Context:
The Bell Zoo and the Carlton Zoo have various animals. The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. It also has three more elephants than the Carlton Zoo has lions and the same number of elephants as the Carlton Zoo has rhinoceroses. Additionally, the Carlton Zoo has two more elephants than rhinoceroses and twice as many monkeys as it has rhinoceroses, elephants, and lions combined. It also has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins, two more penguins than monkeys, and only half as many lions as penguins. The total number of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total number of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.
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Context:
Given a sample of $40$ data points, which is divided into six groups, with the frequencies of the first four groups being $10$, $5$, $7$, and $6$ respectively, and the frequency of the fifth group being $0.10$, what is the frequency of the sixth group?
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Context:
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin 3 \theta$ form an arithmetic progression, in some order.  Find $\cos \theta.$
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Context:
Given the function f(x) = sin(2x) + sqrt(3)cos(2x), find the value of varphi that makes the graph of y = f(x + varphi) symmetric about the line x = 0. Express your answer in radians.
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Context:
Mike had 20 video games but 11 of them weren't working. If he wanted to sell the working games for $8 each and there is a 12% sales tax applied to the selling price, how much money could he earn after considering the sales tax?
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Context:
If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any other student?

(The 7th American Junior High School Mathematics Examination, 1991)
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Context:
设x,y,z∈R* ，满足x+y+z=xyz，则函数

$f ( x , y , z ) = x ^ { 2 } ( y z - 1 ) + y ^ { 2 } ( z x - 1 ) + z ^ { 2 } ( x y - 1 )$

的最小值是
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sum of money is to be distributed among a, b, c, d. If c gets $500 more than d, and d's share is $1500, what is the proportion of their shares?
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Context:
设 $f(x)=x^{3}+3\left(x^{2}+x+\sin \pi x\right)$, 则 $f(1-\pi)+f(\pi-3)=$ $\qquad$.
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Context:
将两个相同的白球、三个相同的红球、四个相同的黑球全部放入三个不同的袋子中. 则没有空袋的放法数为 $\qquad$.
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Context:
若四棱雉 $P-A B C D$ 的棱 $A B, B C$ 的长均为 $\sqrt{2}$, 其他各条棱长均为 1, 则该四棱雉的体积为$\frac{\sqrt{m}}{n}$的形式，请给出$m + n$的值。
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Context:
The sum
\[
\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}
\]
can be expressed as a rational number \(N\). Find the last 3 digits of \(2021! \cdot N\).
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Context:
In $\Delta ABC$, $a+c=6$, and $(3-\cos A)\tan \frac{B}{2}=\sin A$, then the maximum area of $\Delta ABC$ is.
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Context:
2. The numbers from 1 to 8 are arranged in a circle. A number is called large if it is greater than its neighbors, and small if it is less than its neighbors. Each number in the arrangement is either large or small. What is the greatest possible sum of the small numbers?
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Context:
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x^4} + \frac{1}{y^4} + \frac{1}{z^4} = 1.\]Find the minimum value of
\[\frac{x^4 y^4 + x^4 z^4 + y^4 z^4}{x^3 y^2 z^3}.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( AB \) as the diameter of the smallest radius circle centered at \( C(0,1) \) that intersects the graph of \( y = \frac{1}{|x|-1} \), where \( O \) is the origin. Find the value of \( \overrightarrow{OA} \cdot \overrightarrow{OB} \).
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Context:
If the vector $\vec{a} + 3\vec{b}$ is perpendicular to the vector $7\vec{a} - 5\vec{b}$, and the vector $\vec{a} - 4\vec{b}$ is perpendicular to the vector $7\vec{a} - 2\vec{b}$, then the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\qquad$.
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Context:
In triangle ABC, the sides opposite angles A, B, and C are denoted as a, b, and c, respectively. Given that $10a\cos(B) = 3b\cos(A)$ and $\cos(A) = \frac{5\sqrt{26}}{26}$, find the value of angle C.
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Context:
Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription "MAME" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.
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Context:
To hit at least 90% of his targets in 60 rounds, John successfully hits 54 targets. After 40 rounds, he has 20 targets remaining. Calculate the maximum number of rounds he can afford to miss.
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Context:
Let's consider all possible natural seven-digit numbers, in the decimal notation of which the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995^{th}$?
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Context:
The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$
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Context:
Given a triangular pyramid with areas of faces S_1, S_2, S_3, S_4 and distances from a point within the pyramid to the faces H_1, H_2, H_3, H_4, and a constant k such that S_1/1 = S_2/2 = S_3/3 = S_4/4 = k, calculate the value of 1H_1 + 2H_2 + 3H_3 + 4H_4 in terms of the volume V of the pyramid and k. Express your answer as a fraction involving V and k.
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Context:
Given the circle \( O: x^{2}+y^{2}=4 \) and the curve \( C: y=3|x-t| \), and points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^*) \) on the curve \( C \), such that the ratio of the distance from any point on the circle \( O \) to point \( A \) and to point \( B \) is a constant \( k (k>1) \), find the value of \( t \).
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Context:
There are six thin sticks with lengths of \(3, 2 \sqrt{2}, 2, 2, 2, 2\). Using them to form a triangular prism, find the cosine of the angle formed by the lines containing the two longer edges.
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Context:
Given the point \( P \) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)\), and the equation of the line \( l \) is \(x=-\frac{a^{2}}{c}\), and the coordinate of the point \( F \) is \((-c, 0)\). Draw \( PQ \perp l \) at point \( Q \). If the points \( P \), \( Q \), and \( F \) form an isosceles triangle, what is the eccentricity of the ellipse?
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Context:
Given two concentric circles with radii 1 and 3 and common center $O$, a third circle touches both of them. Find the angle between the tangents to the third circle drawn from point $O$.
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Context:
Evaluate the limit $\lim \limits_{n\to \infty} \sum \limits_ {k=0}^n \left(1-\frac{k^2}{2n}\right)^n$.
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Context:
Calculate the real solutions $x\in\mathbb{R}$ to $$ \tag1\sqrt{4^x-6^x+9^x}+\sqrt{9^x-3^x+1}+\sqrt{4^x-2^x+1} = 2^x+3^x+1 $$
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Context:
Given the set M = {1,2,3,4,5,6,7} and the proposition p: ∀n∈M, n > 1, what is the negation of p? Express your answer as a logical proposition.
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Context:
Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y):1\leq x^2+y^2\leq 2 or 3\leq x^2+y^2\leq 4\}$. Then: Find the number of relatively compact connected components of $X\setminus K$ in $X$.
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Context:
Given \(\left[\prod_{k=1}^{n} \frac{(2k+1)^{4}+(2k+1)^{2}+1}{(2k)^{4}+(2k)^{2}+1}\right] = 44\), find \(n = \qquad\), where \([x]\) denotes the greatest integer less than or equal to the real number \(x\).
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Context:
If $\log_{3}2^4\cdot\log_{4}3^5\cdot\log_{5}4^6\cdot....\log_{64} {65}^{66}=y!$, what is the value of $y$?
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Context:
已知直三棱柱 ABC-A_1B_1C_1 的底面为直角三角形，\angle ACB=90^\circ，AC=6，BC=CC_1=\sqrt{2}. P 是 BC_1 上一动点，则 CP+PA_1 的最小值是__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
在平面直角坐标系 $x O y$ 中，圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 则直线 $l$ 的方程为 ax + by + c = 0。请给出 a + |b| + c 的值。
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Context:
The areas of projections of a triangle onto the coordinate planes $Oxy$ and $Oyz$ are equal to $\sqrt{6}$ and $\sqrt{7}$, respectively, while the area of the projection onto the plane $O yz$ is an integer. Find the area of the triangle if it is also known to be an integer.
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Context:
Miki has 15 oranges and 15 pears. She extracts 12 ounces of orange juice from 3 oranges and 10 ounces of pear juice from 4 pears. Miki then makes a juice blend using 5 pears and 4 oranges. Calculate the percentage of the blend that is pear juice.
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Context:
Evan writes a random positive integer on a board: the integer  $k$  has probability  $2^{-k}$  of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote  $5,8,2,3,6,10,2$  in that order then his two sequences would be  $5,8,2,3,6,10$  and  $2,3,5,6,8,10.$ Find the probability that for all  $k \in \{ 1,4,34 \},$  that  $k$  was written, and  $k$  appears in the same position in both sequences.
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Context:
Given points A, B, C, and P all on the surface of a sphere O with PC as its diameter, AB is perpendicular to BC, AB=2, BC=4. If the volume of sphere O is $8\sqrt{6}\pi$, find the tangent value of the angle formed by the skew line PB and AC.
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Context:
Given the distances 1, 2, 4, 8, 16, 32 between points on a plane, what is the smallest number of points required to accommodate all these distances? Express your answer as a whole number.
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Context:
Let $P(3, y)$ be a point on the terminal side of angle $\alpha$, if $\cos\alpha= \frac {3}{5}$, then $y= \_\_\_\_\_\_$.
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Context:
Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]

