inclusionAI/Ring-lite-rl-data huggingface dataset:
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$40$}
设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。第三个函数的表达式为 $y=ax+b\varphi(cx)$，请给出a+b+c的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the angle $C$ of triangle $ABC$ if vertex $A$ is equidistant from the centers of the excircles that touch sides $AB$ and $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^
}
In a group of 100 cars, some cars do not have air conditioning. If at least 51 cars have racing stripes, the greatest number of cars that could have air conditioning but not racing stripes is 49. How many cars do not have air conditioning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
In circle \(\omega\), two perpendicular chords intersect at a point \( P \). The two chords have midpoints \( M_1 \) and \( M_2 \) respectively, such that \( P M_1 = 15 \) and \( P M_2 = 20 \). Line \( M_1 M_2 \) intersects \(\omega\) at points \( A \) and \( B \), with \( M_1 \) between \( A \) and \( M_2 \). Compute the largest possible value of \( B M_2 - A M_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $\triangle ABC$ have median $CM$ ($M \in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Compute the following limit: 

$$ \lim_{n \to \infty} \frac{\left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2}\right)^n }{ 2 \left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 + 1}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 (n^2 +1)}\right)^n } $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
8. An $A M C$ exam consists of 30 questions, with the scoring rules specifying 5 points for each correct answer, 0 points for each wrong answer, and 2 points for each unanswered question. 
The sum of the points a participant earns for each question is called the participant's total score. Assuming there are enough participants, the number of different possible total scores is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{145}
Two cellphones and three laptops are connected to load a certain video. The first cellphone takes nine minutes, the second cellphone takes six minutes, the first laptop takes fifteen seconds, the second laptop takes eleven seconds, and the third laptop takes eight seconds to load that same video. If all devices are connected so that they operate in concert at their respective rates, how many seconds would it take them to load the video, rounded to the nearest thousandth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.483}
Consider a regular  $2n + 1$ -gon  $P$  in the plane, where n is a positive integer. We say that a point  $S$  on one of the sides of  $P$  can be seen from a point  $E$  that is external to  $P$ , if the line segment  $SE$  contains no other points that lie on the sides of  $P$  except  $S$ . We want to color the sides of  $P$  in  $3$  colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to  $P$ , at most  $2$  different colors on  $P$  can be seen (ignore the vertices of  $P$ , we consider them colorless). Find the largest positive integer for which such a coloring is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Two players alternately choose the sign for each of the numbers 1, 2, ..., 20. Once a sign has been chosen for a number, it cannot be changed. The first player tries to minimize the final absolute value of the total, and the second player tries to maximize it. What is the outcome, assuming both players play perfectly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
If $\alpha=3$, then the terminal side of $\alpha$ lies in the $\boxed{\text{answer}}$ quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{II}}
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that  
$P(1) = P(3) = P(5) = P(7) = a$,  and  
$P(2) = P(4) = P(6) = P(8) = -a$.  
Find the smallest possible value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
Given two real numbers \( p > 1 \) and \( q > 1 \) such that \( \frac{1}{p} + \frac{1}{q} = 1 \) and \( pq = 9 \), what is \( q \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9 + 3\sqrt{5}}{2}}
Mary earns $46 cleaning a home. After cleaning a number of homes, she had $276. However, she had to pay a 10% tax on her earnings and has an additional expense of $15 for cleaning supplies and transportation. How many homes did she clean?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
There are 12 students participating in a performance event. The event consists of $n$ programs. Each program is performed by 6 students, and any two programs have at most 2 common performers. Find the maximum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A wheel of radius 8 rolls along the diameter of a semicircle of radius 25 until it bumps into this semicircle. Find the length of the portion of the diameter that cannot be touched by the wheel. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle compare to each other in length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5} : 2\sqrt{2} : 3}
In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3 \sin Y$. Calculate $XY$.

---
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{40\sqrt{2}}{3}}
How do we find the value of $a$ such that the function $f_a(x) = \sin x + \frac{a}{3\pi}x$ has exactly three zeros in the interval $[0, 3\pi]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A circle with a radius of 2 units rolls around the inside of a triangle with sides 9, 12, and 15 units. The circle is always tangent to at least one side of the triangle. Calculate the total distance traveled by the center of the circle when it returns to its starting position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
If the quadratic trinomial $x^{2}+kx+5$ with respect to $x$ can be factored using the cross multiplication method, then the integer $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 6}
Find the exact value of the integral: $$\int_{x}^{+\infty}\frac{e^{-t}}{t}dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E_1(x)}
In a \( 29 \times 29 \) table, the numbers \( 1, 2, 3, \ldots, 29 \) are each written 29 times. It is found that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A rectangular piece of paper has corners labeled $A, B, C$, and $D$, with $BC = 80$ and $CD = 120$. Let $M$ be the midpoint of side $AB$. The corner labeled $A$ is folded along line $MD$ and the corner labeled $B$ is folded along line $MC$ until the segments $AM$ and $MB$ coincide. Let $S$ denote the point in space where $A$ and $B$ meet. If $H$ is the foot of the perpendicular from $S$ to the original plane of the paper, find $HM$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Let \( ABC \) be a triangle with \( AB = 6 \), \( AC = 7 \), and \( BC = 8 \). Let \( I \) be the incenter of \( ABC \). Points \( Z \) and \( Y \) lie on the interior of segments \( AB \) and \( AC \), respectively, such that \( YZ \) is tangent to the incircle. Given a point \( P \) such that
\[
\angle ZPC = \angle YPB = 90^\circ,
\]
find the length of \( IP \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{30}}{2}}
Five teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every other team. However, due to financial difficulties, some games were canceled. 

