inclusionAI/Ring-lite-rl-data huggingface dataset:
Li Ming and Li Hua both arrive at the bus station to wait for either bus number 1 or bus number 2 to go home. Bus number 1 arrives every 8 minutes, while bus number 2 arrives every 10 minutes. What is the probability that Li Ming boards bus number 1 before Li Hua boards bus number 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6}
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. Find the number of two-digit positive integers $n$ such that $R(n) = R(n+1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many solutions does the equation $x = [P_1(x)] + 2[P_2(x)] + 3[P_3(x)] + 4[P_4(x)]$ have, where $P_1(x): x \le 5$, $P_2(x): x \ge 3$, $P_3(x): x$ is odd, and $P_4(x): x \ge 6$ for all integers $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, what number could be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of the car ahead to the front of the car in the same lane is one car length for every 10 kilometers per hour of speed or fraction thereof. Suppose each car is 5 meters long, and a photoelectric eye at the side of the road detects the number of cars that pass by in one hour. Determine the whole number of cars passing the eye in one hour if the speed in the right lane is 50 kilometers per hour. Calculate $M$, the maximum result, and find the quotient when $M$ is divided by 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{338}
A plane contains points \(A\) and \(B\) with \(AB = 1\). Let \(S\) be the union of all disks of radius \(1\) in the plane that cover \(\overline{AB}\). The original answer is in the form \(k\pi - \frac{\sqrt{m}}{n}\). Please find the value of \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$, it is possible to choose four different $a, b, c, d$ such that $a + 2b + 3c = d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{835}
In a class of 50 students, a sample of size 4 is drawn using the systematic sampling method. It is known that students with identification numbers 6, 30, and 42 are included in the sample. Determine the identification number of the fourth student in the sample.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
The ratio of width to length of a book is the golden ratio. Given that the length of the book is 20cm, calculate the width of the book. Express your answer in centimeters (cm), rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.36}
Find
$$
\{\ln (1+e)\}+\left\{\ln \left(1+e^{2}\right)\right\}+\left\{\ln \left(1+e^{4}\right)\right\}+\left\{\ln \left(1+e^{8}\right)\right\}+\cdots,
$$
where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln (e - 1)}
In a chess festival held at a school with $2017$ students, each pair of students played at most one match against each other. By the end of the event, it was observed that for any pair of students who played a match against each other, at least one of them had played at most $22$ matches. What is the maximum possible number of matches in this event?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43890}
10.3. Find the greatest solution of the inequality

$$
\frac{-\log _{3}(100+2 x \sqrt{2 x+25})^{3}+\left|\log _{3} \frac{100+2 x \sqrt{2 x+25}}{\left(x^{2}+2 x+4\right)^{4}}\right|}{3 \log _{6}(50+2 x \sqrt{2 x+25})-2 \log _{3}(100+2 x \sqrt{2 x+25})} \geqslant 0
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12+4\sqrt{3}\approx18.93}
A chemical factory plans to improve the technology for a chemical product and has decided to optimize the processing temperature, with the experimental range set between 60°C and 81°C, requiring an accuracy of ±1°C. If the technician prepares to use the method of fractions to optimize, what is the maximum number of experiments needed to find the optimal temperature?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$  Find the maximum value of
\[7x + 2y + 8z.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
In a campus singing competition, a contestant received scores of $92$, $93$, $88$, $99$, $89$, and $95$ from $6$ judges. The $75\%$ percentile of this data set is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer.  Compute the remainder when $3^n$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Let k and m be non-negative integers, n be a negative integer, and 15^k, 20^m, and 10^n be divisors of 1,031,525. If x = 3^k - k^3 and y = 5^m - m^5, find the value of z = (x + y) - 2^(n+2).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the sequence $\left\{a_{n}\right\}$ that satisfies
$$
a_{n+1} = -\frac{1}{2} a_{n} + \frac{1}{3^{n}}\quad (n \in \mathbf{Z}_{+}),
$$
find all values of $a_{1}$ such that the sequence $\left\{a_{n}\right\}$ is monotonic, i.e., $\left\{a_{n}\right\}$ is either increasing or decreasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{5}}
In an isosceles triangle $ABC$ with an angle $\angle A = 20^\circ$ and base $BC = 12$, a point $E$ on the side $AC$ is chosen such that $\angle ABE = 30^\circ$. Additionally, a point $F$ is on the side $AB$ such that $EF = FC$. Find the length of $FC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that \( f(x) = x^2 + ax + b \) is the common factor of \( x^3 + 4x^2 + 5x + 6 \) and \( 2x^3 + 7x^2 + 9x + 10 \), find the value of \( Q \) if \( f(P) = Q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any other student?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
One hour after Adrienne started walking the 60 miles from point X to point Y, Harold started walking from point X to point Y as well. Adrienne walks at a speed of 3 miles per hour, and Harold walks 1 mile per hour faster than Adrienne. At the same time when Harold starts walking, Christine also begins walking from point X to point Y, and her walking speed is 0.5 miles per hour slower than Harold's. What will be the location for all three of them to meet (the distance from point X) along the line between X and Y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
If $81^4 = 27^x$, what is the value of $3^{-x}$? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3^{\frac{16}{3}}}}
Two bikers, Bill and Sal, simultaneously set off from one end of a straight road. Neither biker moves at a constant rate, but each continues biking until he reaches one end of the road, at which he instantaneously turns around. When they meet at the opposite end from where they started, Bill has traveled the length of the road eleven times and Sal seven times. Find the number of times the bikers passed each other moving in opposite directions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A triangle has an inscribed circle with a radius of 8 cm. Calculate the length of $\overline{AB}$ if the triangle is a right triangle with one angle measuring $30^\circ$. Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\overline{AB} = 16(\sqrt{3} + 1)}
Given an arithmetic progression $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}=\left(x+2y+1\right)3^{n}+\left(x-y-4\right)$ (where $x \gt 0, y \gt 0$). The minimum value of $\frac{1}{x}+\frac{2}{y}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{3}}
设数列 $\left\{x_{n}\right\}$ 满足
$x_{1}=1, x_{n+1}=4 x_{n}+\left[\sqrt{11} x_{n}\right]$ 。
求 $x_{2012}$ 的个位数字.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Professor Oscar has to evaluate his students on the final exams, but this semester, he included a mix of question types and different class sizes.

