparison to 0 (e.g., > 0, < 0, = 0).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{> 0}
Lucas chooses one, two or three different numbers from the list $2, 5, 7, 12, 19, 31, 50, 81$ and writes down the sum of these numbers. (If Lucas chooses only one number, this number is the sum.) How many different sums less than or equal to 100 are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The cells of a $2 \times 2019$ table must be filled with numbers (exactly one number in each cell) according to the following rules. The top row must contain 2019 real numbers, none of which are equal to each other, and the bottom row must contain the same 2019 numbers, but in a different order. Each of the 2019 columns must contain two different numbers, and the sum of these two numbers must be a rational number. What is the maximum number of irrational numbers that could be in the first row of the table? 

(S. Kudrya)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
The circle touches the sides of an angle at the points \(A\) and \(B\). A point \(M\) is chosen on the circle. The distances from \(M\) to the sides of the angle are 24 and 6. Find the distance from \(M\) to the line \(AB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}