inclusionAI/Ring-lite-rl-data huggingface dataset:
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\} = x - \lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying:  
\[\{x\} = \frac{x + \lfloor x \rfloor + \lfloor x + \frac{1}{2} \rfloor}{20}\]  
If $S$ is the sum of all numbers in $A$, find $\lfloor S \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
If the function $y=f(x)$ ($x \in \mathbb{R}$) satisfies $f(x+2)=f(x)$, and when $x \in [-1,1)$, $f(x)=|x|$, then the number of intersection points between the graph of the function $y=f(x)$ and the graph of the function $y=\log_{4}|x|$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $a, b \in \mathbb{R}$, and $f(x) = e^x - ax + b$, if $f(x) \geq 1$ always holds, then the range of $\frac{b - a}{a}$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, +\infty)}
A student drew a $50 \times 50$ empty table and wrote a number above each column and to the left of each row. It turned out that all 100 numbers written are different, with 50 of them being rational and 50 being irrational. Then, in each cell of the table, he wrote the product of the numbers written next to its row and column (like in a multiplication table). What is the maximum number of products in this table that could be rational numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1275}
Find the number of permutations \((b_1, b_2, b_3, b_4, b_5, b_6)\) of \((1,2,3,4,5,6)\) such that
\[
\frac{b_1 + 6}{2} \cdot \frac{b_2 + 5}{2} \cdot \frac{b_3 + 4}{2} \cdot \frac{b_4 + 3}{2} \cdot \frac{b_5 + 2}{2} \cdot \frac{b_6 + 1}{2} > 6!.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{719}
A subset $X$ of $\{1, 2, 3, \ldots, 10000\}$ has the following property: If $a, b$ are distinct elements of $X$, then $ab \not\in X$. What is the maximal number of elements in $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9901}
Find the amount on Rs. 5000 in 2 years, with the rate of interest being 10% for the first year and some percentage for the second year, if the final amount is Rs. 6160. What was the rate of interest for the second year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\%}
Given that points $P$ and $Q$ are located on the circle $({x}^{2}+{{\left( y-6 \right)}^{2}}=2)$ and the ellipse $(\dfrac{{{x}^{2}}}{10}+{{y}^{2}}=1)$ respectively, find the maximum distance between $P$ and $Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \sqrt{2}}
在平面直角坐标系 xOy 中，点集 K=\{(x,y)\ |\ x,y=-1,0,1\}. 在 K 中随机取出三个点，则这三个点两两之间的距离均不超过 2 的概率为________.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{19}
Bogosiov I.I.

On the reverse sides of 2005 cards, different numbers are written (one on each). In one question, it is allowed to point to any three cards and find out the set of numbers written on them. What is the minimum number of questions needed to find out what numbers are written on each card?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
The graphs of the equations
$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$
 are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.\,$ These 63 lines cut part of the plane into equilateral triangles of side length $\tfrac{2}{\sqrt{3}}.\,$ How many such triangles are formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
There are five chain links, each with 3 rings. What is the minimum number of rings that need to be unlinked and relinked to connect these links into one chain?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the product of uncommon real roots of the two polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Five persons wearing badges with numbers  $1, 2, 3, 4, 5$  are seated on  $5$  chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
As shown in the figure, points \( M \) and \( N \) lie on sides \( AB \) and \( AC \) of \(\triangle ABC\) respectively, and \(\overrightarrow{AM} = x \overrightarrow{AB}, \overrightarrow{AN} = y \overrightarrow{AC}\), \( D \) is the midpoint of line segment \( BC \), and \( G \) is the intersection point of line segments \( MN \) and \( AD \). If \(\overrightarrow{AG}=\frac{3}{5} \overrightarrow{AD}\), find the minimum value of \(\frac{1}{x^{2}}+\frac{4}{y^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{85}{9}}
Given that \( a \) is a positive real number and \( b \) is an integer between \( 2 \) and \( 500 \), inclusive, find the number of ordered pairs \( (a,b) \) that satisfy the equation \( (\log_b a)^{1001}=\log_b(a^{1001}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1497}
The number $2027$ is prime. Let $T = \sum \limits_{k=0}^{72} \binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1369}
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
