inclusionAI/Ring-lite-rl-data huggingface dataset:
Define the minimum real $C$ such that for any real numbers $0 = a_0 < a_1 < \dots < a_{1000}$, the inequality $$\min_{0 \le k \le 1000} (a_k^2 + (1000-k)^2) \le C(a_1 + \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
Given $F$ as the focus of the parabola $y^2 = 3x$, $P$ as any point on the parabola, and $A(3,2)$ as a fixed point on the plane, the minimum value of $|PF|+|PA|$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{4}}
已知$x _ { i } \in R ^ { + } ( i = , 2 , \cdots , n ) , \sum _ { i = } ^ { n } x _ { i } = m , p , q \in R ^ { + } , 2 \leq k \in N ^ { + }$.求函数$f ( x ) = \sum _ { i = 1 } ^ { n } \sqrt [ k ] { p x _ { i } + q }$的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt [ k ] { p m n ^ { k - 1 } + q n ^ { k } }$}
A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\pi - \frac{81\sqrt{3}}{4}}
In the arithmetic sequence {a_n}, it is known that a_2 - a_3 - a_7 - a_11 - a_13 + a_16 = 8. Find the value of a_9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:

1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or  
2. From $P$ and $Q$ to $5P-4Q$ and $5Q-4P$.

The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.

At the beginning of the game, the two numbers on the board are $2024$ and $A$. It is known that Kobar does not lose on his first move. Determine the largest possible value of $A$ so that Borah can win this game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2179}
There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other residents are taller than me?"
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{454}
In a $100 \times 100$ grid, the numbers $1, 2, \ldots, 10000$ are placed, each appearing exactly once, such that numbers differing by 1 are located in adjacent cells. After this, the distances between the centers of all pairs of cells whose numbers differ by exactly 5000 are calculated. Let $S$ be the minimum of these distances. What is the maximum possible value of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50 \sqrt{2}}
5. In a row, all natural numbers from 1 to 100 inclusive are written in ascending order. Under each number in this row, the product of its digits is written. The same procedure is applied to the resulting row, and so on. How many odd numbers will be in the fifth row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
$A$ 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 regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even number of goals will be scored during the match.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.502}
On a table, there are five clocks with hands. It is allowed to move any number of them forward. For each clock, the time by which it is moved will be referred to as the translation time. It is required to set all clocks such that they show the same time. What is the minimum total translation time needed to guarantee this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?

A. Gribalko
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Circle \(O\) has chord \(AB\). A smaller circle is tangent to circle \(O\) at \(T\) and tangent to \(AB\) at \(X\) such that \(AX = 2XB\). What is \(\frac{AT}{BT}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: (1,2,5,8),(3,4,6,7),(2,4,5,7),(1,3,6,8),(2,3,7,8), and (1,4,5,6). What is the label of the vertex of the cube that is farthest away from the vertex labelled 2? Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that the real numbers \( a \) and \( b \) satisfy that both quadratic trinomials \( x^2 + ax + b \) and \( x^2 + bx + a \) have two distinct real roots each, and the product of their roots results in exactly three distinct real roots. Find all possible values of the sum of these three distinct roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $\sum_{n=1}^{210} nf(n)$ is divided by $211$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given the parabola \( y^2 = 2px \) (where \( p > 0 \)), two mutually perpendicular chords \( FA \) and \( FB \) pass through the focus \( F \). Find the minimum area of \( \triangle FAB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3 - 2\sqrt{2}) p^2}
There are 6 positive integers \( a, b, c, d, e, f \) arranged in order, forming a sequence, where \( a=1 \). If a positive integer is greater than 1, then the number that is one less than it must appear to its left. For example, if \( d > 1 \), then one of \( a, b, \) or \( c \) must have the value \( d-1 \). For instance, the sequence \( 1,1,2,1,3,2 \) satisfies the requirement; \( 1,2,3,1,4,1 \) satisfies the requirement; \( 1,2,2,4,3,2 \) does not satisfy the requirement. How many different permutations satisfy the requirement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{203}
Three pirates divided the diamonds they collected during the day: Bill and Sam got twelve each, and the rest went to John, who did not know how to count. At night, Bill stole one diamond from Sam, Sam stole one diamond from John, and John stole one diamond from Bill. As a result, the average mass of Bill's diamonds decreased by one carat, the average mass of Sam's diamonds decreased by two carats, but the average mass of John's diamonds increased by four carats. How many diamonds did John receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Sandy walked some distance towards south. Then Sandy turned to her left and walked the same distance. She then turned to her left and walked the same distance. She then turned to her right and walked the same distance. Sandy is 40 meters away from the starting point in the north direction. How far did she walk in each segment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
正整数 $9+95+995+\cdots+\underbrace{99 \cdots 9}_{2016} 5$ 的十进制表示中数值 $1$ 的个数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Given that the line $y=ax+1$ intersects the hyperbola $3x^2-y^2=1$ at points A and B. If the circle with diameter AB passes through the origin, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
Let $f(x) = \cos(\omega x + \varphi)$ (where $\omega > 0$, $|\varphi| < \frac{π}{2}$). If the distance between the axis of symmetry $x = \frac{π}{6}$ of the graph of the function $y = f(x)$ and its center of symmetry is $\frac{π}{8}$, then $f(x) = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos(4x + \frac{\pi}{3})}
Five years ago, the price of a movie ticket was $100. In the first year, the price increased by 12%. In the second year, the price decreased by 5%. In the third year, the price increased by 8%. In the fourth year, the price decreased by 4%. In the fifth and final year, the price increased by 6%. After all these changes, the local government introduced a 7% entertainment tax on the movie ticket price, and a particular movie theater offered a 10% special discount on the ticket's price. What is the final price of the movie ticket after incorporating taxes and discounts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$112.61}
Given a real number $a$ satisfying $a^2 - a - 1 = 0$. Find the value of $a^8 + 7a^{-4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Marisa has two identical cubical dice labeled with the numbers  $\{1, 2, 3, 4, 5, 6\}$ . However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is  $2$  with probability  $0.04$ , and  $12$  with probability  $0.01$ , the maximum possible probability of the sum being  $7$  is  $p$ . Compute  $\lfloor 100p \rfloor$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1221}
Given the power function f(x) = (m^2 - 2m - 2)x^m is monotonically decreasing on (0, +∞), find the value of m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the smallest natural number written in the decimal system such that the product of its digits equals $10! = 1 \times 2 \times 3 \times \ldots \times 9 \times 10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45578899}
Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{520}
Calculate: $|-5|-202{3}^{0}+tan45°+\sqrt{9}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.5}
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.

