inclusionAI/Ring-lite-rl-data huggingface dataset:
June is running for class president. She needs just over 50% of the vote to win. There are 200 students in the school. A certain percentage of students are boys and the rest are girls. If she receives 67.5% of the male vote and 25% of the female vote, what is the smallest percentage of boys in the school for her to win the election?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60\%}
Find the number of solutions in positive integers $(k ; a_{1}, a_{2}, \ldots, a_{k} ; b_{1}, b_{2}, \ldots, b_{k})$ to the equation $$a_{1}(b_{1})+a_{2}(b_{1}+b_{2})+\cdots+a_{k}(b_{1}+b_{2}+\cdots+b_{k})=7$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the area of a trapezoid with bases 11 and 4 and diagonals 9 and 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
A store uses systematic sampling to inspect 20 out of 300 new bags of milk powder. They number all the milk powder bags from 1 to 300 and divide them into 15 groups in numerical order (1-20, 21-40, ..., 281-300). If the number drawn from the first group is 6, then the number drawn from the third group is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
What values of $p \in \mathbb{R}$ ensure that $|x-y|^p$ defines a metric on the set $\mathbb{R}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1]}
The base of a regular quadrilateral pyramid is a square with a side length of 8. The height of the pyramid is 9. A plane is drawn through a side of the base, forming an angle with the base plane equal to \(\arctan \frac{3}{4}\). Find the area of the section of the pyramid created by this plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Miyuki texted a six-digit integer to Greer. Two of the digits of the six-digit integer were 3s. Unfortunately, the two 3s that Miyuki texted did not appear and Greer instead received the four-digit integer 2022. How many possible six-digit integers could Miyuki have texted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Consider a system that fails at each step with probability $p$. Calculate the expected time to failure when the system is conditioned on not failing on the first step.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \frac{1}{p}}
Arrange the numbers $2^{0.3}$, $\log_{0.3}{2}$, $\log_{0.3}{3}$ in increasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log_{0.3}{3} < \log_{0.3}{2} < 2^{0.3}}
Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both
$ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1)}{2}}
Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. For example, $\sqrt{20}$, $\sqrt{16}$, and $\sqrt{6}$ simplify to $2\sqrt{5}$, $4\sqrt{1}$, and $1\sqrt{6}$, respectively. Compute the sum of $a+b$ across all expressions that Rishabh writes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3480}
Let \( r_{1}, r_{2}, \cdots, r_{20} \) be the roots of the polynomial \( x^{20}-7x^{3}+1 \). If \(\frac{1}{r_{1}^{2}+1}+\frac{1}{r_{2}^{2}+1}+\cdots+\frac{1}{r_{20}^{2}+1} \) can be expressed in the form \( \frac{m}{n} \) (with \( m \) and \( n \) coprime), find the value of \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\triangle ABC$ out of paper and then folding $\triangle AFD$ up along $FD$, folding $\triangle BED$ up along $DE$, and folding $\triangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
In a football tournament, 15 teams participated, and each played exactly once with each other. A win was rewarded with 3 points, a draw with 1 point, and a loss with 0 points.

After the tournament concluded, it turned out that some 6 teams each scored at least \( N \) points. What is the largest integer value that \( N \) can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
7-6. In an ornithological park, there are several species of birds, a total of 2021 individuals. The birds sat in a row, and it turned out that between any two birds of the same species, there was an even number of birds. What is the smallest number of bird species that could have been there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
椭圆$:\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1\left( a>b>0 \right)$，${{F}_{1}},{{F2}}$是椭圆的左右焦点，$O$为坐标原点，点$P$为椭圆上一点，$\left| OP \right|=\frac{\sqrt{2}}{4}a$，且$\left| P{{F}_{1}} \right|,\left| {{F1}}{{F}_{2}} \right|,\left| P{{F2}} \right|$成等比数列，则椭圆的离心率为________ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{6}}{4}$}
Given a triangle \(ABC\). An incircle is inscribed in it, touching the sides \(AB\), \(AC\), and \(BC\) at the points \(C_1\), \(B_1\), \(A_1\) respectively. Find the radius of the excircle \(w\), which touches the side \(AB\) at the point \(D\), the extension of the side \(BC\) at the point \(E\), and the extension of the side \(AC\) at the point \(G\). It is known that \(CE = 6\), the radius of the incircle is 1, and \(CB_1 = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Call an $8$-digit number \textit{cute} if its digits are a permutation of $1,2,\ldots,8$. For example, $23615478$ is \textit{cute} but $31234587$ is not. Find the number of $8$-digit \textit{cute} numbers that are divisible by $11$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4608}
On a map, the distance between two mountains is 312 inches. The actual distance between the mountains is some km. Ram is camped at a location that on the map is 25 inches from the base of the mountain. He is 10.897435897435898 km from the base of the mountain. What is the actual distance between the two mountains in km?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle are $1$ and $4$, the side length of this equilateral triangle can be expressed as $\frac{a\sqrt{b}}{c}$, where $(a, c) = 1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Determine the value of $c$ for which the characteristic function of $\frac{S_{n}}{n^{c}}$ converges to a function that is continuous at zero, where $X_{1},...,X_{n}$ are iid with characteristic function $\phi(t)= 1-\sqrt{|t|(2-|t|)}$, $t\in[-1,1]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
At the duck park, there are 25 mallard ducks, 15 wood ducks, ten less than twice as many geese as mallard ducks, and 8 more than thrice the number of wood ducks swans. In the morning, a small flock of 4 mallard ducks, 8 wood ducks, and 7 geese arrive at the park and join the birds already there. Around noon, 9 swans, 5 geese, and 6 wood ducks leave the park. Later in the day, 8 more mallards, 10 wood ducks, and 4 extra swans arrive at the park. In the evening, a group of 15 geese, 11 swans, 5 mallard ducks, and 3 wood ducks return to the park. Overnight, twice the number of mallard ducks plus half the number of swans leave the park, while three-quarters the number of wood ducks and one-fifth the number of geese join other ponds in the area. At dawn, how many more geese than ducks (both mallard and wood ducks) remain at the park?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Let  $	\prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$ where  $a_{1}, a_{1}, . . . , a_{m}$  are nonzero and  $k_{1} < k_{2} <...< k_{m}$ . Find  $a_{1996}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665280}
