inclusionAI/Ring-lite-rl-data huggingface dataset:
Find all prime numbers \( p \) and \( q \) such that \( p q \) divides \( 2^{p} + 2^{q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
Jacob uses the following procedure to write down a sequence of numbers. First, he chooses the first term to be \(6\). To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts \(1\). If it comes up tails, he takes half of the previous term and subtracts \(1\). What is the probability that the fourth term in Jacob's sequence is an integer? 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{13}
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that 
\[ 20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2. \]
If 
\[ \tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019, \]
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201925}
Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?

*Proposed by Anton Trygub*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2{,}042{,}220}
Given an isosceles triangle $\triangle ABC$ where $AB = BC$. A point $K$ is inside $\triangle ABC$ such that $CK = AB = BC$ and $\angle KAC = 30^\circ$. Find $\angle AKB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
The sequence $\{a_n\}_{n\geq 1}$ is defined by the recurrence relation $a_{n+2} = 7a_{n+1} - a_n$ for positive integers $n$, with initial values $a_1 = 1$ and $a_2 = 8$. Another sequence, $\{b_n\}$, is defined by the recurrence relation $b_{n+2} = 3b_{n+1} - b_n$ for positive integers $n$, with initial values $b_1 = 1$ and $b_2 = 2$. Find $\gcd(a_{5000}, b_{501})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define 
\[f(A) = \sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.\]
If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{441}
A volleyball net is in the shape of a rectangle with dimensions of $50 \times 600$ cells.

What is the maximum number of strings that can be cut so that the net does not fall apart into pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30000}
Find the mass percentage of Ca in Calcium oxide
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71.47\%}
How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$144$}
Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are $60, 84$, and $140$ years. The three planets and the star are currently collinear . What is the fewest number of years from now that they will all be collinear again?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
We have four containers. The first three contain water, while the fourth is empty. The second container holds twice as much water as the first, and the third holds twice as much water as the second. We transfer half of the water from the first container, one-third of the water from the second container, and one-quarter of the water from the third container into the fourth container. Now, there are 26 liters of water in the fourth container. How much water is there in total in all the containers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84 \text{ liters}}
Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$298$}
Danil has 6 cards with letters, from which he managed to form the word WNMWNM, depicted in the image. Note that this word has the remarkable property that if you turn it 180 degrees, it becomes the same. How many such words can Danil form using all 6 cards at once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
On the angle bisector of angle \(B A C\) of triangle \(A B C\), a point \(M\) is marked, and on the extension of side \(A B\) beyond point \(A\), a point \(N\) is marked such that \(A C = A M = 1\) and \(\angle A N M = \angle C N M\). Find the length of segment \(A N\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
$设函数f(x)=x-x^3e^{ax+b}，曲线y=f(x)在点(1， f(1))处的切线方程为y=-x+1.$
$求f(x)的极值点个数.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The integers $1, 2, \ldots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Two trains 135 meters and another of unknown length are running in opposite directions, one at the rate of 80 km and the other at the rate of 65 kmph. They will be completely clear of each other from the moment they meet in 7.447680047665153 seconds. How long is the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
In a certain math competition, there are 6 questions, each worth 7 points (each question's score is an integer between 0 and 7). The total score of each participant is the product of the scores of the 6 questions. If two participants have the same total score, the sum of the scores for the 6 questions is calculated to determine the ranking. If still tied, they are considered to have the same rank. In this competition, there are $8^{6}=262144$ participants, and there are no ties. The score of the participant ranked $7^{6}=117649$ is $\qquad$ points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The integers $1$ through $1000$ are arranged on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e., $1, 16, 31, \ldots$) is marked. The marking continues until an already marked number is reached. How many numbers will remain unmarked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{800}
Given a triangular prism $ABC-A_1B_1C_1$, draw lines through the midpoints of any two edges. How many of these lines are parallel to the plane $ABBA_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
As shown in the figure, in the right triangular prism $ABC-A_{1}B_{1}C_{1}$, $\angle ACB=90^\circ$, $BC=CC_1=2$, and $AC=4\sqrt{2}$. Point $P$ is on $BC_1$. Find the minimum value of $CP + PA_1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{13}}
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial  $20^\circ$  incline for  $0.5$  kilometers, a flat area for  $2$  kilometers, and a  $20^\circ$  decline for  $1$  kilometer. If William travels at  $8$  kilometers per hour during uphill  $20^\circ$  sections,  $16$  kilometers per hours during flat sections, and  $20$  kilometers per hour during downhill  $20^\circ$  sections, find the closest integer to the number of minutes it take William to get to school and back.

