inclusionAI/Ring-lite-rl-data huggingface dataset:
Fedja used matches to form a parallelogram, ensuring that the sides are of equal length and the vertices do not lie on a single line. It is observed that exactly 7 or 9 matches fit into the diagonals. Determine the number of matches that make up the perimeter of the parallelogram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[ \sum_{t \mid n} d(t)^3 \] is divisible by $35$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
已知正整数 $n$ 满足 $n \neq 2017$ ，且 $n^{n}$ 与 $2017^{2017}$ 有相同的个位数字，求 $|2017-n|$ 的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
设 $A$ 为 20 ! 的所有因数构成的集合, $x_{i} \in A(1 \leq i \leq n)$ .若对任意的正整数 $y_{i}(1 \leq i \leq n)$, 存在正整数 $z$, 使得 $\frac{y_{i}+z}{x_{i}}$ 为正整数.
求 $n$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$9$}
$$
\begin{aligned}
& \text { [Isosceles, inscribed, and circumscribed trapezoids] } \\
& \text { [ Right triangle with an angle of \$30^\circ\$ ] } \\
& \text { [ Pythagorean theorem (direct and inverse). ] } \\
& \text { [ Equilateral (equiangular) triangle ] }
\end{aligned}
$$

Quadrilateral $A B C D$ is inscribed in a circle with center $O, \angle B O A=\angle C O D=60^{\circ}$. The perpendicular $B K$, dropped to side $A D$, is equal to $6 ; A D=3 B C$.

Find the area of triangle $C O D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{63\sqrt{3}}{4}}
The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
If the solution set of the inequality $x^{2} < |x-1|+a$ is a subset of the interval $(-3,3)$, then the range of the real number $a$ is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 5]}
Given $x \gt 0$, $y \gt 0$, and $\left(x+1\right)\left(y+1\right)=2x+2y+4$, find the minimum value of $xy$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{407}
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
Evaluate: $u =\displaystyle\int_0^\infty\dfrac{dx}{x^4 +7x^2+1}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{6}\)}
Given an arithmetic-geometric sequence $\{a\_n\}$, the sum of its first $n$ terms is denoted as $S\_n$. If $a\_3 = 2S\_2 + 1$ and $a\_4 = 2S\_3 + 1$, find the common ratio $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a finite sequence $\{a_n\}$ with $k$ terms, and it satisfies $a_{n+1}=a_{n-1}- \frac {n}{a_{n}}$ for $n=2,…,k-1$. If $a_{1}=24$, $a_{2}=51$, and $a_{k}=0$, then $k=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
数列 \{a_n\} 定义如下：设 \df{(2n)!}{n!(n+2024)!} 写成既约分数后的分母为 A(n)，a_n 等于 2A(n) 的最大质因数，则 a_n 的最大值等于__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4027}
On a scale, there are 20 carrots, 10 apples, and 5 oranges weighing a total of 8.70 kg. When 4 carrots and 2 apples are removed from the scale, the average weight of the remaining 16 carrots and 8 apples is 206 grams. If the average weight of an apple is 210 grams, what is the average weight (in grams) of the 4 carrots which were removed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{834}
At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.

