inclusionAI/Ring-lite-rl-data huggingface dataset:
A list of positive integers is called *good* if the maximum element of the list appears exactly once. A *sublist* is a list formed by one or more consecutive elements of a list. For example, in the list $10, 34, 34, 22, 30, 22$, the sublist $22, 30, 22$ is good, while $10, 34, 34, 22$ is not.

A list is *very good* if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values in it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
On Namek, each hour has 100 minutes. On the planet's clocks, the hour hand completes one full circle in 20 hours and the minute hand completes one full circle in 100 minutes. How many hours have passed on Namek from 0:00 until the next time the hour hand and the minute hand overlap?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20/19}
Helen the hippo and her friends are preparing for thanksgiving at Helen's house. Helen baked 527 chocolate chip cookies and some raisin cookies yesterday. And she baked the same number of raisin cookies and 554 chocolate chip cookies this morning. Helen baked 1081 chocolate chip cookies. How many raisin cookies did Helen bake yesterday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{527}
Let $X_1$, $X_2$, $X_3$ be independent, identically distributed continuous random variables from the uniform distribution on the interval [0,1]. Find the probability that the median is closer to the largest value $X_($$_3$$_)$ than it is to the smallest value $X_($$_1$$_)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Given that $S_{n}=\sin \frac{\pi}{7}+\sin \frac{2\pi}{7}+\cdots+\sin \frac{n\pi}{7}$, determine the number of positive values in the sequence $S_{1}$, $S_{2}$, …, $S_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
In space, 4 points are given that are not on the same plane. How many planes can be drawn equidistant from these points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Janet pays $40/hour for 3 hours per week of clarinet lessons, $28/hour for 5 hours a week of piano lessons, $35/hour for 2 hours a week of violin lessons, and $45/hour for 1 hour a week of singing lessons. She is given a 10% discount on piano lessons if she takes them for at least 9 months continuously and a 15% discount on violin lessons if taken for the whole year. Additionally, she takes a 2-week holiday break for all lessons except during summer. How much more does she spend on piano, violin, and singing lessons combined than clarinet lessons in a year, taking the discounts and holiday breaks into account?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$5525}
What is the probability that in a KENO draw, the digit 8 does not appear in any of the 20 numbers drawn from the integers ranging from 1 to 80?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.063748}
The integers  $a$ ,  $b$ ,  $c$  and  $d$  are such that  $a$  and  $b$  are relatively prime,  $d\leq 2022$  and  $a+b+c+d = ac + bd = 0$ . Determine the largest possible value of  $d$ ,
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
A $2\times3$ rectangle has vertices at $(0,0)$, $(2,0)$, $(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the x-axis is now back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the format $\frac{m\pi}{n} + k$. Please give the value of m + n + k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the largest natural number \( n \) so that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2004}
A semicircle with a radius of 12 is connected to a regular hexagon such that one side of the hexagon is tangent to the semicircle at its midpoint. If the length of each side of the hexagon is equal to the semicircle's radius, what is the approximate combined perimeter of the two shapes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134}
Compare the sizes: $-\frac{4}{3}$ ____ $-\frac{5}{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{<}
Given that $a > 0$ and $b > 0$, with $m=\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{a}}$, $n=\sqrt{a}+\sqrt{b}$, and $p=\sqrt{a+b}$, determine the order of magnitude of $m$, $n$, and $p$. Express your answer as an inequality using \geqslant or > notation, including all variables m, n, and p.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geqslant n > p}