*Proposed by William Yue*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Nancy's ex owes her child support. He's supposed to pay 30% of his income each year. For the first 3 years, he made $30,000/year, but he also received annual bonuses that varied as follows: $2,000 in the first year, $3,000 in the second year, and $4,000 in the third year. After the first 3 years, he got a 20% raise on his base salary and made $36,000/year for the next 4 years, but he also received year-end bonuses in this period: $5,000 in the fourth year, $6,000 in the fifth year, $7,000 in the sixth year, and $8,000 in the seventh year. During the sixth year, his child support obligation was adjusted to 25% of his income. If he's only ever paid a total of $1,200, how much does he owe Nancy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$75,150}
Find the value of the integral $\int_{0}^{π} \frac{1}{({3+cosθ})^2}dθ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3\pi}{16\sqrt{2}}$}
Find the center of gravity of a plate bounded by the lines \( x=0 \), \( y=0 \), \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), \( y \geqslant 0 \), \( x \geqslant 0 \), given that the surface density \( P = x y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{8a}{15}, \frac{8b}{15} \right)}
Using "学" and "习" to represent two different digits, the four-digit number "学学学学" and "羽习习习习" have a product that is a seven-digit number. And the units digit and the million's place digit of the product are the same as the digit represented by "学". How many different two-digit numbers can "学习" represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the expression $(bcd + acd + abd + abc)^7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630}
There are 5 pots of chrysanthemums: 2 pots of yellow chrysanthemums, 2 pots of white chrysanthemums, and 1 pot of red chrysanthemums. When arranging them in a row, the 2 pots of yellow chrysanthemums must be adjacent to each other, and the 2 pots of white chrysanthemums must not be next to each other. Find the number of different arrangements for these 5 pots of flowers. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the smallest positive integer $m$ such that $529^n + m \cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1984}
Given two non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} + \overrightarrow {b} | = | \overrightarrow {a} - \overrightarrow {b} | = \sqrt {2} | \overrightarrow {a} |$, find the angle between vector $\overrightarrow {b}$ and $\overrightarrow {a} + \overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
7. A sphere is circumscribed around the tetrahedron $ABCD$, and another sphere with radius 1 is tangent to the plane $ABC$ and the two spheres are internally tangent at point $D$. Given $AD=3, \cos \angle BAC=\frac{4}{5}, \cos \angle BAD=\cos \angle CAD=\frac{1}{\sqrt{2}}$, then the volume of the tetrahedron $ABCD$ is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{18}{5}}
Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}} \frac{\cos x \, dx}{5+4 \cos x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{8} - \frac{5}{6} \arctan \frac{1}{3}}
Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.5}
In a convex quadrilateral \(ABCD\), it is known that \(\angle ACB = 25^\circ\), \(\angle ACD = 40^\circ\), and \(\angle BAD = 115^\circ\). Find the angle \(\angle ADB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25^\circ}
Find the indefinite integral of $$\int \tan (x) e^ {(\tan (x))} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tan(x) e^{\tan(x)} - e^{\tan(x)} + C}
Given that the point $(\frac{1}{2}, 2)$ is on the graph of the power function $y = f(x)$ and the point $(-2, \frac{1}{4})$ is on the graph of the power function $y = g(x)$, find the value of $f(2) + g(-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
If $\alpha$ is an angle in the third quadrant, then $180°-\alpha$ is an angle in the \_\_\_\_\_\_ quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{fourth}}
What is the formula for the sum of the form $\frac{\alpha_1 + \alpha_2 + \ldots + \alpha_k}{k}$, where $(\alpha_1, \alpha_2, \ldots, \alpha_k)$ are positive integers satisfying $1 \leq \alpha_1 < \alpha_2 < \ldots < \alpha_k \leq n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n}{k} \cdot \frac{n+1}{2}}
Given a square \(ABCD\) with side length 2, \(E\) is the midpoint of \(AB\). The square is folded along lines \(EC\) and \(ED\) so that \(AE\) coincides with \(BE\), and point \(A\) coincides with point \(B\), named point \(O\). Calculate the volume of the tetrahedron \(O-CDE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, t seconds, where t is a positive integer. What is the product of the smallest and largest possible integer values of t?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3705}
Let  $T$  be the set of ordered triples  $(x,y,z)$ , where  $x,y,z$  are integers with  $0\leq x,y,z\leq9$ . Players  $A$  and  $B$  play the following guessing game. Player  $A$  chooses a triple  $(x,y,z)$  in  $T$ , and Player  $B$  has to discover  $A$ *'s* triple in as few moves as possible.  A *move* consists of the following:  $B$  gives  $A$  a triple  $(a,b,c)$  in  $T$ , and  $A$  replies by giving  $B$  the number  $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$ . Find the minimum number of moves that  $B$  needs to be sure of determining  $A$ *'s*  triple.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If \[x+\sqrt{x^2-1}+\frac{1}{x-\sqrt{x^2-1}}=20,\] then \[x^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}}=\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$51.005$}
If the coefficients $A$ and $B$ of the line $Ax+By=0$ can take different values from the set $\{0, 2, 3, 4, 5, 6\}$, the number of these equations representing different lines is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A number should have been multiplied by 5, but instead, it was processed in a different way. The percentage of error obtained is 96%. What operation was performed on the number instead of multiplication?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2N}
Calculate the limit $$ \lim_{n\to \infty}\left(\dfrac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^{1/n}, $$ given a function $f:\mathbb{R}\to \mathbb{R}$ that is differentiable at $a\in \mathbb{R}$ and satisfies $f(a) > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Candace is trying to decide whether to wear her old shoes or new high-tech shoes for a hike. The high-tech shoes will help Candace walk twice as fast, but they're not broken in yet, so she'll get 1 blister after every 2 hours she spends walking in them. Each blister slows Candance down by 2 miles per hour. If Candance walks 6 miles per hour in the old shoes and plans to hike for 4 hours, how many miles per hour can she go in the new shoes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11 \, \text{mph}}
Let the set \( T = \left\{ x_{1}, x_{2}, \cdots, x_{10} \right\} \) contain five-element subsets such that any two elements of \( T \) appear together in at most two of these subsets. What is the maximum number of such subsets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In the convex pentagon ABCDE, all sides are of equal length, and angles A and B are right angles (90 degrees). Calculate the degree measure of angle E. Express your answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Let \( S_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \) for \( n = 1, 2, \cdots \). Find the smallest positive integer \( n \) such that \( S_{n} > 10 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12367}
3. Initially, the numbers $1, 2, 3, \cdots, 2013$ are written on the blackboard. In each step, you can erase two numbers $a, b$ from the blackboard and write down $a+b$. What is the minimum number of steps required to ensure that the sum of any remaining numbers on the blackboard is not equal to 2014?
(Lin Chang provided the problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
Write a negative number greater than $-\frac{1}{2}$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
In a cube, a line determined by two vertices and a plane containing four vertices can form an "orthogonal line-plane pair". How many such pairs are possible in the cube? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements:
(1)$x_0=0$;
(2)For any integer $i,1\leq i\leq 100$,we have $1\leq x_i-x_{i-1}\leq 2$.
Find the greatest positive integer $k\leq 100$,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have
\[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given that the sequence {a_n} is an increasing arithmetic sequence, and S_n is the sum of the first n terms of the sequence {a_n}. Let's denote the first term of the sequence as a_1 and the common difference as d. 