Class A has 30 students, and their exam consists of 12 multiple-choice questions, 3 essay questions, and 1 presentation. 

Class B has 25 students, and their exam consists of 15 multiple-choice questions, 5 short-answer questions, and 2 essay questions.

Class C has 35 students, and their exam consists of 10 multiple-choice questions, 3 essay questions, and 1 group project presentation per group of 5 students.

Class D has 40 students, and their exam consists of 11 multiple-choice questions, 4 short-answer questions, and 3 essay questions.

Class E has 20 students, and their exam consists of 14 multiple-choice questions, 5 short-answer questions, and 2 essay questions.

Given the varying number of questions, students, and components for each of the 5 classes, how many evaluations must Professor Oscar review in total? Consider the evaluations for each component separately (multiple-choice, short-answer, essay, and presentations).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2632}
A cuckoo clock chimes "cuckoo" on the hour, with the number of "cuckoo" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger until he moved the clock forward by 7 hours. How many times did the clock chime "cuckoo" during this period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
A technician has to visit a total of 5 service centers (A, B, C, D, and E) along the same route, making a round-trip by returning back to center A. The technician completes the drive to center B, then finishes 40% of the journey from center B to center C, and after that, completes 60% of the drive from center C to center D. What percent of the entire round-trip, including the return trip to center A, has the technician completed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Find the derivative.

\[ y = -\frac{1}{3 \sin^{3} x} - \frac{1}{\sin x} + \frac{1}{2} \ln \frac{1+\sin x}{1-\sin x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{ \cos x \sin^4 x}}
How can I solve the following limit problem involving factorials and powers of x: $$\lim_{x\to 1} \frac{(1-x)(1-x^2)\cdots(1-x^{2n})}{[(1-x)(1-x^2)\cdots(1-x^n)]^2}?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\binom{2n}{n}\)}
For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$.

[i]Proposed by Mehtaab Sawhney[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1266}
The regular tetrahedron, octahedron, and icosahedron have equal surface areas. How are their edges related?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{10} : \sqrt{10} : 2}
[ Geometry (miscellaneous).]