If for any positive integer \(n\), it holds that
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the minimum value of the real number \(\lambda\).
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Context:
A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate \(A\) or \(B\) with equal probability. If \(X\) Dalmathians voted for the winning candidate, the expected value of \(X^{2}\) can be expressed as \(\frac{a}{b}\) for positive integers \(a, b\) with \(\operatorname{gcd}(a, b)=1\). Find the unique positive integer \(k \leq 103\) such that \(103 \mid a-bk\).
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Context:
If the power function $f(x)=(m^{2}-9m+19)x^{m-4}$ is monotonically increasing on $(0,+\infty)$, then the real number $m=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the function
$$
f(x) = |a \sin x + b \cos x - 1| + |b \sin x - a \cos x| \quad (a, b \in \mathbf{R})
$$
attains a maximum value of 11, then $a^{2} + b^{2} = \, \, \, $ .
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Context:
Alex is a risk management specialist who helps poker players optimize their winnings and minimize their losses. One of his clients, Jamie, played three rounds of poker. In the first round, Jamie won $120. In the second round, Jamie lost $45. In the third round, Jamie won double the amount he lost in the second round. Alex advised Jamie to save 25% of his total winnings after these three rounds. How much money should Jamie save based on Alex's advice?
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Context:
一个小镇上共有 2023 位居民，他们成立了一些互不相同的俱乐部 A_1, A_2, \cdots, A_n，每个俱乐部至少一人，每个人可以在任意个不同的俱乐部内. 已知对于每个俱乐部 A_i，存在一个不在其中的居民 b_i，考虑所有 b_i 未加入的俱乐部，它们要么为 A_i，要么和 A_i 没有公共成员. 求该小镇上俱乐部个数的最大值.（程国根）
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Context:
The sum of the number of boys and girls in a school is 1150. If the number of boys is a certain number, then the number of girls becomes that number as a percentage of the total number of students. What is the number of boys in the school?
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Context:
[ Angles between angle bisectors $\quad]$