Determine the maximum possible number of positive numbers among these 2018 numbers.

(Walther Janous)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
The U.S. produces about 8 million tons of apples each year. Initially, $30\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\%$ for each additional million tons. Of the remaining apples, $60\%$ is used to make apple juice and $40\%$ is sold fresh. Calculate how many million tons of apples are sold fresh.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.24}
Compute the following limit:
$$ \lim _{n \to \infty} \frac{1}{n}\sum_{k=0}^{n-1}\left[\frac{k+1}{n}-\frac{1}{2}\right]\log(2n-2k-1) .$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{4}}
Consider a scenario where a random variable $X$ represents the number of failures in a sequence of Bernoulli trials with success probability $p$ until the first success occurs. $X$ follows a geometric distribution with parameter $p$: $X \sim Geo(p)$. Now, in this process, after each failure, we toss two fair coins and record the number of heads and tails. When the first success occurs, we check if the number of heads equals the number of tails. Let $B$ be a binomial random variable representing the count of heads out of $2X$ coin tosses with probability $\frac{1}{2}$: $B \sim Bin(2X, \frac{1}{2})$. Determine the probability $P(B = X)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{p}}
已知四面体 $ABCD$ 内接于球 $O$，且 $AD$ 为球 $O$ 的直径。若$\triangle A B C$与$\triangle B C D$均是边长为 1 的等边三角形，求四面体 $ABCD$ 的体积。答案的形式为$\frac{\sqrt{k}}{m}$，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
已知点$M = \{ ( x , y ) \vert \sqrt { 1 - x ^ { 2 } } \cdot \sqrt { 1 - y ^ { 2 } }$

≥xy}，则平面直角坐标系中点集M覆盖区域的面积为 ·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 + \frac { \pi } { 2 }$}
If $9:y^3 = y:81$, what is the value of $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \cdot \sqrt{3}}
For any set \( S \) of five points in the plane, no three of which are collinear, let \( M(S) \) and \( m(S) \) denote the greatest and smallest areas, respectively, of triangles determined by three points from \( S \). What is the minimum possible value of \( \frac{M(S)}{m(S)} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
In each cell of a chessboard, there is a knight. What is the minimum number of knights that should be removed from the board so that there is not a single knight left on the board that can attack exactly four other knights? (A knight attacks cells that are one square away horizontally and two squares vertically, or vice versa.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 knights}
An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Four students from a class are assigned to serve in three different pavilions, A, B, and C, with at least one person assigned to each pavilion. If student A requests not to be assigned to pavilion A, calculate the number of different assignment plans. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Assuming a Poisson process with a rate of 2.5 mistakes per year, calculate the probability that an individual dies before reaching the age of 67.2 years, given that death occurs after 196 mistakes. Use the gamma distribution with parameters 196 and 2.5 to find this probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0228}
Evaluate the following limit using advanced techniques or alternative approaches, without necessarily finding the value of the integral directly:
$$\lim_{p\to0} \left[p \int_{-\infty}^\infty \frac{\ln^2(1+x)}{(p^2 +x^2)^2} \, \mathrm dx\right]$$
Attempts to simplify the integral by manipulating the small parameter $p$ and applying suitable substitutions have been unsuccessful.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{\pi}{2}$$}
10. A rectangle $24 \times 60$ is divided by lines parallel to its sides into unit squares. Into how many parts will this rectangle be divided if we also draw its diagonal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1512}
In the sequence \(\left\{a_{n}\right\}\),
\[
a_{0}=a \quad (a \in \mathbf{Z}_{+}), \quad a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}.
\]