The function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) has how many zeros in the interval \([0, \pi]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^{n}$ contains a $k$-generic subset with at most $d(k, n)$ elements. Calculate the value of $d(k, n)$ when $k = 2$ and $n = 2$, and provide the result of k + n + d(k, n).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If (a certain number ^ 4 * 3.456789) ^ 10 is written as a single term, there would be 20 digits to the right of the decimal place. What is the certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the addition problem, each digit has been replaced by a letter.  If different letters represent different digits then what is the value of $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Sabrina went to the library and found a historical series novel called The Rangers Apprentice. There are 14 books in the series, and each book has 200 pages. She read four books in a month and a certain fraction of the number of books remaining in the second month. Sabrina has to read a total of 1000 pages to finish the whole series. What fraction of the remaining books did she read in the second month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a sequence $\{a_{n}\}$ that satisfies $a_{1}+2a_{2}+4a_{3}+\ldots +2^{n-1}a_{n}=\frac{n}{2}$, rearrange the sequence $\{a_{n}\}$ into a new sequence as follows: $a_{1}$, $a_{2}$, $a_{2}$, $a_{2}$, $a_{3}$, $a_{3}$, $a_{3}$, $a_{3}$, $a_{3}$, $\ldots$, $\underbrace{{a_n},{a_n},…,{a_n}}_{({2n-1}) terms}$, $\ldots$. Find the sum of the first 70 terms of the new sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{47}{16}}
Calculate the variance of the difference between the maximum and minimum of n independent and identically distributed uniform random variables on the interval (0, 1).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n-1)2}{(n+1)^2(n+2)}}
一次会议共有 24 名代表参加，每两名代表之间要么握手一次，要么不握手. 会议结束后发现，总共出现了 216 次握手，且对任意相互握过手的两名代表 P,Q，在剩下的 22 名代表中，恰与 P,Q 之一握过手的代表不超过 10 名. 一个"朋友圈"指的是会议中三名两两之间握过手的代表所构成的集合. 求这 24 名代表中朋友圈个数的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
The Mictlán is an  $n\times n$  board and each border of each  $1\times 1$  cell is painted either purple or orange. Initially, a catrina lies inside a  $1\times 1$  cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)^2}
In a convex pentagon \(ABCDE\), the diagonals \(BE\) and \(CE\) are the angle bisectors at vertices \(B\) and \(C\) respectively, \(\angle A = 35^{\circ}\), \(\angle D = 145^{\circ}\), and the area of triangle \(BCE\) is 11. Find the area of the pentagon \(ABCDE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Consider a random permutation of the set  $\{1, 2, . . . , 2015\}$ . In other words, for each  $1 \le i \le 2015$ ,  $i$  is sent to the element  $a_i$  where  $a_i \in \{1, 2, . . . , 2015\}$  and if  $i \neq j$ , then  $a_i \neq a_j$ . What is the expected number of ordered pairs  $(a_i, a_j )$  with  $i - j > 155$  and  $a_i - a_j > 266$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{651222}
14 bags of candy have an average weight rounded to the nearest tenth of a gram equal to 90.2 grams. Given that the weight of each bag of candy is an integer, find the total weight of these 14 bags of candy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1263}
In a survey on snoring and heart disease, a total of 2000 people were surveyed, and the calculated $\chi^2 = 20.87$. Given the critical values for the chi-squared statistic: 3.841 and 6.635, with corresponding confidence levels of 95% and 99%, determine the confidence level that snoring and heart disease are related based on the calculated chi-squared value. Express your answer as a percentage (%).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99%}
Let $x, y, z, t$ be real numbers satisfying $x y+y z+z t+t x=1$. Find the minimum of the expression

$$
5 x^2+4 y^2+5 z^2+t^2 .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 \sqrt{2}$}
A can give B a start of 50 meters or some seconds in a kilometer race. A takes 190 seconds to complete the race. How many seconds of a head start does A give B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.5}
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
An ice skating rink charges 5 dollars for admission and 2.50 dollars to rent skates. Jill can purchase a new pair of skates for 65 dollars. However, she also needs to consider the sales tax that applies to the purchase of new skates, which is 9%. If she purchases skates, she can only expect them to remain in good condition for a period of two years. Taking into account the cost and life expectancy of new skates as well as the rental fees, how many times would she need to go to the rink to justify buying the skates rather than renting a pair over a period of two years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
A smaller rectangle is made from three squares, each of area 25 cm², and is placed inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle, and the other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. Find the area of the larger rectangle. Express your answer in square centimeters (cm²) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
An industrial vehicle requires maintenance checks every 25 days. Three different protective additives, A, B, and C, can be applied to the vehicle, each increasing the time between maintenance checks as follows:

- Additive A increases the time by 10%.