In triangle $ABC$, the angle bisectors of the angles at vertices $A$ and $C$ intersect at point $D$. Find the radius of the circumcircle of triangle $ABC$, if the radius of the circumcircle of triangle $ADC$ with center at point $O$ is $R=6$, and $\angle ACO=30^{\circ}$.
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Context:
Let \( ABC \) be an isosceles triangle at \( A \) with \( \angle CAB = 20^\circ \). Let \( D \) be a point on the segment \( [AC] \) such that \( AD = BC \). Calculate the angle \( \angle BDC \).
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Context:
Find the area of the cross-section of a rectangular parallelepiped $ABCDEFG$ by a plane passing through the diagonal $AC_1$, parallel to the base diagonal $BD$, inclined to the base plane at an angle of $30^{\circ}$, and forming an angle of $45^{\circ}$ with the diagonal $A_1C$ if the diagonal of the parallelepiped is $d$.
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Context:
A year ago, the total cost of buying a lawnmower was 2/5 times less than the cost it goes for now. If the cost was $1800 a year ago, calculate how much it would cost Mr. Lucian to buy 4 such lawnmowers.
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Context:
Find the sum of all real numbers $x$ for which $$\lfloor\lfloor\cdots\lfloor\lfloor\lfloor x\rfloor+x\rfloor+x\rfloor \cdots\rfloor+x\rfloor=2017 \text { and }\{\{\cdots\{\{\{x\}+x\}+x\} \cdots\}+x\}=\frac{1}{2017}$$ where there are $2017 x$ 's in both equations. ( $\lfloor x\rfloor$ is the integer part of $x$, and $\{x\}$ is the fractional part of $x$.) Express your sum as a mixed number.
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Context:
A person plans to process 408 parts in 15 days. In the first three days, he processes 24 parts each day. Calculate the minimum number of parts he needs to process each day for the remaining days to complete the task ahead of schedule. Express your answer as an integer.
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Context:
Find all prime numbers \( p \) such that there exist integers \( m \) and \( n \) satisfying \( p = m^2 + n^2 \) and \( p \mid (m^3 + n^3 - 4) \).
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Context:
The function $y = f(x)$ is a monotonically decreasing function defined on the interval $[-2, 2]$, and $f(a + 1) < f(2a)$. The range of the real number $a$ is __________.
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Context:
There are 2023 cups numbered from 1 through 2023. Red, green, and blue balls are placed in the cups according to the following rules.
- If cups $m$ and $n$ both contain a red ball, then $m-n$ is a multiple of 2 .
- If cups $m$ and $n$ both contain a green ball, then $m-n$ is a multiple of 3 .
- If cups $m$ and $n$ both contain a blue ball, then $m-n$ is a multiple of 5 .

What is the smallest possible number of empty cups?
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Context:
A train 165 meters long running at the rate of 54 kmph takes 58.9952803775698 seconds to cross a bridge of certain length. How long is the bridge?
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Context:
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$.
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set
$\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing
order are coded as $VYZ, VYX, VVW$, respectively. What is the base-$10$ expression for the integer coded as $XYZ$?
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Context:
Given the one-variable quadratic equation in $x$: $x^{2}+\left(2k+3\right)x+k^{2}=0$ has two distinct real roots $x_{1}$ and $x_{2}$. If $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}=-1$, then the value of $k$ is ____.
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Context:
A red long string is folded in half, and then folded in half again, ..., after being folded $m$ times, the bundle obtained is cut in the middle to get some red short strings; a white long string is folded in half, and then folded in half again, ..., after being folded $n$ times, the bundle obtained is cut in the middle to get some white short strings. If the ratio of the number of red short strings to the number of white short strings is an integer greater than 1, please give the smallest possible value of this integer.
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Context:
Joe B. is frustrated with chess. He breaks the board, leaving a \(4 \times 4\) board, and throws 3 black knights and 3 white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.)
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Context:
Given $f(x) = (e^x - 1)^2 + (e^{-x} - 1)^2$, find the minimum value of $f(x)$.
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Context:
Of the 6,000 employees of company X, 2/10 are clerical, 3/10 are technical, and 5/10 are managerial. If the clerical staff were to be reduced by 1/4, the technical staff by 1/5, and the managerial staff by 1/10, while 50 clerical employees were promoted to managerial positions and 90 technical employees shifted to clerical roles, what percent of the total number of the remaining employees would then be clerical?
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Context:
A thin rectangular slab of potato was cut into two pieces for an osmosis lab. One piece is 50 mm greater than the other. If the original uncut slab is 600 mm, what is the length of the other piece of the potato after it is cut?
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Context:
设 a=\underbrace{66\cdots6}_{10\text{个 }6},b=\underbrace{33\cdots3}_{6\text{个 }3}，则 a,b 的最大公约数为__________.
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Context:
An Olympic track and field athlete is using a new fitness app to optimize their training sessions. The app tracks their running speed, heart rate, and recovery time. During a training session, the athlete runs 400 meters in 50 seconds, then takes a 2-minute rest before running another 400 meters in 45 seconds. The app calculates the athlete's average speed for the entire session, including the rest time.

Calculate the athlete's average speed in meters per second for the session, considering the total time spent running and resting.
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Context:
A group of monkeys is divided into three subgroups, with each subgroup having an equal number of monkeys. They go to the orchard to pick peaches. After finishing the picking, they combine all the peaches and distribute them equally among all the monkeys. If each monkey receives 5 peaches, there are 27 peaches left over; if each monkey receives 7 peaches, then at least one monkey receives fewer than 7 peaches. What is the total number of peaches this group of monkeys has picked?
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Context:
Given a graph $G$ with vertices representing integers $1,2,\dots,99$, two vertices are connected if their greatest common divisor is at least $3$. Determine the chromatic number of $G$.
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Context:
$A$ pair of six-sided dice are labeled so that one die has only even numbers (two each of $2, 4$, and $6$), and the other die has only odd numbers (two of each $1, 3$, and $5$). The pair of dice is rolled. Find the probability that the sum of the numbers on the tops of the two dice is $7$. Express your answer in the form \(\frac{k}{m}\) and provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integer solutions does the equation