Find the largest integer not exceeding \(a_{n}\) when \(0 \leqslant n \leqslant \frac{a}{2}+1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - n}
In a triangle, if two sides have lengths of $3$ and $5$, then the length of the third side can be _______. (Fill in only one number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Using a set of student triangles with internal angles (45°, 45°, 90° and 30°, 60°, 90°), find the number of different angles that can be drawn greater than 0° and less than 176°. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the largest positive integer \( n \) such that for any positive real numbers \( a_1, a_2, \ldots, a_n \) with \( \prod_{i=1}^{n} a_i = 1 \), the inequality \[ \sum_{i=1}^n \frac{a_i}{a_{i+1}} + \sum_{i=1}^n \frac{a_{i+1}}{a_i} \geq 2 \sum_{i=1}^n a_i \] holds, where \( a_{n+1} = a_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Minimize the function \( f(x,y,z,w) = 2x^2 + 3y^2 + 3z^2 + w^2 - 2xy + xz \) subject to the constraints:

\[ \begin{cases} x + 2y - z = 7 \\ x - z + w = 6 \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{449}{24}}
Find the coefficient of degree \(-2\) in the differential form \(\frac{df}{f}\) for the formal Laurent power series \(f=\sum_{i\ge m} a^i t^i\), where \(m\in \mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the total number of matrices that are similar to the matrix \(\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\) over the finite field \(\mathbb{Z}_7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
A and B are taking turns shooting with a six-shot revolver that has only one bullet. They randomly spin the cylinder before each shot. A starts the game. Find the probability that the gun will fire while A is holding it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{11}}
If $x$ and $y$ are real numbers, find the minimum value of $\sqrt{4+y^{2}}+\sqrt{x^{2}+y^{2}-4 x-4 y+8}+\sqrt{x^{2}-8 x+17}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n.$ Evaluate $M(2020). The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4041}
已知幂函数$f(x) = (m - 1 )^2  x^{m^2 - 4m + 2}$在（0，+∞）上单调递减，则$\textit{m}$=______．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have:
$$
\underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0? \quad[20 \text { points }]
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
Let $\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).$ Then $(\mathcal{S} + 1)^3 = \frac{m}{n}$ with $m$ and $n$ coprime positive integers. Find $10m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Margot wants to prepare a jogging timetable. She wants to jog exactly twice a week, and on the same days every week. She does not want to jog on two consecutive days. How many different timetables could Margot prepare?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
40 kids are running a race. 10% of them pass the finish line in less than 6 minutes. A certain multiple of that number finish in less than 8 minutes. 1/6 of the remaining kids take more than 14 minutes, and 4 kids take more than 14 minutes. What is the multiple of the number of kids who finish in less than 6 minutes that finish in less than 8 minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Task 5. In class 5A, a survey was conducted on what fruits the students like. It turned out that 13 students like apples, 11 like plums, 15 like peaches, and 6 like melons. A student can like more than one fruit. Every student who likes plums also likes either apples or peaches (but not both at the same time). And every student who likes peaches also likes either plums or melons (but not both at the same time). What is the minimum number of people that can be in $5 A$? (20 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
21.2.10 ** Divide a circle with a circumference of 24 into 24 equal segments. Choose 8 points from the 24 points such that the arc length between any two points is not equal to 3 or 8. How many different ways are there to select such a group of 8 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{258}
The Razorback shop makes $192 dollars off each t-shirt and a certain amount off each jersey. During the Arkansas and Texas tech game, they sold 157 t-shirts and 19 jerseys. A t-shirt costs $158 more than a jersey. How much does the shop make off each jersey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$34}
在平面直角坐标系中，设点集
$\{P_{1},P_{2},\cdots,P_{4n+1}\}$
$=\left\{(x,y)|x,y\text{ 为整数},|x|\leq n,|y|\leq n,xy=0\right\}$,
其中$n\in\mathbf{N}_+.$求
$$(P_{1}P_{2})^{2}+(P_{2}P_{3})^{2}+\cdots+(P_{n}P_{4n+1})^{2}+(P_{4n+1}P_{1})^{2}$$
的最小值。请提供当 $n=1$ 时的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}