inclusionAI/Ring-lite-rl-data huggingface dataset:
In an acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$, and $\tan C = \frac{4\sqrt{21}}{17}$, the value of $EF$ may be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{267}
Calculate \((8^{2^{6^{4^{2^{5^{8^9}}}}}}) \mod 10000\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8256}
An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.

What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
James injured himself while exercising, pulling his hamstring which caused significant pain for 3 days. Based on his physiotherapist's advice, he realized that this type of injury would actually require at least 5 times as long to fully heal, and he needs to keep in mind that a setback or complication could further delay his recovery. To be cautious, he decided that he would wait another 3 days beyond the initial healing time before even considering light workouts.

When he does get back into it, his physiotherapist wants him to start off with light exercises only for 2 weeks. There's a possibility that complications could arise in this stage, adding an additional week for recovery. If everything goes smoothly, James can then progress to moderate intensity workouts for another week.

Assuming there are no further setbacks, his physiotherapist explained that he would need to wait 3 weeks after transitioning to moderate intensity workouts to start lifting heavy again. 

Taking into account the injury, recovery time, potential setbacks, and various workout intensities, how long could it potentially take before James can resume heavy lifting?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x,z,y$}
设数列 a_{n+1}=\left[\df{a_n}{2}\right]+\left[\df{a_n}{3}\right], n=1,2,\cdots,7，这里 [x] 表示不超过 x 的最大整数. 若 a_8=8，则正整数 a_1 有__________种可能的取值情况.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
求三条直线 $x+y=60, y=\frac{1}{2} x, y=0$ 所围成三角形上的整点个数;
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given two lines $l_{1}$: $(3+m)x+4y=5-3m$ and $l_{2}$: $2x+(5+m)y=8$ are parallel, the value of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{r - p}{q - s - 1}}
On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is the area of triangle \( ADE \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
Determine the Galois group of the polynomial \(x^5 + x - 1\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_2 \times S_3}
A rod has a certain weight. If the weight of a 6 meters long rod is 6.184615384615385 kg, what is the weight of a 13 meters long rod?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.4}
Let $y_n=\sqrt[4]{7+\sqrt[4]{7+\dots+\sqrt[4]{7}}}$ where the expression on the RHS has $n$ roots. Evaluate the limit: $\lim \limits_{n\to \infty}7^n(3-y_n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given functions f(x) and g(x), let a ∈ {x | f(x) = 0} and β ∈ {x | g(x) = 0}. If for all α and β, |α - β| ≤ 1, then f(x) and g(x) are called "adjacent zero-point functions". If f(x) = x - 1 and g(x) = x^2 - ax - a + 3 are "adjacent zero-point functions", find the range of values for the real number a.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2, \frac{7}{3}]}
集 $A=\{0,1,2, \cdots, 9\}, B_1, B_2, \cdots, B_k$ 为 $A$ 的一族非空子集, 并且 $i \neq j$ 时, $B_i \cap B_j$ 至多有两个元素. 求 $k$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{175}
设集合A中元素全为正整数，且对任意x,y∈A(x≠y)，有$\vert x - y \vert \geq \frac { 1 } { 2 5 } x y$，问A中最多有几个数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the improper integral $$\int_\pi^\infty{\frac{x \cos x}{x^2-1}dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a Brownian motion $X_t$, defined for $t\geqslant 0$. Let $T_a$ denote the stopping time at which $X_t$ reaches level $a$, i.e., $T_a = \inf \{t \mid X_t = a\}$. Compute the probability $\mathbb P\{T_{2}< T_{-1} < T_{3}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/12}
Evaluate the line integral \( \int_{C}{z\,\mathrm dx+x\,\mathrm dy-y\,\mathrm dz} \) where \( C \) is the path defined by \( \gamma(t)=(t^2, 1-t, t) \) for \( t \in [-1,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4}{3}}
Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D = 2 \angle B, AD = 5,$ and $ CD = 2.$ Then $ AB$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Given a random variable $X$ with probability density function $f(X)$, let $Y$ be the sum of $N$ independent draws of $X$. Determine the probability density function of $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f^{*N}(y)}
Evaluate the limit: $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + \frac{1}{3}\log{\left(1+3x^3\right)}-1}{2\sin{\left(3x^2\right)}-3\arctan{\left(2x^2\right)}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the minimum distance of the ternary linear code with the following parity check matrix:

$$H=\begin{pmatrix} 1 & 0 & 2 & 1 & 2 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 2 \\ 2 & 2 & 1 & 1 & 0 \end{pmatrix}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(T: \mathbb{R}^4 \to \mathbb{R}^4\) be a linear map with a null space defined by the condition \((x, y, w, z) \in \mathbb{R}^4\) such that \(x + y + w + z = 0\). If the rank of \((T - 4I_4)\) is 3, and the minimal polynomial of \(T\) is \(x(x - 4)^{\alpha}\), find the value of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On an infinite chessboard, some bishops and knights are placed such that:
- Each bishop has a knight on the same diagonal.
- Each knight has a bishop exactly \(\sqrt{5}\) units away.
- Removing any piece disrupts at least one of these conditions.

Determine all possible values of \(n\), the total number of pieces on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the residue of \((1-e^{z})^{-n}\) at \(z = 0\) for any positive integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A positive integer \( N \) is called *apt* if for each integer \( 0 < k < 1009 \), there exists exactly one divisor of \( N \) with a remainder of \( k \) when divided by \( 1009 \). For a prime \( p \), suppose there exists an *apt* positive integer \( N \) such that \( \frac{N}{p} \) is an integer but \( \frac{N}{p^2} \) is not. Determine the number of possible remainders when \( p \) is divided by \( 1009 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Determine the set of values for \( \alpha \) such that the integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 3)}
Calculate the greatest common divisor (GCD) of the complex numbers $6-17i$ and $18+i$ in the ring of Gaussian integers $\Bbb Z[i]$. Provide your answer in the form of a complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2i}
For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\varphi(n)+4 \tau(n)=n$. For example, the number 44 is good because $\varphi(44)+4 \tau(44)=44$. Find the sum of all good positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172}
Evaluate the expression \( s_1 t_k \delta_{ii} \delta_{k1} \delta_{nn} \) where all indices run from 1 to 3. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 s_1 t_1}
What is the probability that a group generated by two random invertible $2 \times 2$ matrices $A$ and $B$ over the integers is isomorphic to the free group $F_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A point is randomly selected on the segment [12, 17]. Let $k$ be the selected value. Determine the probability that the roots of the quadratic equation $(k^2 + k - 90)x^2 + (3k - 8)x + 2 = 0$ satisfy the condition $x_1 \leq 2x_2$. Provide your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
6. In a rectangle $A B C D$ (including the boundary) with an area of 1, there are 5 points, among which no three points are collinear. Find the minimum number of triangles, with these 5 points as vertices, whose area is not greater than $\frac{1}{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the upper bound time complexity (Big-O notation) for the recurrence relation: \[ T(n) = T\left(\frac{n}{3}\right) + T\left(\frac{2n}{3} + 4\right) + a \times n \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n \log n)}
Jonathan wants to buy a dictionary that costs $11, a dinosaur book that costs $19, a children's cookbook that costs $7, an atlas that costs $15, and a set of picture books costing $32. He finds a coupon that offers a 5% discount on his total purchase before taxes. The tax rate is 8%. He has already saved $8 from his allowance. How much more money does Jonathan need to buy all five books, considering the discount and tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$78.18}
A line passing through the focus of the parabola $y^2 = ax$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $x_1 + x_2 = 8$ and the length of segment $AB$ is $12$, then find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知不等式 |ax^2+bx+a|\leq x 对 x\in[1,2] 恒成立，则 3a+b 的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
3. 3.1. Petya marked 5 points on a face of a cube, turned it and marked 6 points on an adjacent face, then turned it again and marked 7 points, and so on. He marked points on each face this way. What is the maximum number of points that can be on two opposite faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the number of integers $2 \le n \le 2016$ such that $n^n - 1$ is divisible by $2$, $3$, $5$, and $7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
How many group homomorphisms are there from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the value of the following infinite series:

$$\sum_{n=1}^\infty \frac{1\cdot3 \cdot ... \cdot (2n-1)}{4^n 2^n n!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2\sqrt{3}}{3} - 1\)}
Given a mechanical device with a lifetime distribution characterized by the hazard rate function \( \lambda(t) = 3.9 \cdot t^2 \) for \( t > 0 \), calculate the probability that the device's lifetime is between 1 and 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2725}
Find the volume of the figure cut off from a cylinder by a plane that passes through the diameter of its base. Given the radius of the cylinder $R$ and the height of the resulting figure $h$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3} h R^{2}}
If a person walks at 35 km / hr instead of 15 km / hr and covers 60% of the distance uphill with a 10% decrease in speed, while the rest of the distance is flat, he would have walked 45 km more. What is the actual distance traveled by him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33.75}
设S={1,2,⋯,7}，则以S的两个不相交的子集为元素的不同2元集的个数为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1093}
Determine the number of palindromes that are less than 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Jade had $38 and her sister Julia had half as much money she had. On New Year's eve, their aunt gave each of them an equal amount of money. How much money did they have in total after their aunt gave them money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$57}
Consider the  $4\times4$  array of  $16$  dots, shown below.

[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]

Counting the number of squares whose vertices are among the  $16$  dots and whose sides are parallel to the sides of the grid, we find that there are nine  $1\times1$  squares, four  $2\times2$  squares, and one  $3\times3$  square, for a total of  $14$  squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the  $14$  squares is missing at least one vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$. 

[i]Mircea Lascu, Dorel Mihet[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lceil \frac{n}{3} \right\rceil}
A pile of candies consists of three types of flavors: chocolate, fruit, and milk. If you take away 5 pieces of chocolate candy, the remaining candies can be divided into 2 identical portions; if you take away 2 pieces of fruit candy, the remaining candies can be divided into 3 identical portions; if you take away 3 pieces of milk candy, the remaining candies can be divided into 5 identical portions. What is the minimum number of candies in this pile?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
John decides to get a new apartment with a roommate. His former rent was $2 per square foot for a 750 square foot apartment. In the first six months of the year, the rent for the new apartment is $2,800 per month and increases by 5% for the last six months. The apartment's utilities cost is $200 per month during the winter months (December, January, and February) and $150 per month during the other months. If John and his roommate split the rent and utilities cost evenly each month, how much money does John save in a year compared to living in his former apartment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$195}
In triangle \(ABC\), angle \(C\) equals \(30^\circ\), and angle \(A\) is acute. A line perpendicular to side \(BC\) is drawn, cutting off triangle \(CNM\) from triangle \(ABC\) (point \(N\) lies between vertices \(B\) and \(C\)). The areas of triangles \(CNM\) and \(ABC\) are in the ratio \(3:16\). Segment \(MN\) is half the height \(BH\) of triangle \(ABC\). Find the ratio \(AH:HC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{AH}{HC} = \frac{1}{3}}
In a particular country, the state of Sunland issues license plates with a format of one letter, followed by three digits, and then two letters (e.g., A123BC). Another state, Moonland, issues license plates where the format consists of two digits, followed by two letters, and then two more digits (e.g., 12AB34). Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates Sunland can issue compared to Moonland, given that Sunland always uses the letter 'S' as the starting letter in their license plates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6084000}
A magician has one hundred cards numbered from 1 to 100. They place each card into one of three boxes - a red box, a white box, and a blue box - in such a way that each box contains at least one card.

