inclusionAI/Ring-lite-rl-data huggingface dataset:
An old water pump can fill up a large trough in 600 sec, but a second, newer pump takes only 200 sec to fill the trough. A third water pump can fill the same trough in 400 sec. How long would it take to fill the trough using all three pumps at the same time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1200}{11}}
Let \( a, b, c, d \) be real numbers defined by

$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$

Calculate their product.
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Answer: \boxed{11}
定义在 \bR 上的奇函数 f(x) 同时是以 2 为周期的周期函数，并且在 x\in(0,1) 时，f(x)=\ln(x^2-2x+\frac{3}{2}). 则函数 f(x) 在 [0,10] 上的零点个数为__________.（李纪琛供题）
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Answer: \boxed{21}
某品牌儿童服装每件售价60元，不征收附加税时，每年销售80万件；若征收附加税，即每销售值100元征收R元（叫做税率R%），则每年的销售量将减少$$\frac{20}{3}$$ R 万件。若在此项经营中，每年征收附加税不少于128万元。问：税率R应怎样确定？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \leq R \leq 8}
5. Place eight balls, each with a radius of 1, in two layers inside a cylinder such that each ball is tangent to its four adjacent balls, and to one base and the side of the cylinder. Then the height of the cylinder is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{8}+2}
The points \((2,3)\) and \((3, 7)\) lie on a circle whose center is on the \(x\)-axis. What is the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{1717}}{2}}
Sasha wrote the numbers $7, 8, 9, \ldots, 17$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the maximum value that the sum of the remaining numbers on the board can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Determine the value of the following expression:

$$
X = \sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \cdots}}}}}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[7]{x^4 y^2 z}}
You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
a , b , c , d and e are 5 consecutive points on a straight line . if bc = 3 cd , de = 8 , a certain distance is given and ac = 11 , the length of ae is 21. What is the length of ab?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
在平面上有一点 O，过点 O 引出 2007 条不同的直线 $l_1, l_2, l_3, \ldots, l_{2007}$，则可形成多少对以 O 为顶点的对顶角？
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Answer: \boxed{4026042}
Consider the cubes whose vertices lie on the surface of a given cube. Which one is the smallest among them?
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Answer: \boxed{\frac{1}{\sqrt{2}}}
Given $p$: $\frac{1}{2} \leq x \leq 1$, $q$: $(x - a)(x - a - 1) > 0$, if $p$ is a necessary but not sufficient condition for $\lnot q$, then the range of values for the real number $a$ is $\_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \frac{1}{2}]}
A set of tiles numbered $1$ through $100$ is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with $1$. Find the number of times the operation must be performed to reduce the number of tiles in the set to one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
It takes 30 days to fill a laboratory dish with bacteria. The bacteria grows at a certain rate each day, and it took 25 days for the bacteria to fill 1/32 of the dish. What is the growth rate of the bacteria each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the sequence ${a_n}$ where ${a_{n+1} + (-1)^n a_n = 2n - 1}$, calculate the sum of the first 16 terms of the sequence. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
To measure the height of tower AB on the opposite bank of a river, a point C is chosen on the bank such that the base of the tower A is exactly to the west of point C. At this point, the elevation angle to the top of the tower B is measured to be 45°. Then, moving from point C in a direction 30° north of east for 30 meters to reach point D, the elevation angle to the top of the tower B from D is measured to be 30°. The height of tower AB is \_\_\_\_\_\_ meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $$ A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}} $$ Find the largest integer less than or equal to $\frac{1}{A}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The Garfield Super Winners play $100$ games of foosball, where teams score a non-negative integer number of points, and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but allow an average of $8$ points per game. Given that the Garfield Super Winners never won or lost by more than $10$, what is the largest possible number of games that they could win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Convert $BD4_{16}$ to base 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{233110_4}
If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If the range of the function $f(x)$ on $[m, n] (m < n)$ is exactly $[m, n]$, then $f(x)$ is called an "equal value mapping interval" of the function. Among the following functions: $① y = x^2 - 1$; $② y = 2 + \log_2 x$; $③ y = 2^x - 1$; $④ y = \frac{1}{x - 1}$. The number of functions that have a unique "equal value mapping interval" is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
As shown in the figure, points $C$ and $D$ divide $AB$ into three equal parts. At 8:00, Person A starts walking uniformly from $A$ to $B$. At 8:12, Person B starts walking uniformly from $B$ to $A$. After a few more minutes, Person C starts walking uniformly from $B$ to $A$. When Person A and Person B meet at point $C$, Person C just reaches point $D$. When Person A and Person C meet at 8:30, Person B just reaches $A$. At what time does Person C start?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8:16}
Out of all parallelograms with a given area, find the one where the longest diagonal is minimized.
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Answer: \boxed{\text{Square}}
When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the remainder when $5^{2021}$ is divided by $17$.
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Answer: \boxed{14}
$已知椭圆C: \frac{x^2}{a^2} + \frac{y^2}{b^2}=1 (a  >  b > 0), F_1，F_2 为C的左，右焦点，点P \left(1,-\frac{3}{2}\right) 为椭圆上一点，且|PF_1|+|PF_2|=4. 过P作两直线与椭圆C相交于相异的两点A,B,直线PA、PB的倾斜角互补，直线AB与x，y轴正半轴相交.$
求直线AB的斜率.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\frac{1}{2}$}
Baker made some cakes and 153 pastries. He sold 8 pastries and 97 cakes. Baker sold 89 more cakes than pastries. How many cakes did he make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
Two cyclists are moving towards each other at a certain speed. They are now 50 miles apart. At this instance, a fly starts from one cyclist and moves towards the other, moving to and fro until the two cyclists meet each other. The fly is moving at 15 miles/hour, and the total distance covered by the fly is 37.5 miles. What is the speed of the cyclists?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
On dividing 635 by a prime number, the quotient is a multiple of 7 and the remainder is 11. Find the divisor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the expansion of $\left(x+1\right)\left(2x+1\right)\cdots \left(nx+1\right)$, where $n$ is a positive integer, find the coefficient of the linear term in $x$. Express your answer as a combination formula (e.g., $C_n^k$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_{n+1}^2}
Calculate