By the end of the tournament, all teams had different scores, and no team had zero points. What is the minimum number of games that could have been played in this tournament, given that:
- A win awards 3 points,
- A draw awards 1 point,
- A defeat awards 0 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
We call a rectangle of the size  $1 \times 2$  a domino. Rectangle of the  $2 \times 3$  removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.

It requires to tile rectangle of size  $2008  \times 2010$  by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Liz is preparing for a huge family Thanksgiving dinner and she decides to roast 4 turkeys. The turkeys weigh 16 pounds, 18 pounds, 20 pounds, and 22 pounds respectively. She can only roast one turkey at a time due to the size of her oven. The first two turkeys need to be roasted at a higher temperature, requiring an additional 20 minutes of preheating time. The other two turkeys do not require additional preheating time.

She roasts each turkey for 15 minutes per pound. If dinner is served at 6:00 pm, what is the latest time she can start preheating her oven and begin roasting the turkeys in order to finish on time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:20}
A $12 \times 12$ grid is colored such that each cell is either black or white. Ensure that within any $3 \times 4$ or $4 \times 3$ subgrid, there is at least one black cell. Find the minimum number of black cells needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
To create a specific type of steel for a particular application, a certain amount of a chemical element must be added during the steelmaking process. It is known that the amount of this element to be added per ton of steel should be within the range of [1000, 2000] grams. Using the 0.618 method to determine the optimal amount to be added, let the amounts of the element added at the first, second, and third trial points be $x_1$, $x_2$, and $x_3$ respectively ($x_1 > x_2$). If the first trial point is better than the second, then find the amount $x_3$ to be added for the third trial point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1764}
Tom originally was only able to lift 100 kg farmer handles per hand. After some intensive training, he was able to increase this number by 150%. He then decided to specialize and was able to get an extra 25% improvement. In addition, he started using a new grip technique which further increased his lifting capacity by 10%. 

He was later introduced to deadlifting, and he found that he could lift 400 pounds with ease. He then learned a new deadlift technique called "sumo style" which improved his lifting ability by 20%. 

