inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that the even function $f(x)$ satisfies $f(x-2)=-f(x)$ for all $x\in R$, and $f(x)=2^{x}$ when $x\in\[-1,0\]$, find the value of $f(2015)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the universal set $A=\{-1,0,1,2,3\}$ and the set $B=\{x| -2 \lt x\leqslant 2, x\in N\}$, find $\bar{B}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1,3\}}
The circle with radius $4$ passing through points $A$ and $B$ is tangent to another circle with radius $3$ passing through points $A$ and $C$, where the distance $|AB| = 2$. If the line $BC$ is tangent to the second circle, what is the value of $|BC|^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If $|x+a| \lt 3$ holds, a sufficient and necessary condition is $2 \lt x \lt 3$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-5,0]}
5. Vova has 19 math grades in his journal, all twos and threes, and the first four grades are twos. It turned out that among the quartets of consecutive grades, all 16 possible combinations of four twos and threes are present. What are Vova's last four grades?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3222}
A circle has 2017 distinct points $A_{1}, \ldots, A_{2017}$ marked on it, and all possible chords connecting pairs of these points are drawn. A line is drawn through the point $A_{1}$, which does not pass through any of the points $A_{2}, \ldots A_{2017}$. Find the maximum possible number of chords that can intersect this line in at least one point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["1018080"]}
Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour.  If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given real numbers $b$ and $c$ such that $b^2+c^2=1$, and let $f(x) = ax + b\sin x + c\cos x$. If there exist two perpendicular tangent lines on the graph of $f(x)$, determine the range of values for $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\sqrt{2}, \sqrt{2}]}
在平面直角坐标系 $x O y$ 中， $\Gamma_{1}$ 是以 $(2,1)$ 为圆心的单位圆， $\Gamma_{2}$ 是以 $(10,11)$为圆心的单位圆。过原点 $O$ 作一条直线 $l$, 使得 $l$ 与 $\Gamma_{1}, \Gamma_{2}$ 各有两个交点，将 $\Gamma_{1}, \Gamma_{2}$共分成四段圆弧，且这四段圆弧中有两段等长. 所有满足条件的直线 $l$ 的斜率之和为 $\qquad$。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{87}
Two ants, one black and one white, start from point \( A \) on a unit cube and move along the edges. Each time they complete traversing an edge, it is considered as completing one segment. The white ant's path is \( AA_1 \rightarrow A_1D_1 \rightarrow \ldots \), and the black ant's path is \( AB \rightarrow BB_1 \rightarrow \ldots \). They both follow the rule: the segment they traverse in the \( (i+2) \)-th move must be non-coplanar with the segment they traversed in the \( i \)-th move (where \( i \) is a natural number). After completing 1990 segments, the black and white ants stop at some vertices of the cube. What is the distance between the two ants at that time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\sqrt{2}\)}
Calculate the value of the integral $I=\int_C \frac{z^2}{(1 + z + z^2)^2} dz$ where C is the unit circle centered at the origin, traversed in the anticlockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What angle is formed by the minute and hour hands at 3:05?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.5^\circ}
正方形内给定 2022 个点，这些点以及正方形的顶点中任意三点不共线. 在这 2022 个点以及正方形的顶点之间连一些仅可能在端点处相交的线段，将正方形划分为一些三角形，则所连线段（不包含正方形的边）的总数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6067}
Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of the three edges connecting to the vertex they are on with equal probability and travel to the other vertex on that edge. They all stop when any two ants reach the same vertex at the same time. What is the probability that all three ants are at the same vertex when they stop?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{16} \)}
In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay  is the income minus the tax on that income.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50000}
A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142857}
For how many positive integers  $n \le 500$  is  $n!$  divisible by  $2^{n-2}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Form a sequence from the consecutive positive integers from 1 to $n$ (where $n > 1$) in such a way that the sum of any two adjacent terms in the sequence is a perfect square. What is the smallest value of the positive integer $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
John had a stock of 1200 books in his bookshop. There were three types of books - novels priced at $10 each, comics priced at $5 each, and biographies priced at $15 each. On Monday, he sold 30 novels, 20 comics, and 25 biographies, and had 5 comics and a novel returned. On Tuesday, he sold 20 novels, 10 comics, and 20 biographies at a 20% discount. On Wednesday, he sold 30 novels, 20 comics, and 14 biographies, and had 5 novels and 3 biographies returned. On Thursday, he sold 40 novels, 25 comics, and 13 biographies at a 10% discount, and on Friday, he sold 55 novels, 40 comics, and 40 biographies, and had 5 comics, 2 novels, and 3 biographies returned. How much revenue did he earn during the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3603$}
A train of a certain length running at the rate of 54 kmph takes 52.66245367037304 seconds to cross a bridge 625 meters in length. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164.94}
5 kilograms of oranges contained 95% of water. If the next day the concentration of water decreased by a certain percentage, the new weight of the oranges was 25 kilograms. By what percentage did the concentration of water decrease?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\%}
在一个国家里，某些城市对之间有双向直飞航班. 已知能通过不多于 100 次航班从任意一个城市飞到任意另一个，同时也可以通过偶数次航班从任意一个城市飞到任意另一个. 求最小的正整数 d，使得对任意两个城市，总能从任意一个城市飞到另一个，满足乘坐的航班数不超过 d 且为偶数.注：飞行路线允许经过某个城市多次.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