$$
\frac{1}{2022}=\frac{1}{x}+\frac{1}{y}
$$

have?
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Context:
Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y)$, such that $f(0,k) = f(k,0) = 2^k$ and
\[ f(a,b) + f(a+1,b+1) = f(a+1,b) + f(a,b+1) \]
for all nonnegative integers $a, b$. Determine the number of positive integers $n \leq 2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b) = n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the first eight prime numbers that have a units digit of 3.
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Context:
A worker makes an error of some percentage in excess while calculating the edge of a square. As a result, the worker makes a 4.04% error in calculating the area. What is the percentage error in calculating the edge?
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Context:
Given that the ellipse $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1$ and the hyperbola $\frac{x^{2}}{p} - \frac{y^{2}}{q} = 1$ $(m, n, p, q\in\mathbb{R}^{+})$ share the same foci $F_1$ and $F_2$, and $P$ is an intersection point of the ellipse and the hyperbola, then the product $|PF_1| \cdot |PF_2| = \_\_\_\_\_\_\_\_\_.$
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Context:
In an acute triangle ABC, let D be a point on [AC] and E be a point on [AB] such that ∠ADB = ∠AEC = 90°. If the perimeter of triangle AED is 9, the circumradius of triangle AED is 9/5, and the perimeter of triangle ABC is 15, calculate the length of BC. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the fractional equation $\frac{m}{x-2}+\frac{2x}{x-2}=1$ with a positive root, find the value of $m$. Express your answer as a single numerical value.
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Context:
In a tournament with 100 wrestlers, all of different strength, the stronger wrestler always wins against the weaker one. The wrestlers are paired up and have matches. Then, they are paired up differently and have matches again. The winners of both matches receive prizes. What is the minimum possible number of prize winners?
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Context:
How many functions $f: \mathbb{Z} \rightarrow \mathbb{R}$ satisfy the following three properties?

(a) $f(1)=1$;

(b) For all $m, n \in \mathbb{Z}, f(m)^{2}-f(n)^{2}=f(m+n) f(m-n)$;

(c) For all $n \in \mathbb{Z}, f(n)=f(n+2013)$.
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Context:
In the plane of square ABCD, how many points P are there such that △PAB, △PBC, △PCD, and △PDA are all isosceles triangles?
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Context:
Let $a, b, c, d$ be four real numbers such that $a + b + c + d = 20$ and $ab + bc + cd + da = 16$. Find the maximum possible value of $abc + bcd + cda + dab$.
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Context:
The base of a triangle is $a$, and the angles adjacent to it measure $45^{\circ}$ and $15^{\circ}$. A circle with a radius equal to the altitude dropped to this base is drawn from the vertex opposite the base. Find the area of the part of the corresponding circle that is contained within the triangle.
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Context:
Eight teams compete in a tournament. Each pair of teams plays exactly one game against each other. There are no ties. If the two possible outcomes of each game are equally likely, what is the probability that every team loses at least one game and wins at least one game? Express your answer as a simplified fraction.
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Context:
Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\cos B$ in terms of $a$ and $r$.
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Context:
Given an arithmetic-geometric sequence $\{a\_n\}$, the sum of its first $n$ terms is $10$, and the sum of its first $2n$ terms is $30$. Find the sum of its first $3n$ terms.
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Context:
Let \( n \) be the product of 3659893456789325678 and 342973489379256. Determine the number of digits of \( n \).
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Context:
Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is the length of $|BC|$?
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Context:
A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calculate the sum of all possible numbers of members in this club.
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Context:
A train 2500 m long can cross an electric pole in 100 sec. What is the speed of the train?
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Context:
Given the line $y=a (0 < a < 1)$ and the function $f(x)=\sin \omega x$ intersect at 12 points on the right side of the $y$-axis. These points are denoted as $(x\_1)$, $(x\_2)$, $(x\_3)$, ..., $(x\_{12})$ in order. It is known that $x\_1= \dfrac {\pi}{4}$, $x\_2= \dfrac {3\pi}{4}$, and $x\_3= \dfrac {9\pi}{4}$. Calculate the sum $x\_1+x\_2+x\_3+...+x\_{12}$.
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Context:
In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$.
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Context:
When a circular sector is rotated about one of its boundary radii, it forms a solid whose spherical surface area is equal to the conical surface area. Find the sine of the central angle of the circular sector.
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Context:
Let
\[
S = \left\{ \{a_n\} \mid a_1 = 1, a_{n+1} = 2a_n \text{ or } 4a_n + 1, \; n \in \mathbf{Z}_{+} \right\}.
\]
Let \( T \) be the set of all elements formed by the sequences in \( S \). Consider an arbitrary positive integer \( t \) in the interval \([1, 2021]\). What is the probability that \( t \in T \)?
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Context:
Given \( x \in[0, \pi] \), compare the sizes of \( \cos (\sin x) \) and \( \sin (\cos x) \).
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Context:
To walk 4 km, ride a bicycle for 6 km, and drive a car for 40 km, Uncle Vanya takes 2 hours and 12 minutes. If he needs to walk 5 km, ride a bicycle for 8 km, and drive a car for 30 km, he needs 2 hours and 24 minutes. How much time will Uncle Vanya need to walk 8 km, ride a bicycle for 10 km, and drive a car for 160 km?
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\boxed{5 \text{ hours } 48 \text{ minutes}}


Context:
Find the smallest natural number $m$ for which there exists a natural number $n$ such that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are identical, with the condition that $a<b$.
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Context:
Misha and Sasha left city $A$ for city $B$ on bicycles at noon. Simultaneously, Vanya left city $B$ for city $A$. All three travel at constant but different speeds. At 1 PM, Sasha was exactly halfway between Misha and Vanya, and at 1:30 PM, Vanya was halfway between Misha and Sasha. When will Misha be exactly halfway between Sasha and Vanya?
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Context:
A store sells three types of products, A, B, and C, with prices of 2 yuan, 3 yuan, and 5 yuan per item, respectively. A person bought some quantity of these three products, paying a total of 20 yuan. They discovered that they had bought too many of one type of product and returned two items of that product. However, the cashier only had 10 yuan notes and no change, so the person adjusted the quantities of the other two products (while ensuring every product was bought at least once) to keep the total cost the same. What is the quantity of product B that the person finally bought?
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Context:
Jenny places a total of 20 red Easter eggs in several green baskets and a total of 30 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket?
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Context:
Let $f(n) = \sum_{gcd(k,n)=1,1\le k\le n}k^3$ . If the prime factorization of $f(2020)$ can be written as $p^{e_1}_1 p^{e_2}_2 ... p^{e_k}_k$, find $\sum^k_{i=1} p_ie_i$.
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Context:
Let $L, E, T, M, \text{ and } O$ be digits that satisfy the equation:
\[
LEET + LMT = TOOL
\]
Given that $O = 0$, digits may be repeated, and $L \neq 0$, determine the value of the 4-digit integer $ELMO$.
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Context:
In the expression 8k8 + k88 - 16t6, k and t represent non-zero digits within the integers. If k and t represent non-zero digits within the integers above, what is the value of t?
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Context:
Given the sequence $\{a_n\}$ where $a_1 = \frac{1}{2}$ and $a_{n+1} = \frac{1+a_n}{1-a_n}$ for $n \in N^*$, find the smallest value of $n$ such that $a_1+a_2+a_3+…+a_n \geqslant 72$.
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Context:
Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally?
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Context:
A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$.
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Context:
Irene just bought a new lamp for her bedside table. The old lamp was 1 foot (12 inches) tall, and the new lamp's height is given as 55.56666666666667 centimeters. Irene's friend from Europe wants to know the difference in height between the lamps in meters. How much taller is the new lamp than the old lamp when comparing their heights in meters?
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Context:
Let f(x) be a function with the domain of R. Proposition p: "∀x > 0, f(x) > 0". What is the negation of proposition p? Express your answer as a logical statement with correct quantifiers and conditions.
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Context:
A container is shaped like a square-based pyramid where the base has side length  $23$  centimeters and the height is  $120$  centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon  $5$  centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.
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Context:
Given natural numbers \( M \) and \( N \), both greater than ten, consisting of the same number of digits, and such that \( M = 3N \). To obtain the number \( M \), one of the digits in \( N \) is increased by 2, and each of the remaining digits is increased by an odd digit. What digit could the number \( N \) end with? Find all possible answers.