Let  $P(x)=x^{2020}+x+2$ , which has  $2020$  distinct roots. Let  $Q(x)$  be the monic polynomial of degree  $\binom{2020}{2}$  whose roots are the pairwise products of the roots of  $P(x)$ . Let  $\alpha$  satisfy  $P(\alpha)=4$ . Compute the sum of all possible values of  $Q(\alpha^2)^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020 \times 2^{2019}}
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
It is known that the numbers EGGPLANT and TOAD are divisible by 3. What is the remainder when the number CLAN is divided by 3? (Letters denote digits, identical letters represent identical digits, different letters represent different digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
You are planning to divide the tofu into three pieces, each in a rectangular shape, measuring 54 centimeters (cm) long and 20 centimeters (cm) wide when viewed from above. You want to cut all 3 pieces to have the same size of a square when viewed from above. Find the length in centimeters (cm) of one side to make the largest square. (However, tofu is only cut from top to bottom.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
How many integers between  $123$  and  $789$  have at least two identical digits, when written in base  $10?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Evaluate the following expression using mathematical manipulations:

$$2^{n-1} \left(\cos\theta - \cos\left( \frac{\pi}{n} \right)\right)\left(\cos\theta - \cos\left( \frac{2\pi}{n} \right)\right)...\left(\cos\theta - \cos\left( \frac{(n-1)\pi}{n} \right)\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sin(n\theta)}{\sin\theta}}
A colony of bees can contain up to 80,000 individuals. In winter, they are more exposed to death, particularly during colder winters. During a particularly harsh winter, it is observed that the colony loses bees at a rate of 3/5 of a percent per day. How many days will pass until the colony reaches a fourth of its initial population?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231}
Let $f$ be an entire function such that $f(x+ix)\in\mathbb{R}$ for all $x\in \mathbb{R}$. If $f(2)=1-i$, find $f(2i)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+i}
A thin sheet of an equilateral triangle with a side length of $1$ is cut into two parts along a line parallel to its base, with one of the parts being a trapezoid. Let $S = \frac{\text{perimeter of the trapezoid}}{\text{area of the trapezoid}}$. Determine the minimum value of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{6}}{3} + 2\sqrt{3}}
Find the minimum value of $\displaystyle \sum_{k=1}^n \frac{x_k^k}{k}$, where $x_1, x_2, \ldots, x_n$ are positive real numbers satisfying $\displaystyle \sum_{k=1}^n \frac{1}{x_k} = n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_n}
Benjamin is tracking how many miles he walks in a week. He walks to work and home five days a week, walks his dog twice a day every day, walks to his best friend’s house once a week, and walks to the convenience store twice a week. Work is six miles away, dog walks are two miles, his best friend’s house is one mile, and the convenience store is three miles. How many miles does Benjamin walk in a week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Bill picked 50 apples from the orchard with his wife and two children. He sends each of his kids to school with 3 apples for their two favorite teachers. His wife Jill bakes two apple pies, using 10 apples per pie. How many apples does Bill have left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A certain plant's main stem grows several branches, each branch grows the same number of small branches. The total number of main stems, branches, and small branches is $91$. If each branch grows $x$ small branches, then $x=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Some years ago, Lucy was three times as old as Lovely. Ten years later, Lucy would be twice as old as Lovely. Lucy is now 50 years old. How many years ago was Lucy three times as old as Lovely?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
Find the number of ways to color  $n \times m$  board with white and black
colors such that any  $2 \times 2$  square contains the same number of black and white cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest prime  $p$  for which there exist positive integers  $a,b$  such that 
	\[
		a^{2} + p^{3} = b^{4}.
