inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the Laplace transform of the function $$f(x) = (1 + 2ax)x^{-\frac{1}{2}}e^{ax}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{s\sqrt{\pi}}{(s - a)^{\frac{3}{2}}}}
The marked price of a book was 30% less than the suggested retail price.  Alice purchased the book for half the marked price at a Fiftieth Anniversary sale.  What percent of the suggested retail price did Alice pay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35\%}
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of this ellipse is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given a positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil \frac{n}{2} \rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $x$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{450}
A rectangle is inscribed within a semi-circle with a perimeter of 126 cm. One pair of opposite sides of the rectangle are tangent to the semi-circle. The length of the rectangle is three times its width. What is the radius of the semi-circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.625}
Mona has 12 match sticks of length 1, and she has to use them to make regular polygons, with each match being a side or a fraction of a side of a polygon, and no two matches overlapping or crossing each other. What is the smallest total area of the polygons Mona can make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Given the function $f(x) = x^3 + ax + \frac{1}{4}$, if the x-axis is the tangent to the curve $y = f(x)$, then the value of $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{4}}
Find the smallest positive real number $\lambda$ such that for any three complex numbers $\mathrm{z}_{1}$, $\mathrm{z}_{2}$, $\mathrm{z}_{3} \in \{\mathrm{z} \in \mathbb{C} \mid |\mathrm{z}| < 1\}$, if $\mathrm{z}_{1} + \mathrm{z}_{2} + \mathrm{z}_{3} = 0$, then $\left| \mathrm{z}_{1} \mathrm{z}_{2} + \mathrm{z}_{2} \mathrm{z}_{3} + \mathrm{z}_{3} \mathrm{z}_{1} \right|^{2} + \left| \mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}_{3} \right|^{2} < \lambda$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of triangle $ABC$?

Enter your answer in the form $\frac{a \sqrt{b} - c}{d},$ and simplified as usual.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{2} - 3}{2}}
If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the trees: two configurations are considered identical if one can be converted into the other by swapping the trees (along with the corresponding groups) or by rotating each group around its tree. In how many ways can the children be divided into groups?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given the total number of stations is 6 and 3 of them are selected for getting off, calculate the probability that person A and person B get off at different stations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
The starting and ending points of a tram route are Station A and Station B, respectively. A tram departs from Station A for Station B every 5 minutes, and the entire journey takes 15 minutes. A person starts cycling from Station B towards Station A along the tram route. When he starts, a tram just arrives at Station B. On the way, he encounters 10 trams coming towards him before reaching Station A. Upon reaching Station A, there is a tram just departing from Station A. How many minutes did it take for him to cycle from Station B to Station A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
12. Given $S$ as a binary string of $10^{4}$ bits containing only $0$ and $1$, a positive integer $k \leqslant 10^{4}$, a $k$-block of $S$ is a substring of $S$ consisting of $k$ consecutive bits. Two $k$-blocks $a_{1} a_{2} \cdots a_{k}=b_{1} b_{2} \cdots b_{k}$ are equal if and only if $a_{i}=b_{i}(i=1$, $2, \cdots, k)$. Consider all $10^{4}$-bit binary strings that contain at most seven different 3-blocks. Find the maximum number of different 10-blocks such a string can contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Plot points $A, B, C$ at coordinates $(0,0),(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $A B$ and $B C$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise 45 degrees about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise 45 degrees about point $A$. Find $\frac{X_{1}+X_{2}}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{4}"]}
How many pairs of integers \((x, y)\) are there such that \(x \geq y\) and \(\frac{1}{x}+\frac{1}{y}=\frac{1}{211}\) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the numeral system in which the following multiplication is performed: \(352 \cdot 31 = 20152\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $C$ be the unit circle $x^{2}+y^{2}=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$-and $y$-axes with diagonal $p q$. What is the probability that no point of $R$ lies outside of $C$? The original answer is in $\frac{n}{\pi^m}$format, please give the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the limit of the following sum as $n$ approaches infinity?
$$
\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight line and a curve. Find the sum of the coordinates of the intersection points of a straight line and a curve, rounded down.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Petya, Sasha, and Misha are playing knockout tennis. A knockout game means that in each round, two players play while the third waits. The loser of the round gives place to the waiting player and becomes the one waiting in the next round. Petya played a total of 12 rounds, Sasha played 7 rounds, and Misha played 11 rounds. How many times did Petya win against Sasha?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If x, y, and z are positive integers, and 6z = some value = 2x, then the least possible value of x + y + z is 26. What is the value that 6z is equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Solve the equation $$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\ldots+\sqrt{4^{2008}x+3}}}}}-\sqrt{x}=1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2^{4016}}}
The sale price of an article including the sales tax is some amount. The rate of sales tax is 10%. If the shopkeeper has made a profit of 16%, and the cost price of the article is Rs. 531.03, what is the sale price of the article including the sales tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{677.59}
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A school arranges four classes in the morning, each lasting 40 minutes. The first class starts from 8:00 to 8:40, with a 10-minute break between classes. If a student arrives late, and he arrives at the classroom between 9:10 and 10:00, what is the probability that he listens to the second class for no less than 10 minutes? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a *single* and that between a boy and a girl was called a *mixed single*. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and $2S_{n}$ is the arithmetic mean of $6$ and $a_{n}$. If for any positive integer $n$, $3S_n-\frac{1}{S_n} \in [k,t]$, then the minimum value of $t-k$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}}
Jacob's water tank can hold up to 350 liters of water. Jacob collects water from the river and rain to fill his water tank. The daily water collection varies based on weather conditions and river flow. On average, he can collect 300 to 600 milliliters of water from the rain and 900 to 1500 milliliters of water from the river every day. How many days does Jacob need to fill up his water tank, considering the variable rate of daily water collection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{213}
设 $a_{1}, a_{2}, \cdots, a_{20} \in\{1,2, \cdots, 5\}, b_{1}, b_{2}, \cdots, b_{20} \in\{1,2, \cdots, 10\}$, 集合 $X=\{(i, j) \mid$ $\left.1 \leqslant i<j \leqslant 20,\left(a_{i}-a_{j}\right)\left(b_{i}-b_{j}\right)<0\right\}$, 求 $X$ 的元素个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)