[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil\frac{n}{2}\rceil}
Given the series representation $$\sum_{k=0}^n \frac{A_k}{x+k}=\frac{n!}{x(x+1)(x+2)(x+3)\cdots(x+n)}$$ find the value of $A_7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
During the holidays, Jordan read 130 French novels, while his brother Alexandre read 1/10 of what Jordan read. Their sister Camille read 2 times as many novels as Alexandre, and their friend Maxime read 5 less than half the total number of novels read by Jordan, Alexandre, and Camille combined. How many more novels did Jordan read than Maxime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Among all triangles formed by selecting any 3 vertices from the 8 vertices of a rectangular prism, calculate the number of acute-angled triangles. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
12.074. A sphere is inscribed in a cone. The ratio of the radius of the circle of contact between the spherical and conical surfaces to the radius of the base of the cone is $k$. Find the cosine of the angle between the slant height of the cone and the plane of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-k}
Fill the natural numbers \(1 \sim 22\) in the boxes below (each box can hold only one number). Among the 11 fractions formed, the maximum number of fractions with integer values is \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the smallest composite number that has no prime factors less than 20.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{529}
In rectangle $ABCD$, $AB = 4$ cm, $BC = 10$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fourth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{10}}
Two swimmers start at opposite sides of a 90-foot swimming pool. One swims at 3 feet per second, while the other swims at 2 feet per second. They swim back and forth for 12 minutes without taking any time to turn around. How many times do they meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c, a \neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
8. Given that the three sides of $\triangle A B C$ are $10$, $12$, and $14$, respectively, and that $\triangle A B C$ is folded into a tetrahedron along its three midlines. Then the diameter of the circumscribed sphere of the tetrahedron is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{55}}
Given the sets $P=\{0, m\}$ and $Q=\{x|2x^2-5x<0, x\in \mathbb{Z}\}$, if $P\cap Q \neq \emptyset$, then find all possible values of $m$. Express your answer as a list of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2}
Find a number that is both the sum of two perfect squares and the difference of two perfect cubes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
A pool contains one-eighteenth of its capacity in water. Two water pipes begin to fill the pool simultaneously. When the pool reaches two-ninths of its capacity, the first pipe starts to fill the pool alone and takes 81 minutes to fill an amount equal to what the second pipe had already added. Then, the second pipe fills the pool alone for 49 minutes, and at this point, the total amount of water added by both pipes is the same. After that, both pipes continue to fill the pool. How many additional minutes are needed for both pipes to completely fill the pool?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231 \text{ minutes}}
在某片树林中有n(≥3)个鸟巢，而且它们之间的距离不相等.每一个巢中各有一只鸟，在某一时刻某些鸟离开自己的巢而飞落到其他的巢中.如果某一对鸟之间的距离小于另外一对鸟之间的距离（一只鸟可以与其他任何已知鸟结成“一对”），那么在飞落之后，第一对鸟之间的距离就大于第二对鸟之间的距离.问：当n取何值时能做到这一点？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly $5$ values of $j \ge 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Select 3 out of 5 members from the class committee to serve as the class's Academic Affairs, Entertainment, and Sports commissioners. Given that members A and B cannot serve as the Entertainment commissioner, determine the number of different methods for this selection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the number of 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) such that each element \(a_i\) is chosen from the set \(\{1,2,3,4\}\) and the expressions \[ a_j^2 - a_j a_{j+1} + a_{j+1}^2 \] for \(j = 1, 2, 3, 4, 5, 6\) (with \(a_7 = a_1\)) are all equal to each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 4 : 5}
Given the function $y=\log_{a}(x-3)+2$ $(a > 0, a \neq 1)$, its graph passes through a fixed point $A$. If point $A$ is also on the graph of the power function $f(x)$, then $f(2)=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Carla is taking attendance for her sixth-grade class. She knows that 2 students are in the restroom, and a certain multiple of that number of students minus one are absent. There are four rows with six desks each that are all 2/3 full. Carla teaches 23 students, including the ones in and out of the classroom. What is the multiple of the number of students in the restroom to find the number of absent students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If
\[ 1 \cdot 1995 + 2 \cdot 1994 + 3 \cdot 1993 + \dots + 1994 \cdot 2 + 1995 \cdot 1 = 1995 \cdot 997 \cdot y, \]
compute the integer \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665}
Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D$. Then, if all measurements are in degrees, angle $BDC$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}(180-A)}
In the incircle of triangle ABC, the line AA1 intersects the incircle again at Q. The lines A1C1 and A1B1 intersect the line passing through A and parallel to BC at P and R respectively. Given that angle PQC1 = 45 degrees and angle RQB1 = 65 degrees, calculate the angle PQR. Express your answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110^\circ}
5. A $20 \times 19$ rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
We will say that a list of positive integers is *admissible* if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that from each admissible list, some numbers can be crossed out such that the sum of the numbers left uncrossed is greater than or equal to $1810-d$ and less than or equal to $1810+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $\frac{c}{d}$ is the length of a side of the square, find the value of $c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200689}
Determine all real values of $p$ for which the function $d(x, y) = |x-y|^p$ is a metric on $\mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
A math class has fewer than 50 students. When the students try to sit in rows of 8, 5 students are left in the last row. When the students try to sit in rows of 6, 3 students remain in the last row. How many students are in this class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
In a $4 \times 4$ grid, place 4 pieces such that there is exactly one piece in each row, each column, and each diagonal. How many ways are there to do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In how many ways can the numbers \(1, 2, \ldots, 2002\) be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4004}
对正整数 n, 考虑含有 n 阶元的有限非交换群, 记 $a_{n}$ 为这些群的阶的最小可能值.则 $a_{1}+a_{2}+a_{3}+a_{4}=\qquad.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Determine the value of $p$ for which the series $\sum_{n=1}^{\infty} (-1)^n \cdot \arctan \left(\frac{1}{\sqrt{n^p}}\right) \cdot \log \left(1+\frac{1}{n^{2p}}\right)$ converges conditionally.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/5}
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of  $50$  meters per second, while each of the spiders has a speed of  $r$  meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum  $c$  so that for any  $r<c,$   the fly can always avoid being caught?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
从一个正九边形的 9 个顶点中选 3 个使得它们是一个等腰三角形的三个顶点的方法数是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
There are 300 integers between a certain number and 1000 such that their unit digit is odd. What is the starting number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Find the smallest positive integer value of n such that the expansion of (3x + 1/x sqrt(x))^n contains a constant term. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If the graph of the power function $y=f(x)$ passes through the point $\left( -2,-\frac{1}{8} \right)$, find the value(s) of $x$ that satisfy $f(x)=27$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Find the smallest positive integer \( n \) such that there exists a complex number \( z \), with positive real and imaginary part, satisfying \( z^{n} = (\bar{z})^{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let  $ a,b$  be integers greater than  $ 1$ . What is the largest  $ n$  which cannot be written in the form  $ n \equal{} 7a \plus{} 5b$  ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
A top hat contains $3$ red chips and $2$ green chips. Chips are drawn randomly, one at a time without replacement, until all $3$ of the reds are drawn or until both green chips are drawn. What is the probability that the $3$ reds are drawn? Express your answer in the form of \(\frac{k}{m}\) as a common fraction, and give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Eight-sided polygon ABCDEFGH has integer side lengths. It can be divided into a rectangle and a square. The area of the square is greater than the area of the rectangle. The product of the two areas is equal to 98. What could be the perimeter of ABCDEFGH?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le  m \le n$. Find the integer nearest to $b_{36}/b_{25}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5394}
In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\sqrt{10}$. Determine the area of quadrilateral $AEDC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Point  $P$  lies outside a circle, and two rays are drawn from  $P$  that intersect the circle as shown. One ray intersects the circle at points  $A$  and  $B$  while the other ray intersects the circle at  $M$  and  $N$ .  $AN$  and  $MB$  intersect at  $X$ . Given that  $\angle AXB$  measures  $127^{\circ}$  and the minor arc  $AM$  measures  $14^{\circ}$ , compute the measure of the angle at  $P$ .