	\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
The main structure of the Chinese space station includes the Tianhe core module, the Wentian experiment module, and the Mengtian experiment module. Assuming that the space station needs to arrange 6 astronauts, including astronauts A and B, to conduct experiments, with each module having at least one person and at most three people, find the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{450}
Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged, it is possible to travel between any two cities in Graphistan using only GA flights, provided that the absolute value of the difference between the number of flights originating from and terminating at any city is not more than $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
From a copper ingot in the shape of a rectangular parallelepiped with dimensions $80 \times 20 \times 5 \text{ cm}$, a sheet is rolled to a thickness of 1 mm. Determine the area of this sheet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80000 \; \text{cm}^2}
Given that the median of $18$ integers is $5$, and the $75$th percentile is also $5$, then the minimum possible number of occurrences of $5$ among these $18$ numbers is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $S$ be a set composed of positive integers with the property that for any $x \in S$, removing $x$ from $S$ results in the remaining numbers having an arithmetic mean that is a positive integer. Additionally, $1 \in S$ and $2016$ is the largest element in $S$. Find the maximum value of $|S|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Alice conducted a survey among a group of students regarding their understanding of snakes. She found that $92.3\%$ of the students surveyed believed that snakes are venomous. Of the students who believed this, $38.4\%$ erroneously thought that all snakes are venomous. Knowing that only 31 students held this incorrect belief, calculate the total number of students Alice surveyed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
There exists a positive number $m$ such that the positive roots of the equation $\sqrt{3} \sin x - \cos x = m$ form an arithmetic sequence in ascending order. If the point $A(1, m)$ lies on the line $ax + by - 2 = 0 (a > 0, b > 0)$, find the minimum value of $\frac{1}{a} + \frac{2}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
Let \( n \in \mathbb{N}^{*} \) and \( P \in \mathbb{R}[X] \) be a polynomial of degree at most \( n \) such that \( P(k) = \frac{k}{k+1} \) for all \( k \in \llbracket 0, n \rrbracket \). What is the value of \( P(n+1) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n+1+(-1)^{n+1}}{n+2}}
Find the coefficient of $x^{50}$ in the expansion of the series: 
$$\sum_{k=0}^{1000} (k+1)x^k(1+x)^{1000-k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{1002}{50}}
In triangle \(ABC\), the angles \(A\) and \(B\) are \(45^{\circ}\) and \(30^{\circ}\) respectively, and \(CM\) is the median. The circles inscribed in triangles \(ACM\) and \(BCM\) touch segment \(CM\) at points \(D\) and \(E\). Find the area of triangle \(ABC\) if the length of segment \(DE\) is \(4(\sqrt{2} - 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16(\sqrt{3}+1)}
Suppose $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ are unit vectors, and $\overrightarrow{a}= \overrightarrow{b} + \overrightarrow{c}$. Then, the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
If the line $x+ay+6=0$ is parallel to the line $(a-2)x+3y+2a=0$, determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
a person saved $ 10 in buying an item on sale . if he spent $ 100 for the item , approximately how much percent he saved in the transaction ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\%}
There are two identical magnets in the shape of a rectangle. If you attach two magnets horizontally, the total circumference is 70 centimeters (cm) and the length is 15 centimeters (cm). Find the area of one magnet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Sean enters a classroom in the Memorial Hall and sees a 1 followed by 2020 0's on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the \(n\)th digit from the left with probability \(\frac{n-1}{n}\). (In particular, the 1 is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-3 number. (For example, if the remaining number on the board is 1000, then its value is 27.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{681751}
Compute the  $\textit{number}$  of ordered quadruples  $(w,x,y,z)$  of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1
wxy^2 + wx^2z + w^2yz + xyz^2 &=2
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A pyramid-like stack with a rectangular base containing $6$ apples by $9$ apples is constructed, with each apple above the first level fitting into a pocket formed by four apples below, until no more apples can be fit in a new layer. Determine the total number of apples in the completed stack.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{154}
Sides $\overline{AM}$ and $\overline{CD}$ of regular dodecagon $ABCDEFGHIJKL$ are extended to meet at point $P$. What is the degree measure of angle $P$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
16. Let $a_{1}, a_{2}, \ldots, a_{24}$ be integers with sum 0 and satisfying $\left|a_{i}\right| \leq i$ for all $i$. Find the greatest possible value of $a_{1}+2 a_{2}+3 a_{3}+\cdots+24 a_{24}$.