In quadrilateral \(ABCD\), \(m\angle B = m\angle C = 120^\circ\), \(AB = 3\), \(BC = 4\), and \(CD = 5\). Find the area of \(ABCD\) in the form \(\frac{k\sqrt{3}}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
The points \( K, L, \) and \( M \) are the midpoints of the sides \( AB, BC, \) and \( CD \) of the parallelogram \( ABCD \). It turns out that the quadrilaterals \( KBLM \) and \( BCDK \) are inscribed. Find the ratio \( AC : AD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac27$}
In a certain country, there are 100 cities. Some pairs of cities are connected by two-way flights, with no more than $n$ flights per city. A ticket can only be purchased for a route connecting two different cities consisting of two flights with a layover in a third city (routes differing in travel direction or layover city are considered different). It turned out that tickets are sold for a total of 1000 different routes. What is the minimum value of $n$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two squirrels are located at points \( A \) and \( B \) and start simultaneously jumping along lines \( A O \) and \( B O \) toward point \( O \). After passing point \( O \), each continues to move along their respective lines. The distances \( A O = 120 \) meters and \( B O = 80 \) meters, and the angle \(\angle A O B = 60^\circ\). The squirrels move at the same constant speed. What is the minimum distance between the squirrels during their movement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20 \sqrt{3}}
Chord $[AD]$ is perpendicular to the diameter $[BC]$ of a circle. Let $E$ and $F$ be the midpoints of the arcs $AC$ and $CD$, respectively. If $AD \cap BE = \{G\}$, $AF \cap BC = \{H\}$ and $m(AC) = \alpha$, find the measure of angle $BHC$ in terms of $\alpha$. The answer is in the form $x^\circ$, where $x = k - \frac{m\alpha}{2}$. Please find the value of $k + m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Find the value of the 15th term in the expansion of $(-1+i)^{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-38760}
Jonah bought 6 pineapples for $3 each. Each pineapple could be cut into 12 pineapple rings. He sold 4 pineapple rings for $5 each. How much profit did Jonah make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Find the set of all real numbers 'a' such that the inequality x^2 - 2(a-2)x + a > 0 holds for all x in (-∞, 1) ∪ (5, +∞). Express your answer using interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 5]}
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative and the sum of any 11 consecutive terms is positive. How many terms can such a sequence have at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
已知数列 $\left\{a_{n}\right\}$ 满足 $a_{n}=(\sqrt{2}+1)^{n}-(\sqrt{2}-1)^{n}(n \in \mathbf{N})$ ，用 $[x]$ 表示不超过实数 $x$ 的最大整数，则 $\left[a_{2017}\right]$ 的个位数字是 ______。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Linda had some savings and decided to spend 60% of it on various items. She bought furniture worth 40% of her savings. With the remaining 20% of her savings, she plans to buy a TV and a blender. The TV costs $230 and is 70% of the remaining budget, and the blender has a 15% discount applied to it. If the original cost of the blender was $120, what were Linda's original savings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1642.86}
What is the minimal value of $n$ such that the symmetric group $S_n$ contains an even permutation of order 312?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
A starts a business with Rs. 3500 and after 6 months, B joins with A as his partner. B invests an amount that changes on a monthly basis. In the first month, B contributes Rs. 2000, in the second month Rs. 3000 and this continues in an arithmetic progression. At the same time, three months after B joins, C enters the partnership by investing Rs. 5000 initially and adds Rs. 500 each month thereafter. After a year, the profit is divided in the ratio 3:3:4. What is B’s total contribution in the capital over the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27000}
In $\triangle ABC$, the side lengths opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $c - a$ is equal to the height $h$ from vertex $A$ to side $BC$, then calculate the value of $\sin \frac{C - A}{2} + \cos \frac{C + A}{2}$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The average weight of a class is some value. When a new student weighing 90 pounds joins the class, the average decreases by 1 pound. In a few months, the student's weight increases to 110 pounds and the average weight of the class becomes that value + 4 pounds. None of the other students' weights changed. What is the initial average weight of the class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21600}
Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
8. Find the sum of all integer values of $\mathrm{h}$ for which the equation ||$r+h|-r|-4 r=9|r-3|$ in terms of $r$ has no more than one solution.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-93}
A school plans to select 8 students from six senior classes to participate in a citywide high school English speech contest, with at least one student from each class. The total number of ways to allocate these 8 spots is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
A group of students decided to collect as many paise from each member of the group as is the number of members. The number of members in the group is 68. How much is the total collection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46.24}
$n$ teams are to hold a double round-robin tournament (each pair of teams plays two matches, each having a home match). Each team can play multiple away matches in a week (from Sunday to Saturday). However, if a team has a home match in a given week, no away matches can be scheduled for that team in that week. If all matches are to be completed within 4 weeks, find the maximum possible value of $n$.