(N. Agahanov)
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Context:
Let $P_{1}, P_{2}, \ldots, P_{8}$ be 8 distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these 8 points, such that: (1) each $P_{i}$ is the endpoint of at most one segment and (2) two no segments intersect. (The configuration with no edges drawn is allowed.)
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Context:
A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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Context:
An inspector rejects 0.02% of the meters as defective. How many meters will he examine to reject a certain number of meters, if he examined 10,000 meters to reject 2?
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Context:
Given proposition p: ∃x ≥ 0, 2^x = 3, what is its negation? Express your answer as a logical proposition.
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Context:
Given non-zero real numbers \( x \) and \( y \) that satisfy
\[
(5x + y)^{2019} + x^{2019} + 30x + 5y = 0,
\]
find the value of \(\frac{x}{y}\).
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Context:
A person buys an article at a certain price. He sells the article at a price that gives him a profit of 35%, which is Rs. 675. What was the original price of the article?
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Context:
A solid rectangular block is created using $N$ congruent 1-cm cubes adhered face-to-face. When observing the block to maximize visibility of its surfaces, exactly $252$ of the 1-cm cubes remain hidden from view. Determine the smallest possible value of $N.$
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Context:
Felicity, Adhira, and Benjamin took separate road trips in their respective vehicles. Felicity's car runs on gasoline, Adhira's on diesel, and Benjamin's on a mix of ethanol and biodiesel. The total fuel consumption for their trips are related as follows: Felicity used 2 times the amount of gasoline as Adhira used diesel, while Benjamin used a mix of 35% ethanol and 65% biodiesel, wherein the combined volume was 5 gallons less than Adhira's diesel consumption.

Their vehicles have different fuel consumption rates: Felicity's car uses 35 miles per gallon (mpg) for gasoline, Adhira's uses 25 mpg for diesel, and Benjamin's uses 30 mpg for ethanol and 20 mpg for biodiesel.

Each of them drove different distances, with Adhira covering 150 miles more than Felicity, who in turn covered 50 miles more than Benjamin. Together, they covered a total distance of 1750 miles. Calculate the number of gallons of gasoline Felicity used on her trip.
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Context:
In the Cartesian coordinate system $x O y$, consider the set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly chosen from $K$, what is the probability that there exists a pair of points among them with a distance of $\sqrt{5}$?
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Context:
Given that the geometric sequence $\{a\_n\}$ has a common ratio of $q$, and $S\_n$, $S\_{n-1}$, $S\_{n+1}$ form an arithmetic sequence, find the value of $\frac{a\_5 + a\_7}{a\_3 + a\_5}$ .
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Context:
Without stoppages, a train travels a certain distance with an average speed of some km/h, and with stoppages, it covers the same distance with an average speed of 125 km/h. The train stops for 30 minutes per hour. What is the average speed of the train without stoppages?
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Context:
Compute $(2 \cos 30^\circ + 2i \sin 30^\circ)^{10}.$ Enter your answer in rectangular form.
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Context:
Three fair six-sided dice, each numbered 1 through 6, are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?
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Context:
一个"位置"是指直角坐标平面上的一个点 (x,y)，其中 x,y 都是不超过 20 的正整数.最初时，所有 400 个位置都是空的. 甲乙两人轮流摆放石子，由甲先进行. 每次轮到甲时，他在一个空的位置上摆上一个新的红色石子，要求任意两个红色石子所在位置之间的距离都不等于 \sqrt{5}. 每次轮到乙时，他在任意一个空的位置上摆上一个新的蓝色石子，蓝色石子所在位置与其它石子所在位置之间的距离可以是任意值. 如此这般进行下去直至某人无法再摆放石子.求最大的整数 k，使得无论乙如何摆放蓝色石子，甲总能保证至少摆放 k 个红色石子.
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Context:
The height \( PO \) of the regular quadrilateral pyramid \( PABC D \) is 4, and the side of the base \( ABCD \) is 6. Points \( M \) and \( K \) are the midpoints of segments \( BC \) and \( CD \). Find the radius of the sphere inscribed in the pyramid \( PMKC \).
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Context:
Evaluate the following limit: $$\lim \limits_{n \to \infty}\sum_{k=1}^{n}\frac{k}{n^2}\sin{\frac{k}{n^2}}\cos{\frac{k}{n}}.$$
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Context:
1. It is known that there is a prime number between $10^{17}$ and $10^{17}+10$. Try to find this prime number.
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Context:
On the lateral side \( CD \) of trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is drawn to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
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Context:
The radius of the base of a cone is $R$. Two mutually perpendicular generators divide the lateral surface area of the cone into parts with the ratio $1: 2$. Find the volume of the cone.
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Context:
24. Brother Bear and Little Bear went on an adventure to find treasure and obtained a treasure chest. This treasure chest requires a ten-digit password consisting only of the digits 0 and 1 to open. Brother Bear first entered “1000000000”, and the treasure chest indicated that 1 digit was incorrect; Little Bear then entered “0110000000”, and the treasure chest indicated that 2 digits were incorrect. If the treasure chest no longer gives hints for subsequent password attempts, how many more times do the clever Brother Bear and Little Bear need to try to definitely open the treasure chest?
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Context:
Find the limit of the sequence $\{a_n\}$ defined by $$a_1 = 1, a_{n+1} = \frac{3n-1}{3n} a_n + \frac{1}{n^2}, \quad \forall n\ge 1.$$
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Context:
2. [7] Suppose that $a, b, c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of
$$
20\left(a^{2}+b^{2}+c^{2}+d^{2}\right)-\sum_{\text {sym }} a^{3} b,
$$
where the sum is over all 12 symmetric terms.
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Context:
4. In how many ways can two knights, two bishops, two rooks, a queen, and a king be arranged on the first row of a chessboard so that the following conditions are met:
1) The bishops stand on squares of the same color;
2) The queen and the king stand on adjacent squares. (20 points).
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Context:
Given points O(0, 0), A(2, 0), B(1, $-2\sqrt{3}$), and P, a moving point on the curve $y = \sqrt{1 - \frac{x^2}{4}}$, determine the range of values for $\overrightarrow{OP} \cdot \overrightarrow{BA}$.
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Context:
Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\ldots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?
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Context:
For how many three-element sets of distinct positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?
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Context:
If a rectangle is formed by a wire with a length of $32cm$, then the maximum value of the area of the rectangle is ______ $cm^{2}$.
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Context:
Given vectors $\overset{⇀}{m}=(\cos x,-\sin x)$ and $\overset{⇀}{n}=(\cos x,\sin x-2 \sqrt{3}\cos x)$, where $x\in\mathbb{R}$, let $f(x)= \overset{⇀}{m}\cdot \overset{⇀}{n} = \frac{24}{13}$, and $x\in\left[ \frac{\pi}{4}, \frac{\pi}{2}\right]$, then the value of $\sin 2x$ is __________.
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Context:
Find the largest 12-digit number \( N \) that satisfies the following two conditions:

- The decimal representation of \( N \) contains six digits "4" and six digits "7";
- In the decimal representation of \( N \), no four consecutive digits form the number "7444".
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Context:
已知$f ( x ) = a \sin x + b \sqrt [ 3 ] { x } + c \ln ( x + \sqrt { x ^ { 2 } + 1 } ) + 1 0 0 3 ( a 、 b 、$c为实数），且$f ( \lg \log _ { 3 } 1 0 ) = 1$,则$f ( \lg 1 g 3 ) = \underline { }$ .
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Context:
某国有 36 座城市和 5 家航空公司，任意两座城市之间恰由一家航空公司执行双向飞行的航班. 如果一家航空公司在城市 A, B 与 B, C 之间均执行航班，则称三元组 (A,B,C) 是"恰当连通的". 求最大的正整数 k，使得至少有 k 个三元组是恰当连通的.
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Context:
In convex quadrilateral  $ABCD$ ,  $\angle BAD = \angle BCD = 90^o$ , and  $BC = CD$ . Let  $E$  be the intersection of diagonals  $\overline{AC}$  and  $\overline{BD}$ . Given that  $\angle AED = 123^o$ , find the degree measure of  $\angle ABD$ .
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Context:
Let \(ABCDEF\) be a regular hexagon and let point \(O\) be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn't exist any equilateral triangle with vertices of all the same color?
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Context:
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the sum of the first 12 terms being 60, find the minimum value of $\left|a_{1}\right| + \left|a_{2}\right| + \cdots + \left|a_{12}\right|$.
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Context:
Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.
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Context:
Given that Jo climbs a flight of 8 stairs, and Jo can take the stairs 1, 2, 3, or 4 at a time, or a combination of these steps that does not exceed 4 with a friend, determine the number of ways Jo and the friend can climb the stairs together, including instances where either Jo or the friend climbs alone or together.
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Context:
How many fractions less than 1 are there, where the numerator and the denominator are single-digit positive integers?
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Context:
Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, \( \frac{4}{3} \) seconds, \( \frac{5}{3} \) seconds, and 2 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 1 minute? (Include first and last.)
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Context:
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. Find the area of the region bounded by these $6$ reflected arcs. The original answer is in the form of k\sqrt{m} - n\pi, please provide the value of k + m + n.
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Context:
方程 \df{1}{x^2}+\df{1}{y}+\df{1}{z}=\df{4}{3} 有__________组整数解.
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Context:
Find all positive integers \( k \) such that for any positive numbers \( a, b, c \) satisfying \( k(ab + bc + ca) > 5(a^2 + b^2 + c^2) \), there exists a triangle with side lengths \( a, b, c \).
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Context:
A sequence consists of the digits \(122333444455555 \ldots\) such that each positive integer \(n\) is repeated \(n\) times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence.
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Context:
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
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Context:
已知正整数 $x_{1}<x_{2}<\cdots<x_{9}$ 且 $x_{1}+x_{2}+\cdots x_{9}=220$, 则在 $x_{1}+x_{2}+\cdots+x_{5}$ 取得最大值的情况下, $x_{9}-x_{1}$ 的最大值为 $\qquad$.
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Context:
In cyclic pentagon \(ABCDE\), \(\angle ABD = 90^\circ\), \(BC = CD\), and \(AE\) is parallel to \(BC\). If \(AB = 8\) and \(BD = 6\), find \(AE^2\).
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Context:
There are 100 people in a room with ages 1, 2, ..., 100. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?
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Context:
## Task 10/75

Given is a regular pyramid with a square base. Each side face forms an angle $\alpha$ with the base. Two spheres are inscribed in this pyramid such that one sphere touches all five faces of the pyramid, while the other touches the four side faces of the pyramid and the surface of the first sphere.