(2 marks)設 $a_{1} 、 a_{2} 、 \cdots 、 a_{24}$ 為整数, 它們之和為 0 , 且對所有 $i$ 皆有 $\left|a_{i}\right| \leq i$ 。求 $a_{1}+2 a_{2}+3 a_{3}+\cdots+24 a_{24}$ 的最大可能值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1432}
In how many ways can a \(3 \times 8\) rectangle be divided into L-shaped trominoes (shapes consisting of three square cells)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Fold an equilateral triangle $ABC$ with side length 2 along the height $AD$ from $A$ to $BC$ to form a dihedral angle $B-AD-C$. Find the surface area of the circumscribed sphere of the tetrahedron $B-ACD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi}
Two circles touch each other externally at point \(K\). One line touches these circles at different points \(A\) and \(B\), and another line touches them at different points \(C\) and \(D\). The common tangent to the circles, passing through point \(K\), intersects these lines at points \(M\) and \(N\). Find \(MN\) if \(AC = a\) and \(BD = b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a + b}{2}}
One million bucks (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are honest bucks, who always tell the truth, and the remaining bucks are dishonest bucks, who never tell the truth. Each of the million bucks claims that "at most one of my neighboring bucks is an honest buck." A pair of neighboring bucks is said to be buckaroo if exactly one of them is an honest buck. What is the minimum possible number of buckaroo pairs in the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200000}
Four, (50 points) Find the maximum value of a positive integer $n$, such that for any simple graph of order $n$ with vertices $V_{1}, V_{2}, \cdots, V_{n}$, there always exist $n$ subsets $A_{1}, A_{2}, \cdots, A_{n}$ of the set $\{1,2, \cdots, 2014\}$, satisfying: $A_{i} \cap A_{j} \neq \varnothing (i \neq j)$ if and only if $V_{i}$ is adjacent to $V_{j}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
A circle is divided into $n$ equal parts. Marceline sets out to assign whole numbers from $1$ to $n$ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $887$, $217$, and $1556$ occupy consecutive positions. How many parts was the circumference divided into?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
Problem 5. (5 points) In a tournament, 55 boxers participated, and after each match, the winner left the tournament. The matches were held consecutively. It is known that in each match, boxers whose difference in the number of victories (at that moment) was no more than 1, met. How many matches could the winner of the tournament have played at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a differential form $\omega=2xdx\wedge dy + y^2dx\wedge dz$ and vector fields $X=x^2y\frac{\partial}{\partial y} + x\frac{\partial}{\partial z}$ and $Y=x\frac{\partial}{\partial y}$, calculate $\omega(X,Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
对于给定的正整数$k$，定$义 f _ { 1 } ( k )$为$k$的数字和的平方，并令

$f _ { n + 1 } ( k ) = f _ { 1 } ( f _ { n } ( k ) ) ,$

求：$f _ { 1 9 9 1 } ( 2 ^ { 1 9 9 0 } )$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .

(You may use the fact that   $\sin{n^\circ}$  is irrational for positive integers  $n$  not divisible by  $30$ .)

*Ray Li*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178}
Two siblings sold their flock of sheep. Each sheep was sold for as many florins as the number of sheep originally in the flock. They divided the revenue by giving out 10 florins at a time. First, the elder brother took 10 florins, then the younger brother, then the elder again, and so on. In the end, the younger brother received less than 10 florins, so the elder brother gave him his knife, making their earnings equal. How much is the knife worth in florins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Numbers from 1 to 8 are arranged in a circle. We call a number "big" if it is greater than its neighbors, and "small" if it is less than its neighbors. In the arrangement, each number is either big or small. What is the smallest possible sum of the big numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Given that $\sin\alpha + \cos\alpha = \frac{1}{5}$, and $0 \leq \alpha < \pi$, find the value of $\tan\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \frac {4}{3}}
There are $N$ lockers, labeled from 1 to $N$, placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker $n$ and there are more than $n$ open lockers, she keeps locker $n$ open and closes the next $n$ open lockers, then repeats the process with the next open locker. If she is at locker $n$ and there are at most $n$ lockers still open, she keeps locker $n$ open and closes all other lockers. She continues this process until only one locker is left open. What is the smallest integer $N>2021$ such that the last open locker is locker 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046}