(Note: A match held at team A's venue between team A and team B is considered a home match for team A and an away match for team B.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A train of a certain length is running at a speed of 58 kmph. It takes 17.998560115190788 seconds to pass a man who is running at 8 kmph in the same direction as the train. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
由数字 $1,2, \cdots, 6$ 组成无重复数字的六位数 $N=\overline{a_{1} a_{2} \cdots a_{6}}$ ，满足条件 $\left|a_{k+1}-a_{k}\right| \neq 1$ $(k \in\{1,2, \cdots, 5\})$. 则这样的六位数的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
There are 6 numbers including 4, 314, 710, and x. The range w of these numbers is 12. What is the difference between the greatest possible value of x and the least possible value of x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{682}
Consider a polynomial function $p(z)$ of degree $n$. If all the coefficients of $p(z)$ have absolute value less than or equal to $k$, where $k$ is a positive constant, find an upper bound for the absolute value of the zeros of $p(z)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+k}
Let $\triangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\triangle AB'C'$ with $AB' = 7\sqrt{6}$, $B'C' = 15\sqrt{6}$, $AC' = 20\sqrt{6}$ such that $\angle BAB' = \frac{5\pi}{12}$. The lines $BB'$ and $CC'$ intersect at point $D$. Let $O$ be the circumcenter of $\triangle BCD$, and let $O'$ be the circumcenter of $\triangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
A train of certain length running at 84 kmph crosses a platform in 16 sec. The length of the platform is 233.3632 m. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{139.97}
Let $y \neq -1$. Define

$$
x_{1} = \frac{y-1}{y+1}, \quad x_{2} = \frac{x_{1}-1}{x_{1}+1}, \quad x_{3} = \frac{x_{2}-1}{x_{2}+1}, \ldots
$$

What is the value of $y$ if $x_{1978} = -\frac{1}{3}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The simple interest on a certain sum of money at the rate of 4% p.a. for 5 years is a certain amount. The same amount of interest can be received on the same sum after 4 years at a rate of 5%. What was the amount of interest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$20}
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{442}}
For how many integers $n$ between $1$ and $50$, inclusive, is
<cmath>\frac{(n^2-1)!}{(n!)^n}</cmath>
an integer? (Recall that $0! = 1$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Given points O(0,0) and A(1,1), the slope angle of line OA is $\boxed{\text{answer}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{4}"]}
In equilateral $\triangle ABC$ with side length $2$, determine the projection of $\overrightarrow{AB}$ onto $\overrightarrow{BC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Suppose point \(P\) is inside triangle \(ABC\). Let \(AP, BP\), and \(CP\) intersect sides \(BC, CA\), and \(AB\) at points \(D, E\), and \(F\), respectively. Suppose \(\angle APB=\angle BPC=\angle CPA, PD=\frac{1}{4}, PE=\frac{1}{5}\), and \(PF=\frac{1}{7}\). Compute \(AP+BP+CP\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{12}}
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 a buckaroo pair if exactly one of them is an honest buck. What is the minimum possible number of buckaroo pairs in the grid?

Note: Two bucks are considered to be neighboring if their cells \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) satisfy either: \(x_{1} = x_{2}\) and \(y_{1} - y_{2} \equiv \pm 1 \pmod{1000}\), or \(x_{1} - x_{2} \equiv \pm 1 \pmod{1000}\) and \(y_{1} = y_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,200,000}
Mr. Yadav spends 60% of his monthly salary on consumable items and 50% of the remaining on clothes and transport. He saves the remaining amount. His savings at the end of the year were 48456. How much amount per month would he have spent on clothes and transport?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4038}
Say an odd positive integer $n > 1$ is \textit{twinning} if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\pi}
Find the smallest \( n > 4 \) for which we can find a graph on \( n \) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $f(x)=x^{2}+ax+b$ where $a, b \in \mathbb{R}$. If the solution set of the inequality $0 \leq f(x) \leq 6-x$ with respect to $x$ is $\left[2,3\right]\cup \{6\}$, then $a+b=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
John hits 70% of his free throws. For every foul, he gets 2 shots. He gets fouled a certain number of times a game. He gets 112 free throws if he plays in 80% of the 20 games the team plays. How many times does he get fouled per game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
A circle passes through the midpoints of the hypotenuse $AB$ and the leg $BC$ of the right triangle $ABC$ and touches the leg $AC$. In what ratio does the point of tangency divide the leg $AC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Aida made three cubes with positive integer side lengths $a$, $b$, and $c$. She decided to dismantle these cubes into unit cubes and tried to construct a larger cube with side length $a+b+c$. Unfortunately, she was 648 blocks short. How many possible ordered triples $(a, b, c)$ exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the value of $\alpha$ in the following equation involving Beta functions for all real numbers $m > 0$: $$\frac{\mathrm{B}(m, \frac{1}{2})}{\mathrm{B}(m, m)}=2^{\alpha}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m-1}
Find all real numbers \( k \) such that the inequality
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
holds for any \( a, b, c, d \in [-1, +\infty) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
f(x)=(x-1)(x-2)⋯⋯(x-2004)·

问：有多少个实数x，使得 f(x)|=1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4008}
What is the maximum number of edges of a regular 10-sided prism that can be intersected by a plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}