A sphere of radius $\sqrt{5}$ with center at point $O$ touches all sides of triangle $ABC$. The point of tangency $N$ bisects side $AB$. The point of tangency $M$ divides side $AC$ such that $AM=\frac{1}{2} MC$. Find the volume of the pyramid $OABC$, given that $AN=NB=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let  $r$  be a positive integer. Find the smallest positive integer  $m$  satisfying the condition: For all sets  $A_1, A_2, \dots, A_r$  with  $A_i \cap A_j = \emptyset$ , for all  $i \neq j$ , and  $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$ , there exists  $a, b \in A_k$  for some  $k$  such that  $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023r}
Given the function $y=\cos(2x- \frac{\pi}{6})$, find the horizontal shift required to transform the graph of $y=\sin 2x$ into the graph of $y=\cos(2x- \frac{\pi}{6})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
If $a$, $b$, and $c$ are positive numbers such that $ab = 24\sqrt{3}$, $ac = 30\sqrt{3}$, and $bc = 40\sqrt{3}$, find the value of $abc$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120\sqrt{6}}
Sherry is waiting for a train. Every minute, there is a 75% chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a 75% chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \left(\frac{13}{16}\right)^5}
Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105717501}
Piazzanos Pizzeria has a special offer on triple cheese pizzas, and a buy-2-get-1-free special on meat lovers pizzas. If the standard price for a pizza is $5, it costs $55 to purchase some triple cheese and 9 meat lovers pizzas using the special pricing. What is the ratio of the number of triple cheese pizzas you need to buy to get one free?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Two trains of length 100 m and 280 m are running towards each other on parallel lines at a certain speed and 30 kmph respectively. From the moment they meet, they will be clear of each other in 18.998480121590273 seconds. What is the speed of the first train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Consider the collection of all 5-digit numbers whose sum of the digits is 43. One of these numbers is chosen at random. What is the probability that it is a multiple of 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
How many odd, 5-digit palindromic numbers are there with all unique digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Let
\[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ]. \]

Find the remainder when $N$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
若锐角 A,B,C 满足 \sin^2A+\sin^2B+\sin^2C=2，则 \df{1}{\sin^2A\cos^4B}+\df{1}{\sin^2B\cos^4C}+\df{1}{\sin^2C\cos^4A} 的最小值是__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Reduce the fraction \(\frac{2x^{6} + 5x^{4} - 3x^{3} + 2x^{2} - 12x - 14}{4x^{6} - 4x^{4} - 6x^{3} - 3x^{2} + 25x - 28}\). As a result of the reduction, the degrees of the polynomials in the numerator and the denominator should decrease.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x^3 + 2x + 2}{2x^3 - 3x + 4}}
Let \(\alpha\) and \(\beta\) be angles such that 
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of 
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50308}
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
For an annuity with $3n$ equal payments and a periodic interest rate of $i$, it is observed that the present value of the first $n$ payments, one period before the first payment, equals the present value of the last $2n$ payments. Determine the value of $v^n$ where $v = \left(\frac{1}{1+i}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{5} - 1}{2}\)}
Given \( n \) sticks. From any three, it is possible to form an obtuse triangle. What is the maximum possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let f(x) be a function defined on R with a period of 2, and for any real number x, it always holds that f(x) - f(-x) = 0. When x is in [0, 1], f(x) = -sqrt(1-x^2). Find the number of zeros of the function g(x) = f(x) - e^x + 1 in the interval [-2017, 2017].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Replace the symbols $\$ last\$ with the same numbers so that the equation becomes true: $\$ \$ |\frac{20}{x} - \frac{x}{15} = \frac{20}{15} \$ \$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Maya loves to read books. Last week she read 5 books. Each book had 300 pages of text. This week she read a certain amount compared to last week. Maya read a total of 4500 pages. What is the ratio of the number of books she read this week to the number of books she read last week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues until someone cannot say a new number, and the person who cannot say a new number loses the game. Who will be the last person in the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Plane $S$ makes an angle of $30^{\circ}$ with the horizontal. Plane $S$ is then rotated by $120^{\circ}$ around a vertical line as an axis. What is the angle of inclination between the resulting plane and plane $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
4. What is the greatest length that a closed, non-self-intersecting broken line can have, which runs along the grid lines of a square grid of size $8 \times 8$?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
On the bisector of angle \( B A C \) of triangle \( A B C \), there is a point \( M \), and on the extension of side \( A B \) beyond point \( A \), there is a point \( N \) such that \( A C = A M = 1 \) and \( \angle A N M = \angle C N M \). Find the radius of the circumcircle of triangle \( C N M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the value of N such that the number of odd integers between 4 and N is equal to the number of even integers between 1 and 103. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
In a middle school math competition, there were three problems labeled A, B, and C. Among the 25 students who participated, each student solved at least one problem. Among the students who did not solve problem A, the number who solved problem B was twice the number of those who solved problem C. The number of students who solved only problem A was one more than the number of students who solved A along with any other problem. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}