- Additive B increases the time by 15%.
- Additive C increases the time by 5%.

Working conditions and the manufacturer's recommendations can also affect the time between maintenance checks:

- In harsh working conditions, time between maintenance checks must be reduced by 5%.
- The vehicle's manufacturer recommends increasing the time between maintenance checks by an additional 3% with the combined use of the additives.

Assuming that the vehicle is treated with all three additives in consecutive order and works under harsh conditions, by what percent is the time between maintenance checks increased in total, taking into account the manufacturer's recommendation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29.97\%}
Among the numbers $1, 2, 3, \cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{22}{41}}
已知抛物线 \Omega:y=x^2，动线段 AB 在直线 y=\sqrt{3}x-3 上（B 在 A 右侧），且 |AB|=2\sqrt{3}. 过 A 作 \Omega 的切线，取左边的切点为 M. 过 B 作 \Omega 的切线，取右边的切点为 N. 当 MN\parallel AB  时，求点 A 的横坐标.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the plane rectangular coordinate system $xOy$, it is known that $MN$ is a chord of the circle $C: (x-2)^{2} + (y-4)^{2} = 2$, and satisfies $CM\perp CN$. Point $P$ is the midpoint of $MN$. As the chord $MN$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $2x-y-3=0$, such that $\angle APB \geq \frac{\pi}{2}$ always holds. Find the minimum value of the length of segment $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6\sqrt{5}}{5} + 2}
Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes,
which form $1$ degree angles with one another at the origin. Determine the sum of the x-coordinates
of the points of intersection of these lines with the line $y = 100-x$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8950}
Given circle $C$: $(x-5)^{2}+(y+1)^{2}=m(m > 0)$, there is exactly one point on the circle that has a distance of $1$ to the line $4x+3y-2=0$. Determine the value of the real number $m$. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
某个国家有2023个岛和2022座桥，任意一座桥连接两个不同的岛，任意两个岛之间至多有一座桥相连，且可以从任意一个岛通过若干座桥到达其他任何岛。若三个岛中的某个岛与另两个岛都有桥连接，则称这三个岛组成“岛群”.已知任两个“岛群”中都有相同的岛，那么恰有一座桥的岛最少有多少个？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
Find the number of integers $n$ with $1 \leq n \leq 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of 1001.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
A positive integer is *bold* iff it has  $8$  positive divisors that sum up to  $3240$ . For example,  $2006$  is bold because its  $8$  positive divisors,  $1$ ,  $2$ ,  $17$ ,  $34$ ,  $59$ ,  $118$ ,  $1003$  and  $2006$ , sum up to  $3240$ . Find the smallest positive bold number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1614}
求最大的正整数 m，使得存在集合 \{1,2,\cdots,100\} 的 m 个子集 A_1, A_2, \cdots, A_m，满足对任意 1\leq i<j\leq m，都有|A_i\cup A_j|-|A_i\cap A_j|=2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Part of an \(n\)-pointed regular star is shown. It is a simple closed polygon in which all \(2n\) edges are congruent, angles \(A_1,A_2,\cdots,A_n\) are congruent, and angles \(B_1,B_2,\cdots,B_n\) are congruent. If the acute angle at \(A_1\) is \(10^\circ\) less than the acute angle at \(B_1\), then \(n=\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
4.1. A train of length $L=600$ m, moving by inertia, enters a hill with an angle of inclination $\alpha=30^{\circ}$ and stops when exactly a quarter of the train is on the hill. What was the initial speed of the train $V$ (in km/h)? Provide the nearest whole number to the calculated speed. Neglect friction and assume the acceleration due to gravity is $g=10 \mathrm{m} /$ sec $^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