Alexandru Mihalcu
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a cube with edge length 1, two cross-sections in the form of regular hexagons are made. Find the length of the segment at which these cross-sections intersect.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
7 distinct primes can be written as a, b, c, a + b + c, a + b - c, a - b + c, -a + b + c, where two of a, b, c sum to 800. Let d be the difference between the largest and the smallest of the 7 primes. What is the largest possible value of d?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1594}
A cricket bat is sold for $850, making a profit of $205. What is the profit percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.78\%}
Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{6}}{9}}
The domain of the inverse function of $f(x) = 2 - x$ $(0 < x \leq 3)$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 2)}
Let $S$ be a set. We say $S$ is $D^\ast$-finite if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a witness of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Ravi has some coins. He has 2 more quarters than nickels and 4 more dimes than quarters. If he has 6 nickels, how much money does he have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.50}
John gets his pool cleaned every 3 days throughout the year. In summer (3 months), it costs $150 per cleaning, and in the remaining 9 colder months, it costs only $120 per cleaning. He gives a 10% tip to the cleaner every time he comes. In addition to cleaning, he needs to use $250 worth of chemicals twice a month during the summer and $150 worth of chemicals twice a month during the rest of the year. Furthermore, he spends $100 a month on equipment rental for pool maintenance tools, and his pool pump consumes an average of $75 worth of electricity per month. John also subscribes to a quarterly $200 repair and inspection service.