Find a new $\delta$ for the function $f(x)=3x^{2}-2x+1$ at $x=2$ such that $|x-2|<\delta$ implies $|f(x)-f(2)|<\frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/26}
Richard has a four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars.  (One dollar is 100 cents.) What is the expected value of this final sum of money, in cents?

*Proposed by Lewis Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1025}
Given {(x1,x2,x3,x4) ⊆ {x | (x-3) * sin(πx) = 1, x > 0}}, find the minimum value of x1+x2+x3+x4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
On Christmas Eve, she received 77 gifts to put under her Christmas tree. This year, she also decided to buy 33 gifts for her family, friends, and some orphans. She had to ensure that 9 orphans received an equal number of gifts from the 33 bought. After distributing the gifts among the orphans, she placed the remaining gifts for her family and friends under the tree. She also wanted to make other kids happy, so she decided to send 40% of her original received gifts to the orphanage downtown. How many gifts were left under her Christmas tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length $\frac{1}{2}$ and edges parallel to those of the unit cube that contains all three points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
How to evaluate the following binomial sum that arises in the analysis of an algorithm:
$$\sum_{i = 1}^{n-k+1} i \binom{n-i}{k-1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\binom{n+1}{k+1}$$}
The score of a math test for a class of students, denoted as ξ (full marks: 100 points), follows a normal distribution: ξ ~ N(85, σ^2), and P(83 < ξ < 87) = 0.3, P(78 < ξ < 83) = 0.13. Calculate the probability P(ξ < 78). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.22}
The line $l_{1}$: $x+my+6=0$ is parallel to the line $l_{2}$: $(m-2)x+3y+2m=0$. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmost position (the units position), the result is $\frac{n}{3}$. Find the smallest possible value of the sum of the digits of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
For every $a \in \mathbb{N}$, denote by $M(a)$ the number of elements in the set
\[ \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \}. \]
Find $\max_{a \leq 1983} M(a).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Every asterisk in the equation \(2 * 0 * 1 * 5 * 2 * 0 * 1 * 5 * 2 * 0 * 1 * 5 = 0\) is to be replaced with either \(+\) or \(-\) so that the equation is correct. What is the smallest number of asterisks that can be replaced with \(+\) to make the equation true? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{406}
Determine the order of the function $\int_1^{k\epsilon} \frac{e^{-2x}+e^{-2k+2x}-2e^{-2k}}{x}\,dx$ with respect to $k$ as $k$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(1)}
The domain of the function $f(x)=\log_{\frac{1}{2}}(\log_4(\log_{\frac{1}{4}}(\log_{16}(\log_{\frac{1}{16}}x))))$ is an interval of length $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers.  What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
Count the number of polynomials of the form $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$ that satisfy the condition $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|=3$, where $n$ is a non-negative integer, $a_0$ is a positive integer, and the remaining $a_i$ are integers or zero. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1 \le x_2 \le \cdots \le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{601}
You make a cube that is made of 10 x 10 x 10 smaller cubes which sums up to a total of 1000 smaller cubes. Now, if someone takes a certain number of layers of cubes, 512 cubes will remain. How many layers of cubes were taken?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
$n$ teams are to have a double round-robin tournament (each team plays every other team twice, once at home and once away). Each team can play multiple away games in a week (from Sunday to Saturday), but if they have a home game in a week, they cannot have any away games that week. If all games can be completed within 4 weeks, find the maximum value of $n$. Note: A game held at team A's venue between teams A and B is called a home game for team A and an away game for team B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Kyuhyung has several square-shaped cards. The cards are laid out neatly to form a large square, and there are 3 cards left. It is said that 44 more cards are needed to increase to fill the outer perimeter with cards. How many cards does Kyuhyung have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
14. (9th grade) In a trapezoid, the smaller base is 1 dm, the angles adjacent to it are $135^{\circ}$, and the angle between the diagonals facing the base is $150^{\circ}$. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
A set \( \mathcal{T} \) of distinct positive integers has the following property: for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 1024 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Three regular polygons are inscribed in a circle, the number of sides of each subsequent one being twice that of the previous one. The areas of the first two are $S_{1}$ and $S_{2}$. Find the area of the third.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{2S_{2}^{3}}{S_{1}+S_{2}}}}
Let  $f(n)$  denote the largest odd factor of  $n$ , including possibly  $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1365}
Molly and her parents love to hang out at the beach on weekends. Molly spent the better part of the day at the beach and saw some people join them. At 5:00, 40 people left the beach. The total number of people at the beach if they all stayed until evening is 63. How many people joined them at the beach?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Let  $x_1 \dots, x_{42}$ , be real numbers such that  $5x_{i+1}-x_i-3x_ix_{i+1}=1$  for each  $1 \le i \le 42$ , with  $x_1=x_{43}$ . Find all the product of all possible values for  $x_1 + x_2 + \dots + x_{42}$ .