$$
\sqrt{1+2 \sqrt{1+3 \sqrt{1+\ldots+2017 \sqrt{1+2018 \cdot 2020}}}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$ ($n=1,2,3,\ldots$). When the first term $a_1$ and the common difference $d$ change, if $a_5+a_8+a_{11}$ is a constant, then among $S_n$ ($n=1,2,3,\ldots$), the constant value is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_{15}}
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( 0 < x < \frac{\pi}{2} \). Determine the range of the function \( y = \sin x - 2 \cos x + \frac{32}{125 \sin x (1 - \cos x)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{2}{5}, +\infty \right)}
若 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2 a+3 b+5 c$ 的最大值和最小值之差。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A sequence $a_1,a_2,\dots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq 1$. If $a_1=999$, $a_2<999$ and $a_{2006}=1$, how many different values of $a_2$ are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{648}
Given several numbers, one of them,  $a$ , is chosen and replaced by the three numbers  $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are  $1000$  ones, and we apply the process several times. A number  $m$  is called *good* if there are  $m$  or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{667}
Find any five consecutive natural numbers less than 100, whose product is divisible by 2014.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53, 54, 55, 56, 57}
If 30 sheep eat 45 bags of grass in 40 days, with the fastest sheep eating at twice the rate of the slowest sheep, in how many days will one slowest sheep eat one bag of grass?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1594}
Let \(X_1, X_2, \ldots, X_n, \ldots\) be independent random variables, where each \(X_n\) is uniformly distributed on \([0, n]\). Find a sequence \((a_n)\) such that \(\frac{1}{a_n} \sum_{k=1}^n X_k^2 \to 1\) in probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n^3}{9}}
Find the number of 12-cycles in the tent map.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{335}
A convex polyhedron is bounded by 4 regular hexagonal faces and 4 regular triangular faces. At each vertex of the polyhedron, 2 hexagons and 1 triangle meet. What is the volume of the polyhedron if the length of its edges is one unit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{23\sqrt{2}}{12}}
Using only pennies, nickels, dimes, and quarters, calculate the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin^{2}\left(\frac{x}{2}\right) \cos^{6}\left(\frac{x}{2}\right) \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 \pi}{8}}
If \( N=\frac{1987^{4(\sqrt{3}+2)^{1987}+1988}}{1987^{(\sqrt{3}+2)^{1988}}+(\sqrt{3}+2)^{1988}} \), what is the last digit of \( N \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a certain football competition, a victory is worth 3 points, a draw is worth some points, and a defeat is worth 0 points. Each team plays 20 matches. If a team scored 8 points after 5 games, it has to win at least 9 of the remaining matches to reach the 40-point mark by the end of the tournament. How many points is a draw worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$191$}
Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{819}
Given positive integer  $n$ , find the biggest real number  $C$  which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than  $1$  is less than  $C$ , then we can divide the set of numbers into no more than  $n$  groups so that the sum of reciprocals of every group is less than  $1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n+1}{2}}
For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, $n$-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
John bought a shirt on sale for 25% off the original price and another 25% off the discounted price. The final price was $17. What was the price before the first discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$30.22}
Let  $n$  be a positive integer. Each of the numbers  $1,2,3,\ldots,100$  is painted with one of  $n$  colors in such a way that two distinct numbers with a sum divisible by  $4$  are painted with different colors. Determine the smallest value of  $n$  for which such a situation is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
1. The numbers from 1 to 2000 are written on a board. In one move, it is allowed to erase any number $n$, as well as no more than $n$ numbers not less than $n$. In what minimum number of moves can all the numbers be erased?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
For how many values of q, does | | | q - 5 | - 10 | - a certain number | equals 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt{3}$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$). Compute the square of the area of $X_1X_2X_3X_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), and \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C \), and \( M \) and \( I \) are the centroid and incenter of the triangle \( \triangle P F_{1} F_{2} \) respectively. If \( M I \perp x \)-axis, then find the radius of the incircle of \( \triangle P F_{1} F_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
Find the dimension of the null space of the linear transformation $T: P_{n} \rightarrow P_{n-1}$ defined by $T(p(x)) = p'(x) - \int_{0}^{x} p(t) dt$, where $P_{n}$ is the vector space of all polynomials with real coefficients of degree at most $n$.
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Answer: \boxed{0}
Consider the four points \(A=(2,4)\), \(B=(0,0)\), \(C=(6,0)\), and \(D=(6,3)\) in the plane. For which point \(P\) in the plane is the sum \(PA+PB+PC+PD\) minimized?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(4, 2)}
Let $ABC$ be a triangle with $AB=11$, $BC=12$, and $CA=13$. Let $M$ and $N$ be the midpoints of sides $CA$ and $AB$, respectively. The incircle touches sides $CA$ and $AB$ at points $X$ and $Y$, respectively. Suppose that $R$, $S$, and $T$ are the midpoints of line segments $MN$, $BX$, and $CY$, respectively. Then $\sin \angle{SRT} = \frac{k\sqrt{m}}{n}$, where $k$, $m$, and $n$ are positive integers such that $\gcd(k,n) = 1$ and $m$ is not divisible by the square of any prime. Compute $100k + 10m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5033}
Given that the cubic function $f(x)= \frac {1}{3}x^{3}-(4m-1)x^{2}+(15m^{2}-2m-7)x+2$ is increasing on $x∈(-∞,+∞)$, the range of values for $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2,4]}
8,9