In what ratio do the volumes of the two spheres stand to each other?
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Context:
In the triangle \(ABC\), medians \(AN\) and \(CM\) are drawn, and \(\angle ABC = 120^\circ\). A circle passing through points \(A\), \(M\), and \(N\) also passes through point \(C\). The radius of this circle is 7. Find the area of triangle \(ABC\).
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Context:
On a table, there are three cones standing on their bases, touching each other. The radii of their bases are 6, 24, and 24. A truncated cone is placed on the table with its smaller base downward, and it has a common generatrix with each of the other cones. Find the radius of the smaller base of the truncated cone.
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Context:
A target consisting of five zones is hanging on the wall: a central circle (bullseye) and four colored rings. The width of each ring equals the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye scores 315 points. How many points does hitting the blue (second to last) zone score?
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Context:
A circle constructed on the smaller leg of an isosceles trapezoid as its diameter touches the longer leg, which is equal to \(a\).

Find the midline of the trapezoid.
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Context:
Real numbers $x, y$, and $z$ are chosen from the interval $[-1, 1]$ independently and uniformly at random. What is the probability that

$$\vert x\vert +\vert y\vert +\vert z\vert +\vert x+y+z\vert=\vert x+y\vert +\vert x+z\vert +\vert y+z\vert$$
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Context:
已知数列 $\left\{(-1)^{n} \frac{3 n+2}{n(n+1) \cdot 2^{n+1}}\right\}$ 的前 $n$ 项和为 $S_{n}$. 计算 $S_{18}$ 的值。标准答案格式为$\frac{a}{b^{c} \times d}-\frac{e}{f}$，计算a+b+c+d+e+f的值。
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Context:
Let $ABCD$ be a convex quadrilateral with $AB=2$, $AD=7$, and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}$. Find the square of the area of $ABCD$.
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Context:
Let positive integers a, b, and c have the following properties:

When a is divided by 29, the remainder is 7.
When b is divided by 47, the remainder is 22.
When (a + b) is divided by 23, the remainder is c.

Determine how many possible combinations of a, b, and c are less than 1000.
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Context:
Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} $$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.
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Context:
Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Find all pairs  $(a,b)$  of positive integers  such that  $a!+b$  and  $b!+a$  are both powers of  $5$ .

*Nikola Velov, North Macedonia*
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Context:
Find the residue of the function f(z) at z = 2, where f(z) is defined as: $$f(z)=(ze^{z-3}+(3-z)e^z)e^{(z-2)^{2}+(z-2)^{-2}}$$
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Context:
Intersect an octahedron with a plane so that the cross-section is a regular hexagon. Calculate the area of this hexagon if the edge length of the octahedron is \(a\).
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Context:
Given a non-constant sequence $\left\{a_{i}\right\}$ satisfying $a_{i+1}^{2} - a_{i} a_{i+1} + a_{i}^{2} = 0$, and $a_{i+1} \neq a_{i-1}$ for $i = 1, 2, \cdots, n$, with the conditions that for a given natural number $n$, $a_{1} = a_{n+1} = 1$, find the value of $\sum_{i=0}^{n-1} a_{i}$.
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Context:
Determine all primes \( p \) such that
\[ 5^p + 4 \cdot p^4 \]
is a perfect square, i.e., the square of an integer.
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Context:
Determine how many zeros precede the last digit of \(2009^{2011}\).
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Context:
97**. A piece is randomly broken off from each of three identical rods. What is the probability that these three pieces can be used to form an acute triangle?
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Context:
How many positive integers divide $5n^{11} - 2n^5 - 3n$ for all positive integers $n$?
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Context:
Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$
for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.
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Context:
Let $ABC$ be a triangle with area $5$ and $BC = 10$. Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G$. Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2 + AC^2$.
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Context:
The set of five-digit numbers $\left\{ N_1, N_k \right\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, shares at least one digit in at least one position with at least one of the numbers $N_1, N_k$. Find the smallest possible value of $k$.
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Context:
On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ to $B$ were the same as on the section from $B$ to $C$, the entire journey from $A$ to $C$ would take 5.5 hours. How much time would the boat take to travel upstream from $C$ to $A$ under these conditions? The boat's own speed remains unchanged in all cases.
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Context:
There are some guys who have an equal number of bullets. All of them shoot 4 bullets. The remaining bullets are equal to the total number of bullets each had after division. Each one originally had 25 bullets. How many guys are there?
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Context:
Simplify the following expression:
\[ \frac{1-\sqrt{2 t}}{\frac{1-\sqrt[4]{8 t^{3}}}{1-\sqrt[4]{2 t}}-\sqrt{2 t}} \cdot\left(\frac{\sqrt[4]{\frac{1}{\frac{1}{2}}+\sqrt[4]{4 t^{2}}}}{1+\sqrt[4]{\frac{1}{2 t}}}-\sqrt{2 t}\right)^{-1}. \]
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Context:
Suppose that property P is satisfied by at least three of $a_1,b_1,c_1$, by at least three of $a_2,b_2,c_2,\cdots$, and by at least three of $a_5,b_5,c_5$. Determine the largest value of $m$ for which it is guaranteed that at least $m$ of the $a's$, or $m$ of the $b's$, or $m$ of the $c's$ satisfy property P.
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Context:
18. $[\mathbf{9}]$ Let $f$ be a function that takes in a triple of integers and outputs a real number. Suppose that $f$ satisfies the equations
$$
\begin{aligned}
f(a, b, c) & =\frac{f(a+1, b, c)+f(a-1, b, c)}{2} \\
f(a, b, c) & =\frac{f(a, b+1, c)+f(a, b-1, c)}{2} \\
f(a, b, c) & =\frac{f(a, b, c+1)+f(a, b, c-1)}{2}
\end{aligned}
$$
for all integers $a, b, c$. What is the minimum number of triples at which we need to evaluate $f$ in order to know its value everywhere?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the least common multiple (L.C.M.) of 76 and \( d \) is 456 and the highest common factor (H.C.F.) of 76 and \( d \) is \( c \), determine the value of the positive integer \( d \).
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Context:
The domain of the function $f(x)$ is $\{x|x\in R$, and $x\neq 1\}$. It is known that $f(x+1)$ is an odd function. When $x \lt 1$, $f(x)=2x^{2}-x+1$. Then, when $x \gt 1$, the decreasing interval of $f(x)$ is ____.
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Context:
Stephanie has several household bills to settle. Her electricity bill is $180 and she's paid 75% of that. Her gas bill is $120 out of which she's paid 60%. She has an outstanding water bill of $200, of which she has only paid 25%. Steph has also been paying down her internet bill of $100 in six payments of $5, leaving the remaining balance with her.  Thereafter, her ISP offered a discount of 15% on the remaining balance. Stephanie received a subsidy from her housing association that reduced her heating bill by 30%; the initial bill amount was $150. She paid 45% of the adjusted bill striving to pay off her debts. Furthermore, she has a garbage disposal bill of $60 with a standing payment of half of the bill so far. In addition, she has to pay a phone bill of $90, but she's off to a start by paying 20% of that. Finally, she has a cable bill of $80, but she only has to pay 60% of the cost due to a promotional offer. With all these financial commitments, how much does Stephanie still have to pay?
Please reason step by step, and put your final answer within \boxed{}.