Given that S_14 = 14 and S_9 = 18 - λa_10, calculate the value of λ. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
What is the maximum number of natural numbers that can be written in a row such that the sum of any three consecutive numbers is even, and the sum of any four consecutive numbers is odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Assume that the targeted respondents belong to three different demographics: demographic A, demographic B, and demographic C. The response rates for these demographics are 70%, 60%, and 50%, respectively. The population proportion of each demographic is 40%, 35%, and 25%. In order to receive 210 responses, what is the minimum number of questionnaires that should be mailed out, taking into consideration the different demographics and their respective response rates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{342}
全体正整数的集合可以分成两个互不相交的正整数子集

{f(1),f(2),⋯,f(n),⋯},{g(1),g(2),⋯,g(n),⋯}

其中 f(1)&lt;f(2)&lt;⋯&lt;f(n)&lt;⋯

g(1)&lt;g(2)&lt;⋯&lt;g(n)&lt;⋯

且有 g(n)=f(f(n))+1,n≥1

求f(240).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{388}
$已知数列a_n的奇数项是公差为d_1的等差数列，偶数项是公差为d_2的等差数列，S_n是数列a_n的前n项和，a_1=1,a_2=2.$
$若S_4=11,a_{10}=2a_{7}，求a_{12}；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$17$}
The quiz participants were asked four questions: 90 participants answered the first question correctly, 50 answered the second correctly, 40 answered the third correctly, and 20 answered the fourth correctly. No one was able to answer more than two questions correctly. What is the minimum number of participants in the quiz under these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Compute the following limit: $$\lim_{x \to \infty}\dfrac {f^{-1}(8x)-f^{-1}(x)}{x^{1/3}},$$ where $f: \mathbb R \to \mathbb R$ is defined as $f(x)=8x^3+3x$, and it is known that $f$ is continuous, strictly increasing, and satisfies $\lim _{x\to \infty}f(x)=\infty$ and $\lim_{x \to -\infty}f(x)=-\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
8-1. Find all common points of the graphs

$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text { and } \quad y=\cos 9 \pi x
$$

with abscissas belonging to the segment $x \in[0 ; 1]$. In your answer, specify the sum of the abscissas of the found points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
A bag of potatoes weighs 36 kg, which is 1.5 times the weight of a bag of carrots. Two bags of tomatoes weigh an additional 16 pounds. Convert the weights of the bags of potatoes and carrots to pounds and combine them with the weight of the tomatoes to find the total weight of all the produce in pounds. How much do the bags of potatoes, carrots, and tomatoes weigh together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148.28}
In trapezoid ABCD, AB is parallel to DC, angle ABC = 90 degrees, and E is the midpoint of leg AD. If EC = sqrt(13) and AB + BC + CD = 2*sqrt(26), then calculate the measure of angle BCE in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Consider a fair coin and a fair 6-sided die.  The die begins with the number 1 face up.  A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step.  After 5 such steps, what is the probability that the number 1 is face up on the die?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{192}}