inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{28743}{12}}
A **binary palindrome** is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). Leading zeroes are not permitted in the standard representation. For example, 2015 is a binary palindrome because in base 2 it is \(11111011111\). How many positive integers less than 2015 are binary palindromes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
Find the volume of a regular triangular prism if the side of its base is \(a\) and the lateral surface area is equal to the sum of the areas of the bases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^3}{8}}
A man is rowing a flat bottom canoe down a river which varies in width. In still water, he can row at a speed of 24 kmph. He realizes that it takes him thrice as long to row up the river as it does to row down the river. On top of that, he has to deal with cross-currents that alter his speed by a factor of 15%. Assuming the cross-currents affect his speed equally in both directions and that the speed of the stream changes linearly with the width, calculate the rate of the stream in a section with double the width of the narrowest section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.4}
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Let $f$ be a Lebesgue integrable function on $[0,1]$ with $\lim_{x\rightarrow 1}f(x) = c$, where $c$ is a constant. Determine the value of $\int_0^1 nx^n f(x) \, dx$. \(n \in \mathbb{N}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c}
At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the nine-sided regular polygon $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1,A_2,\dots,A_9\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Peter knows that the distance an airplane travels between two countries is 7019 km, while the distance between Spain and Germany is 1615 km. Peter took a flight from Spain, which made a stopover in Germany, while being there, he receives a call from his brother asking him how many kilometers he has to fly to get to the other country and then return to Spain. What is the distance between Spain and the other country?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5404}
Calculate the limit \(\lim_{x \to 0} \left(\frac{1 + \sin x}{1 - \cos x}\right)^{\frac{1}{x}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
What is the standard notation or terminology for the set \( S = \{ a \in A \mid \sigma(a) = a \} \) if \( A \) is an algebra over a field \( F \) and \( \sigma: A \rightarrow A \) is an involution of the second kind?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H(A, \sigma)}
Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3-3^{-999}}{4}}
Find the integer value of $x$ that maximizes the function \( f(x) = a\left(1-\dfrac{a}{x}\right)\left(1-\dfrac{a}{bx}\right)^{n-2} \) given the constraints: \( a \in \{1, 1/4, 1/2, 3/4\} \), \( 50 \leq n \leq 500 \), \( 5 \leq b \leq 44 \), and \( 5 \leq x \leq 64 \). Use the derived formula \( x = \dfrac{a(n-1)}{n+b-2} \) as a starting point, and ensure that the final value of \( x \) is an integer within the specified range.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
In a farm, there are ducks, cows, and spiders. The total number of legs is 72 more than twice the number of heads. Additionally, there are twice as many spiders as ducks, and the total number of animals does not exceed 40. Find the total number of cows in the farm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the derivative of $a^x$ using the limit definition: \( \frac{\mathrm{d}}{\mathrm{d}x}a^x = \lim_{h\to0}\frac{a^{x+h}-a^{x}}{h} \). Specifically, determine \( \lim_{h\to0}\frac{a^{h}-1}{h} \) for \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^x \ln a}
Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
There are 240 residents in the village of Sosnovka. Some people are acquainted with each other, and some are not. It is known that any five residents can be seated at a round table so that each of them is acquainted with both of their neighbors. What is the smallest number of pairs of acquainted residents in Sosnovka?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28440}
If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
27
}
Determine the maximum possible value of the expression