If \( a \) and \( b \) are distinct prime numbers and \( a^2 - aQ + R = 0 \) and \( b^2 - bQ + R = 0 \), find the value of \( R \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Using "学" and "习" to represent two different digits, the product of the four-digit numbers "学学学学" and "习习习习" is a seven-digit number, and its units and millions digits are the same as the digit represented by "学". How many possible two-digit numbers can "学习" represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a $7 \times 7$ grid, some squares are painted black and the rest are painted white, with at least one square of each color. A pattern is called "symmetric" if it looks the same when rotated $90^{\circ}$ and if it can be folded along either diagonal such that the two halves match. Find the total number of symmetric patterns. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1022}
Using the concept of indicator random variables, find the expected value of the product of the number of red and black balls drawn from a box containing 15 green, 10 red, and 10 black balls, when 12 balls are drawn with replacement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{528}{49}}
Find the expected number of times the pattern "21" appears in a sequence of iid random variables $(X_i)_{i\geq 1}$ with $X_i\sim X$, where $X\sim p_1\delta_1+p_2\delta_2+p_3\delta_3$, before seeing 3 for the first time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{p_2\cdot p_1}{p_3}$}
Find the number of lattice points (including boundary points) inside a square with vertices at \( A(5,0) \), \( B(0,5) \), \( C(-5,0) \), and \( D(0,-5) \) in the Cartesian coordinate plane \( Oxy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not available.
- On Thursday, neither those involved earlier in the week nor Wednesday's helpers are willing to participate, but four new individuals are ready to help.
- On Friday, Tim decides he could either do it himself or get help from one previous assistant who has volunteered again.
How many different combinations of people could be involved in the prank over the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given $\triangle ABC$, where $3\sin^{2}B + 7\sin^{2}C = 2\sin A\sin B\sin C + 2\sin^{2}A$, find the value of $\sin(A + \frac{\pi}{4})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\sqrt{10}}{10}}
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? Express your answer in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers. Please provide the value of a + b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{356}
Petya and Vasya participated in the elections for the position of president of the chess club. By noon, Petya had 25% of the votes, and Vasya had 45%. After noon, only Petya's friends came to vote (and accordingly, only voted for him). As a result, Vasya ended up with only 27% of the votes. What percentage of the votes did Petya receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55\%}
Given a set of data: $2$, $6$, $5$, $4$, $7$, $9$, $8$, $10$, what is the $80\%$ quantile of this data set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Laurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
The coefficients of the polynomial  $P(x)$  are nonnegative integers, each less than 100. Given that  $P(10) = 331633$  and  $P(-10) = 273373$ , compute  $P(1)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A square is drawn on a plane with its sides horizontal and vertical. Several line segments parallel to its sides are drawn inside the square, such that no two segments lie on the same line and do not intersect at a point that is internal for both segments. The segments divide the square into rectangles, with each vertical line crossing the square and not containing any of the segments intersecting exactly $k$ rectangles, and each horizontal line crossing the square and not containing any of the segments intersecting exactly $l$ rectangles. What could be the number of rectangles in the partition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{kl}