[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(40,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label(" $A$ ",A,plain.NE);label(" $B$ ",B,plain.NW);label(" $M$ ",M,SE);
label(" $P$ ",P,E);label(" $N$ ",N,dir(250));label(" $X$ ",X,plain.N);[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Find the remainder when $712!$ is divided by $719$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{718}
Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 \mid (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1729}
In a binary search tree, when searching for the key value 30, nodes containing the key values 10, 20, 40, 50, 70, 80, and 90 are traversed, not necessarily in the order given. How many different orders are possible in which these key values can occur on the search path from the root to the node containing the value 30?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Calculate the limit of the function:

$$\lim_{x \rightarrow \frac{1}{3}} \frac{\sqrt[3]{\frac{x}{9}}-\frac{1}{3}}{\sqrt{\frac{1}{3}+x}-\sqrt{2x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2 \sqrt{2}}{3 \sqrt{3}}}
Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only condition is that if the union of two different subsets $A$ and $B$ equals $M$, then $A$ and $B$ must be given different colors. What is the least possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Sum up those natural numbers less than 1980, in which the sum of the digits is even!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{979605}
A sphere with a radius of 2017 is placed at the base of a tower. On top of this sphere, three hemispheres are stacked, each with the same radius of 2017. The centers of the sphere and the hemispheres are collinear. Calculate the maximum height of this tower.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10085}
Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1 \leq x,y \leq 1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit goes to the point $(0,0)$ strictly before Ayush does can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The numbers \(1,2,\cdots,9\) are randomly placed into the \(9\) squares of a \(3 \times 3\) grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is odd. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In a triangle $\Delta ABC$, it is given that \( \frac{2\cos A}{a} +\frac{\cos B}{b} +\frac{2\cos C}{c}=\frac{a}{bc} +\frac{b}{ca} \). Find the angle $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Provide an example of a simple equation with rational coefficients whose graph has no rational coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 = 3}