$$
27abc + a\sqrt{a^2 + 2bc} + b\sqrt{b^2 + 2ca} + c\sqrt{c^2 + 2ab}
$$

where \(a, b, c\) are positive real numbers such that \(a + b + c = \frac{1}{\sqrt{3}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3 \sqrt{3}}}
The eight-digit number 757AB384 is divisible by 357. Get B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A river 42 feet across is to be used for a boat race. Each boat is 3 feet wide and must have at least 2 feet of space between them or the riverbank. There is also a 2 mph current flowing downstream that affects the boats' speed. If each boat's individual speed without any current is 6 mph, calculate the minimum time it takes for all boats to reach the finish line located 1 mile downstream.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such that $\tfrac{1}{E_n}
> 2017$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13446}
A glazier has 15 windows of various sizes and shapes, each requiring one pane of glass. The panes have been prepared to match the windows exactly, but the glazier is unaware of this. He selects panes by finding one that is large enough for each window, either exactly fitting or one that can be cut to fit. If no suitable pane is found, he moves to the next window. It is not allowed to assemble a pane from multiple parts. What is the maximum number of windows that can remain unglazed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Chloe wants to purchase a jacket that costs $\$45.50$. She checks her purse and finds she has four $\$10$ bills, ten quarters, and some nickels and dimes. What is the minimum number of dimes that she must have if she has 15 nickels?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let $R$ be an equivalence relation on $P(\Bbb Z)$. Determine the number of elements in the equivalence class of a set $S$ that contains exactly $k$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Two table tennis players, \(A\) and \(B\), are preparing to play 7 matches against each other. Player \(A\) is offered 100 times the money if he wins all 7 matches. Given that past experience shows \(A\) has twice the chance of winning 4 matches and losing 3, as opposed to winning 5 matches and losing 2, determine the fair multiplier instead of 100 times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \dfrac{11}{5} \right)^7}
Determine the number of permutations \( \alpha \in S_9 \) such that \( \alpha \omega \alpha^{-1} = \gamma \), where \( \omega = (13624)(587)(9) \) and \( \gamma = (15862)(394)(7) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $F$ be a focus of the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$. If there exists a point $P$ on $C$ such that the midpoint of the line segment $PF$ is exactly one endpoint of its imaginary axis, then the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Determine the values of $K$ such that all roots of the polynomial \( m^3 + m^2 + 2m + K = 0 \) are negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
A grocery store sells three different types of fruit: apples, oranges, and bananas, with varying quantities in each box. The markup on a box of apples is 10 percent of the cost, the markup on a box of oranges is 15 percent of the cost, and the markup on a box of bananas is 20 percent of the cost. The costs of the boxes are $30 for apples, $40 for oranges, and $50 for bananas. The box of apples contains 25 apples, the box of oranges contains 20 oranges, and the box of bananas contains 15 bananas.

Calculate the weighted average markup percentage of the total selling price for all three boxes of fruits, considering the quantity of each fruit type. (Note: Markup = Selling Price - Cost)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.83\%}
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The sequence $\{a\_n\}$ consists of positive numbers and satisfies $a\_1=1$, $\sqrt { \frac {1}{ a\_{ n }^{ 2 }}+3}= \sqrt { \frac {1}{ a\_{ n+1 }^{ 2 }}}$. Let $b\_n= \frac {1}{ a\_n^{ 2 } a\_{ n+1 }^{ 2 }}$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is denoted as $S\_n$. If $S\_n < t$ holds true for any $n∈N^{*}$, then the range of the real number $t$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac {1}{3},+\infty)}
Determine the minimal number \( M \) of telephone connections needed to ensure that no more than 1% of calls from city A to city B are blocked. Assume city A has 2000 phones, each user needs a connection for 2 minutes on average during 10 working hours, and calls are random. Use a Gaussian distribution with a mean of \( 6.6667 \) connections. Ignore impossible cases of less than 0 or more than 2000 calls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The volume of a truncated cone circumscribed around a sphere is twice the volume of the inscribed sphere. By how many times is the radius of the base circle of the truncated cone the radius of the top circle's radius?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3 + \sqrt{5}}{2}}
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{42}}
Evaluate the limit: \[ \lim_{n \to \infty} \sqrt[n]{\frac{(2n - 1)!}{n!}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the values of $x$ for which the series \( \sum_{n=0}^\infty \frac{x^n}{n\#} \) converges, where \( n\# \) is the product of all primes less than or equal to \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-e, e)}
Evaluate the sum \( \sum_{n=1}^{\infty}\frac{n^n}{(H_n)^2n!}\) where \(H_n=\sum\limits_{k=1}^n\frac1k\) is the \(n\)th harmonic number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the integral \( \int_{\gamma}\omega \) on the curve \( \gamma(t)=(t\cos(2\pi t), t\sin(2\pi t), t^2) \), where \( t \in [0,1] \), and the curve is contained in the paraboloid \( S \) defined by \( S = \{(x,y,z) \mid z = x^2 + y^2\} \). The differential form is \( \omega = ydx + xdy + (x^2 + y^2)dz \). Evaluate the integral and find the value of \( \int_{\gamma} \omega \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $P(x)$ be a polynomial of degree 3 with real coefficients and leading coefficient 1. Let the roots of $P(x)$ be $a, b, c$. Given that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a, b, c, d$ such that $\gcd(a, b, d) = 1$ 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{109}
In rectangle $ABCD$, where $AB = 6$ and $BC = 3$, a point $P$ is chosen on $AB$ such that $\angle APD = 2 \angle CPB$. Determine the length of $AP$. \( \text{Express your answer as a decimal.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.354}
A train of length 400 m moving at a speed of 72 kmph takes 69.99440044796417 seconds to cross another train of a certain length, moving at 36 kmph in the same direction. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{299.944}
已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是。原始答案为 \frac{a}{b} 格式，请给出 a + b 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{289}
For which natural numbers \( n \), does there exist a bijection \( f : \mathbb{N} \to \mathbb{Q}^n \) and a norm \( \|\cdot\| \) on \( \mathbb{R}^n \) such that \( \|f(k+1)-f(k)\|=1 \) for all \( k \in \mathbb{N} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 2}
A king has eight sons, and each night he sends three of them to guard the golden apples from the Firebird. No two sons agree to guard together more than once. What is the maximum number of nights this arrangement can continue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 13 weights arranged in a row on a table, ordered by mass (the lightest on the left, the heaviest on the right). It is known that the mass of each weight is an integer number of grams, the masses of any two adjacent weights differ by no more than 5 grams, and the total mass of the weights does not exceed 2019 grams. Find the maximum possible mass of the heaviest weight under these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{185}
In a drawer, there are 5 white socks, 3 brown socks, and 4 blue socks. How many ways can you choose a pair of socks such that the socks are of different colors and at least one of them is blue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
If five pairwise coprime distinct integers \( a_{1}, a_{2}, \cdots, a_{5} \) are randomly selected from \( 1, 2, \cdots, n \) and there is always at least one prime number among them, find the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Natalia is participating in a week-long cycling competition. On Monday, she rode 40 kilometers and achieved a 5% bonus for her speed. On Tuesday, she rode 50 kilometers and earned a 3% bonus. On Wednesday, she rode 50% fewer kilometers than she did on Tuesday and gained a 7% bonus for her speed. On Thursday, she rode as many kilometers as the sum of the distances covered on Monday and Wednesday, and earned a 4% bonus for her speed.