In trapezoid $ABCD$, side $AD$ is the larger base. It is known that $AD=CD=4 \frac{2}{3}$, $\angle BAD=90^{\circ}$, and $\angle BCD=150^{\circ}$. On base $AD$, triangle $AED$ is constructed, and points $B$ and $E$ lie on the same side of line $AD$ with $AE=DE$. The height of this triangle, drawn from vertex $E$, is $1 \frac{2}{5}$. Find the area of the common part of trapezoid $ABCD$ and triangle $AED$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49(3\sqrt{3}-5)}{3}}
If 1 is an element of the set $\{a+2, (a+1)^2, a^2+3a+3\}$, then the real number $a=$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The digits 2, 3, 5, 7, and 8 can be used, each exactly once, to form many five-digit integers. Of these integers, find the one that is as close as possible to 30000. What is the tens digit of this integer? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
2020 magicians are divided into groups of 2 for the Lexington Magic Tournament. After every 5 days, which is the duration of one match, teams are rearranged so no 2 people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10095}
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131070}
A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1698$}
What is the maximum number of rooks that can be placed on a $3n \times 3n$ chessboard so that each of them is attacked by no more than one of the others?
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Answer: \boxed{4n}
Find $\frac{a^{8}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}$, given that $\frac{a}{3}-\frac{3}{a}=4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
What is the maximum number of disjoint groups into which all the integers from 1 to 25 can be divided so that the sum of the numbers in each group is a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}