Given that 1 kg is approximately equal to 2.20462 pounds, calculate Tom's total lifting capacity (in kg) for the farmer handles and deadlift combined.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{905.22}
Find the smallest natural  $ k $  such that among any  $ k $  distinct and pairwise coprime naturals smaller than  $ 2018, $  a prime can be found.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The negation of the proposition "$\forall x \gt 0$, $e^{x} \gt x+1$" is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exists x > 0, e^{x} < x+1}
How many ways are there to color every integer either red or blue such that \( n \) and \( n+7 \) are the same color for all integers \( n \), and there does not exist an integer \( k \) such that \( k, k+1 \), and \( 2k \) are all the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle $ABC$ with angle $C$ equal to $30^{\circ}$, a median $AD$ is drawn. Angle $ADB$ equals $45^{\circ}$. Find angle $BAD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Given a quadrilateral \(PQRS\) made from two similar right-angled triangles \(PQR\) and \(PRS\), where \(PQ = 3\), \(QR = 4\), and \(\angle PRQ = \angle PSR\), calculate the perimeter of quadrilateral \(PQRS\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled \(1, 2, \ldots\). Each ball has a probability \(\frac{1}{2^n}\) of being put into box \(n\). The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{7}}
Let $\alpha$ and $\beta$ be such that $\pi<\alpha-\beta<3\pi$, and $cos\alpha+cos\beta=-27/65$ and $sin\alpha+sin\beta= -21/65$, find $cos(\alpha-\beta)/2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{\sqrt{130}}}
There are 8 football teams playing a round-robin tournament. The winning team gets 1 point, the losing team gets 0 points, and in the case of a draw, each team gets 0.5 points. After the tournament, the scores of the teams are ranked from highest to lowest, and it is found that no two teams have the same score. Additionally, the score of the second-place team is equal to the total score of the last four teams combined. Determine the score of the team that came in second place in this tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \(m\) be a positive integer, and let \(T\) denote the set of all subsets of \(\{1, 2, \ldots, m\}\). Call a subset \(S\) of \(T\) \(\delta\)-good if for all \(s_1, s_2 \in S\), \(s_1 \neq s_2\), \(|\Delta(s_1, s_2)| \geq \delta m\), where \(\Delta\) denotes symmetric difference (the symmetric difference of two sets is the set of elements that are in exactly one of the two sets). Find the largest possible integer \(s\) such that there exists an integer \(m\) and a \(\frac{1024}{2047}\)-good set of size \(s\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Given the function $f(x)=\frac{|x|-\sin{x}+1}{|x|+1}$ ($x\in\mathbb{R}$), its maximum value is $M$ and its minimum value is $m$. Find the value of $M+m$ $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a parabolic curve $y^{2}=2x$ with a point $P$ on it, points $B$ and $C$ are on the y-axis. A circle described by $(x-1)^{2}+y^{2}=1$ is inscribed within the triangle $PBC$. Find the minimum area of triangle $PBC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Three lines are drawn parallel to the sides of a triangle through a point inside it, dividing the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range within which the ratio of the area of each inner triangle to the area of the original triangle can lie.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/9}
某校三个年级举办乒乓球比赛，每个年级选派 4 名选手参加比赛. 组委会随机将这 12 名选手分成 6 组，每组 2 人，则在上述分组方式中每组的 2 人均来自不同年级的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{449}
Find the odd integer that contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Evaluate the sum:
\[ \sum_{x=0}^\infty x(x-1) {2+x \choose x}(0.008)(0.8)^x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
求最小的实数 C>1，满足如下条件：对任意整数 n\geq 2 和满足 \df{1}{a_1}+\df{1}{a_2}+\cdots+\df{1}{a_n}=1 的非整正实数 a_1, a_2, \cdots, a_n，都可以找到正整数 b_i，使得 (1) 对 1\leq i\leq n，b_i=[a_i] 或 [a_i]+1；(2) 1<\df{1}{b_1}+\df{1}{b_2}+\cdots+\df{1}{b_n}\leq C.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate $$\int_{0}^{\infty}\frac{x}{1+x^3}dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{3\sqrt3}\)}
Marie has 75 raspberry lollipops, 132 mint lollipops, 9 blueberry lollipops, and 315 coconut lollipops. She decides to distribute these lollipops equally among her 13 friends, distributing as many as possible. How many lollipops does Marie end up keeping for herself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The display has a row of 4 parallel holes, each of which can display 0 or 1. If each time it displays two of the holes, but two adjacent holes cannot be displayed at the same time, the total number of signals that the display can show is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally likely. If each train takes 45 minutes to clear the intersection, what is the probability of a collision today?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{48}}
Someone arranged a 10-volume collection of works in a random order. Let's call a "disorder" a pair of volumes where the volume with the higher number is to the left. For a given arrangement of volumes, the number \( S \) of all disorders is calculated. What values can \( S \) take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 1, 2, \ldots, 45}
4. Determine the minimum value of the quantity $|x+y|$ given that the numbers $x$ and $y$ satisfy the relation $5 \cos (x+4 y)-3 \cos (x-4 y)-4 \sin (x-4 y)=10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{8}-\frac{3}{8}\arccos\frac{3}{5}}
There are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we take some balls from a non-empty box and put them into an empty box. On a sheet of paper, we write down the product of the resulting number of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{65}}
Find the number of counterexamples to the statement: 
\[ \text{If } N \text{ is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then } N \text{ is prime.} \]
Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( f(x) = \frac{1}{x^3 + 3x^2 + 2x} \). Determine the smallest positive integer \( n \) such that 
\[ f(1) + f(2) + f(3) + \cdots + f(n) > \frac{503}{2014}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
When Qiuqiu serves beer, one bottle can fill a certain number of cups. Each cup always has half a cup of foam, and the volume of the beer foam expands to 3 times its original volume. If one bottle of beer can fill 4 cups when there is no foam, how many cups can Qiuqiu fill with one bottle of beer? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The number $734,851,474,594,578,436,096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3004}
Determine the dimension of the nullspace of the matrix $B_n$ over the field $\mathbb{F}_2$, where:
$$ B_n = \begin{bmatrix} J_n & J_n & \dots & J_n \\ J_n & J_n & \ddots & \vdots \\ \vdots & \ddots & \ddots & J_n \\ J_n & \dots & J_n & J_n \end{bmatrix} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - 1}
In trapezoid \(ABCD\), \(BC\) is parallel to \(AD\), \(AB = AD\), \(\angle ABC = \frac{2\pi}{3}\), and \(\angle BCD = \frac{\pi}{2}\). \(\triangle ABD\) is folded along \(BD\) such that point \(A\)'s projection onto plane \(BCD\) is point \(P\). Given that the cosine of the angle between \(AB\) and \(CD\) is \(\frac{\sqrt{3}}{6}\), find the cosine of the angle between \(BP\) and \(CD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
A point \(P\) is chosen at random in the interior of equilateral triangle \(ABC\). What is the probability that \(\triangle ABP\) has a greater area than each of \(\triangle ACP\) and \(\triangle BCP\)? Express your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime positive integers. Find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the maximal positive integer $p$ such that $5^7$ is the sum of $p$ consecutive positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{20}{33}$}