On Friday, she increased her ride distance by 20% compared to the distance she covered on Thursday, and earned a 6% speed bonus. On Saturday, she covered 75% of the distance she covered on Friday but faced a challenging elevation gain, resulting in a 2% bonus. Finally, on Sunday she managed to ride as many kilometers as the difference between the distances she covered on Saturday and Wednesday, and due to an easier elevation, earned a 10% bonus for her speed.

Considering the base distances ridden each day, the bonuses earned for speed, and additional weekend rides with fluctuating elevation gains, how many kilometers did Natalia effectively ride in total during the week-long competition, including those earned from her speed bonuses?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{367.05}
Let \(\omega\) be a complex number such that \(\omega^{2013} = 1\) and \(\omega^m \neq 1\) for \(m = 1, 2, \ldots, 2012\). Determine the number of ordered pairs of integers \((a, b)\) with \(1 \le a, b \le 2013\) such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is an algebraic integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2251205}
Evaluate \( \lim_{n\to \infty} \frac{a_n}{b_n} \) given that \( a_n \to \infty \), \( b_n \to \infty \), and \( a_n + b_n \) is bounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the limit: \[ \lim_{x \rightarrow 0} -2\int_{0}^{x^{2}} \cos t^{2} \, dt \] and express the result in terms of a Maclaurin series expansion up to the order of $x^2$. Ignore terms of order $o(x^4)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2x^2}
A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\frac{1}{4}$}
Find all polynomials  $P$  with real coefficients such that   $$ \frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x) $$   holds for all nonzero real numbers  $x,y,z$  satisfying  $2xyz=x+y+z$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P(x) \equiv c(x^2 + 3)}
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? Provide your answer as a simplified fraction in the form \frac{k}{m}, and find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
已知 $p$ 为质数, $\sqrt{p}$ 的小数部分为 $x, \frac{1}{x}$ 的小数部分为 $\frac{\sqrt{p}-31}{75}$. 求所有满足条件的质数 $p$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
Evaluate the limit: \[ \lim_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \] where \( \delta_{a} \) is the Dirac distribution at the point \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta''}
If the direction vector of line $l$ is $\overrightarrow{e}=(-1,\sqrt{3})$, calculate the inclination angle of line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
$\prod_{k=1}^{6}\left(x^{5}-5 x+k\right)=0$ 有 $\qquad$个实根.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Ted loves creating haikus, which are Japanese three-line poems with syllable patterns of 5, 7, and 5 respectively. However, his vocabulary is limited. Specifically, for integers $1 \le n \le 7$, he knows $2n$ words that each contain $n$ syllables. Words cannot cross between lines but may be repeated.

If Ted can create $N$ distinct haikus, find the remainder when $N$ is divided by $100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
定义函数 A(x)=\ds{m=1}{\infty}\ds{n=0}{\infty}\df{m!n!}{(m+n+1)!}x^m，计算 [100A(\frac{2}{3})].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109}
For a positive integer $n$, consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. 

For example, if $n = 117$, it has six divisors: $1, 3, 9, 13, 39, 117$. Two of these divisors, namely $3$ and $13$, have unit digits equal to $3$. Hence, for $n = 117$, $p = 33.33\ldots$.

Find the maximum possible value of $p$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
On a circle of length 2013, there are 2013 marked points dividing it into equal arcs. Each marked point has a chip. We call the distance between two points the length of the smaller arc between them. For what largest $n$ can the chips be rearranged so that there is again a chip at each marked point, and the distance between any two chips, initially no more than $n$ apart, increases?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}