An audience member selects two of the three boxes and takes one card from each. They then announce the sum of the numbers on the two cards. Knowing this sum, the magician can determine from which box no card was taken.

How many ways are there to distribute the cards into the boxes so that this trick always works? (Two distributions are considered different if at least one card is placed into a different box.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the additive group of ordered pairs of integers $(m,n) [$with addition defined componentwise: $\left.(m,n)+\left(m^{\prime},n^{\prime}\right)=\left(m+m^{\prime}, n+n^{\prime}\right)\right]$ consider the subgroup $H$ generated by the three elements$$(3,8),\quad(4,-1),\quad(5,4).$$Then $H$ has another set of generators of the form$$(1,b),\quad(0,a)$$for some integers $a,b$ with $a>0$. Find $a$.  
[Elements $g_{i},\ldots,g_{k}$ are said to generate a subgroup $\boldsymbol H$ if $\mathrm{(i)}$ each $g_{i}\in\boldsymbol H$, and $\mathrm{(ii)}$ every $h \in\boldsymbol H$ can be written as a sum $h=n_{1}g_{1}+\cdots+n_{k}g_{k}$ where the $n_{i}$ are integers (and where, for example, $3g_{1}-2g_{2}$ means $\left.\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\right).\right]$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the village, there are 100 houses. What is the maximum number of closed, non-intersecting fences that can be built so that each fence encloses at least one house and no two fences enclose the same set of houses?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
A parliament consists of $n$ senators. These senators are organized into 10 parties and 10 committees. Each senator belongs to exactly one party and one committee. 

Determine the smallest possible value of $n$ such that it is possible to label the parties and the committees with numbers from 1 to 10, ensuring that there are at least 11 senators for which the party and committee numbers match.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tree, and so on, marking every 7th tree. How many trees are unmarked when he returns home?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
How long will it take a train traveling at 68 kmph to pass an electric pole if the train is 170 m long, given a 5% incline on the track and a headwind of 20 kmph affecting the train's speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.75}
Given that the volume of pool \(M\) is 1.5 times that of pool \(N\), there are three water pipes \(A\), \(B\), and \(C\). Pipe \(A\) alone can fill pool \(M\) in 5 hours; pipe \(B\) alone can fill pool \(N\) in 5 hours; pipe \(C\) alone can fill pool \(N\) in 6 hours. If pipes \(A\), \(B\), and \(C\) are opened simultaneously, with \(A\) always filling pool \(M\), \(C\) always filling pool \(N\), and \(B\) starting by filling pool \(M\) and then switching to fill pool \(N\), the two pools are both filled exactly at the same time. How many hours does pipe \(B\) fill pool \(M\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{8}}
A parallelepiped $A B C D A_1 B_1 C_1 D_1$ is given. Points $M, L$, and $K$ are taken on the edges $A D, A_1 D_1$, and $B_1 C_1$ respectively, such that $B_1 K = \frac{1}{3} A_1 L$ and $A M = \frac{1}{2} A_1 L$. It is known that $K L = 2$. Find the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Determine the number of orbits for the action of $G$ on the fields with nine elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Write the numbers \(1234567891011 \cdots \cdots 19941995\) on the blackboard to form the integer \(N_{1}\). Erase the digits at the even positions of \(N_{1}\). The remaining digits form the integer \(N_{2}\). Erase the digits at the odd positions of \(N_{2}\). The remaining digits form the integer \(N_{3}\). Erase the digits at the even positions of \(N_{3}\). The remaining digits form the integer \(N_{4}\). This process continues until only one digit remains on the blackboard. Determine this digit (Note: Counting positions from left to right, for example, in 12345, 1 is in the 1st position, 2 is in the 2nd position, and so on).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A 是 10 阶实矩阵, 且 $A^{5}=O$, 则 $\operatorname{rank}(A)$ 的所有可能值之和为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A train of a certain length moving at a speed of 72 kmph takes 105.99152067834574 seconds to cross another train of length 640 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{419.92}