*Proposed by Michael Ma*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{588}
Four cars \( A, B, C, \) and \( D \) travel at constant speeds on the same road (not necessarily in the same direction). Car \( A \) passed \( B \) and \( C \) at 8am and 9am, respectively, and met \( D \) at 10am. Car \( D \) met \( B \) and \( C \) at 12pm and 2pm, respectively. Determine at what time \( B \) passed \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:40 \text{ a.m.}}
The vertices of a regular  $2012$ -gon are labeled  $A_1,A_2,\ldots, A_{2012}$  in some order. It is known that if  $k+\ell$  and  $m+n$  leave the same remainder when divided by  $2012$ , then the chords  $A_kA_{\ell}$  and  $A_mA_n$  have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled  $A_1$  and  $A_4$ . How is the tenth vertex labeled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A_{28}}
Consider a rectangular sheet of paper measuring 11 inches by 11 inches, a rectangular sheet of paper measuring 5.5 inches by 11 inches, an equilateral triangle with side length of 6 inches, and a circle with a diameter of 4 inches. What is the difference in combined areas (in square inches) between the two rectangular sheets and the triangle and the circle together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153.35}
Given the equation of a hyperbola, $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), the distance from one focus of the hyperbola to an asymptote is $\frac{\sqrt{5}}{3}c$ ($c$ is the semi-focal length). Determine the eccentricity of the hyperbola. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
There are 2012 dwarves, each of whom is either good or bad. Every day they attend a meeting in groups of 3 or 5. During each meeting, if the majority of the attendees are good, all attendees of that meeting become good; if the majority are bad, all attendees become bad. If after the third day's meetings all 2012 dwarves have become good, what is the minimum number of good dwarves before the first day's meetings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{435}
An angle's complement is $10^{\circ}$ larger than its supplement. If the complement of the angle is $\frac{1}{3}$ smaller than its supplement, then the measure of the angle is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi \sqrt{15}}{16}}
At a tanning salon, customers are charged $10 for their first visit in a calendar month and $8 for each subsequent visit. However, customers who visit more than twice in a month receive a 10% discount on each visit after the first two. In addition, the salon offers a $15 premium tanning service, which 20% of customers choose to upgrade to for their first visit. Last month, 150 customers visited the salon, of which 40 made a second visit, 15 made a third visit, and 5 made a fourth visit. All other customers made only one visit. If those visits were the only source of revenue for the salon, what was the revenue for the last calendar month at the salon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2114}
An electronic clock displays time from 00:00:00 to 23:59:59. How much time throughout the day does the clock show a number that reads the same forward and backward?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96 \text{ seconds}}
The cost price of a certain product is 1360 yuan. When it is sold at 80% of the marked price, the profit margin is 15%. What should be the marked price of the product in yuan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1955}
Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$.  Compute the largest possible value for the circumradius of $\mathcal T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4 + \sqrt{3}}}
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_{i}$ denotes the $i$ th digit, then $d_{i} \leq d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers 43210, 132, and 3, but not the numbers 1337 and 123. How many 5-digit numbers does Kelvin like?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14034}
Find the sum of the prime factors of $67208001$, given that $23$ is one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{781}
When Spencer jumps rope, he starts with a speed of 4 jumps per minute on the first day and doubles his speed daily. On each day, he jumps for 10 minutes in the first session and increases the duration of his second session by 5 minutes each day. Given that Spencer practices on days 1, 2, 4, 5, and 6 within a 7-day period, how many jumps will he perform in total during this time frame?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8600}
Find the common ratio of the infinite geometric series: $$\frac{7}{8} - \frac{5}{12} + \frac{35}{144} - \dots$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{10}{21}}