Taking all these factors into account, calculate John's total pool maintenance and operation expenses for an entire year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$23,930}
The opposite sides of a quadrilateral inscribed in a circle intersect at points \( P \) and \( Q \). Find the length of the segment \( |PQ| \), given that the tangents to the circle drawn from \( P \) and \( Q \) are \( a \) and \( b \) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a^2 + b^2}}
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:30 PM}
Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its 4 digits when expressed in mmdd format (e.g. the digit sum of 13 May is $0+5+1+3=9$). Find the number of dates in the year 2021 with digit sum equal to the positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(E,O,E)}
Consider the set of all bijective functions from the set of non-negative integers to the set of rational numbers. For each function, define its maximum step size as the maximum absolute difference between the values of the function at consecutive integers. What is the greatest lower bound of the maximum step sizes of all such functions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A beadshop has a varied profit distribution throughout the week: On Monday, it earns 20% of its profit, followed by 18% on Tuesday, 10% on Wednesday, 25% on Thursday, and 15% on Friday. However, on Saturday, there are no sales, but the shop offers a 20% discount on their items, resulting in a 5% loss in overall profit. The shop makes a total profit of $3,500 during the six days (from Monday to Saturday; Sunday is not included). Calculate the actual profit made on Friday, in dollars, after factoring in the discount applied on Saturday that affects the overall profit obtained.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$552.63}
Audrey has 26 peaches and decides to buy 3 times as many. Paul has 48 peaches and decides to buy twice as many. Meanwhile, their friend Maya has 57 peaches and buys an additional 20. What's the difference between the total number of peaches Audrey, Paul, and Maya have now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{325}
Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Is there a closed form for the general recurrence relation \( f(n) = an + b\sum_{k=0}^{n-1}f(k) \), where \( a \) and \( b \) are constants?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{b}\left((b+1)^n - 1\right)}
The wholesale department operates a certain product, with a wholesale price (selling price) of 500 yuan per unit, and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank, with a monthly interest rate of 4.2‰. The storage and operating cost of the product is 0.30 yuan per unit per day. Find the maximum average storage period for the product without incurring a loss. Express your answer in days as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{502}
Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Find the range of the independent variable x in the function y = 1 / (x - 1). Express your answer as an inequality or a statement about x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x ≠ 1}
We wrote the reciprocals of natural numbers from 2 to 2011 on a board. In one step, we erase two numbers, \( x \) and \( y \), and replace them with the number

$$
\frac{xy}{xy + (1 - x)(1 - y)}
$$

By repeating this process 2009 times, only one number remains. What could this number be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2010! + 1}}
Trapezoid $EFGH$ has sides $EF=100$, $FG=55$, $GH=23$, and $HE=75$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the length of $EQ$ if it is expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{750}{13}}
Indecisive Andy starts out at the midpoint of the 1-unit-long segment \(\overline{H T}\). He flips 2010 coins. On each flip, if the coin is heads, he moves halfway towards endpoint \(H\), and if the coin is tails, he moves halfway towards endpoint \(T\). After his 2010 moves, what is the expected distance between Andy and the midpoint of \(\overline{H T}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
A 90-degree angle is covered by some angles with the same vertex in two or three layers, and the sum of the angles is 290 degrees. Find the measure of the angle covered in three layers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20^\circ}
已知集合S={1,2,⋯,2005}.A⊆S,A中任两数之和不被117整除，试求｜A｜的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. Calculate the ratio of the perimeter of the rectangle to the perimeter of the rhombus. Express your answer as a ratio (e.g., 1:1, 2:1, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
$已知f(x)=|x-a|x+|x-2|(x-a).$
$若x\in (-\infty ,1)时， f(x)<0，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1，+\infty )}
Two circles \(C_{1}\) and \(C_{2}\) touch each other externally and the line \(l\) is a common tangent. The line \(m\) is parallel to \(l\) and touches the two circles \(C_{1}\) and \(C_{3}\). The three circles are mutually tangent. If the radius of \(C_{2}\) is 9 and the radius of \(C_{3}\) is 4, what is the radius of \(C_{1}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the volume of a regular quadrilateral prism if its diagonal makes an angle of \(30^{\circ}\) with the plane of the lateral face, and the side of the base is equal to \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^3 \sqrt{2}}
Among the 12 lines where the edges of a hexagonal pyramid are located, how many pairs of skew lines exist? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
记如下关于实数变量 $x, y$ 的函数
$$
f(x, y)=\sqrt{4 y^{2}-12 y+10}+\sqrt{18 x^{2}-18 x+5}+\sqrt{18 x^{2}+4 y^{2}-12 x y+6 x-4 y+1}
$$
的最小值为 $a$. 求取整 $\left[8 a^{2}\right]$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}