Context:
52. \(ABCD\) is a circumscribed quadrilateral. The lengths of the segments from \(A\) to the points of tangency are equal to \(a\), and the lengths of the segments from \(C\) to the points of tangency are equal to \(b\). In what ratio does the diagonal \(AC\) divide the diagonal \(BD\)?
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Context:
Evaluate the integral $$J=\int _0 ^\infty \frac{\log(x^2+1/x^2)}{1+x^2}dx.$$
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Context:
At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $\frac{a}{3}$ thrice instead. We say that a positive integer $n$ is persistent if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.
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Context:
Given the line $l: \sqrt{3}x-y-4=0$, calculate the slope angle of line $l$.
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Context:
A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?
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Context:
Let \(ABC\) be a triangle with incenter \(I\), centroid \(G\), and \(|AC| > |AB|\). If \(IG \parallel BC\), \(|BC| = 2\), and \(\text{Area}(ABC) = \frac{3\sqrt{5}}{8}\), then find the value of \(|AB|\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
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Context:
Evaluate the following limit without using L'Hôpital's Rule:
$$\lim_{x \rightarrow 0}\frac{\cos(1-\frac{\sin x}{x})+\cos(2-\frac{\sin(2x)}{x})+\cdots+\cos(k-\frac{\sin(kx)}{x})-k}{\cos(1-\cos(x))+\cos(2-2\cos(2x))+\cdots+\cos(k-k\cos(kx))-k}$$
This question was proposed to the Romanian Math Magazine by Jalil Hajimir.
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Context:
A boat crossed a lake from north to west at a certain speed, entered a river and covered twice as much distance going upstream at 4 km/h. It then turned around and stopped at the south shore of the lake. The boat averaged 3.6 km/h that day. What was its approximate speed while crossing the lake if its downstream speed was 6 km/h?
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Context:
If $a$ is an element randomly selected from the set $\{1, 2, 3, 4, 5, 6, 7\}$, then the probability that the circle $C: x^2 + (y-2)^2 = 1$ is contained inside the circle $O: x^2 + y^2 = a^2$ is ______.
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Context:
Given a continuous function $f(x)$ where $f(0)<0$ and $f(1)>0$, and the evaluation time for $f(x)<0$ is $1$ unit while for $f(x)>0$ it's $C$ units ($C>1$), what is the optimal point to choose within the interval $[0,1]$ to perform bisection in order to minimize the expected time to reach a desired accuracy, assuming only the bisection method can be used?
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Context:
There is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the following properties:

1. Every two rows are different.
2. Each row contains exactly $6$ cells that contain $1$.
3. For every three rows, there exists a column such that the intersection of the column with the three rows (the three cells) all contain $0$.

What is the greatest possible value of $n$?
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Context:
\(ABCD\) is a convex quadrilateral and \(E, F\) are the mid-points of \(BC\) and \(CD\) respectively. The line segments \(AE, AF\) and \(EF\) divide \(ABCD\) into four triangles, whose areas are four consecutive integers. Find the greatest possible area of \(\triangle ABD\).
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Context:
Find the upper bound of the function g(x), where g'(x) = 1/(x^2 + g^2(x) + 1) and g(1) = 1.
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Context:
On the sides of the acute angle $ABC$, points $A$ and $C$ are taken. One circle touches the line $AB$ at point $B$ and passes through point $C$. The second circle touches the line $BC$ at point $B$ and passes through point $A$. Point $D$ is the second common point of the circles. It is known that $AB = a$, $CD = b$, $BC = c$. Find $AD$.
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Context:
In a row, there are $n$ integers such that the sum of any three consecutive numbers is positive, and the sum of any five consecutive numbers is negative. What is the maximum possible value of $n$?
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Context:
设$a$，$b$为正实数$\frac { 1 } { a } + \frac { 1 } { b } \leq 2 \sqrt { 2 } , ( a - b ) ^ { 2 } = 4 ( a b ) ^ { 3 }$，则

$\log _ { a } b $值为多少？ .
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Context:
Jake splits some shots of vodka with his friend. Each shot of vodka is 1.5 ounces. The vodka is 50% pure alcohol. Jake drank 3 ounces of pure alcohol. How many shots of vodka did Jake split with his friend?
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Context:
A dozen eggs and a certain amount of pounds of oranges are currently at the same price. If the price of a dozen eggs rises by 9 percent and the price of oranges rises by 6%, it will cost $15 more to buy a dozen eggs and the oranges. How many pounds of oranges are in the group?
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Context:
Two trains of length 120 m and a certain length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. They will be clear of each other from the moment they meet in 20.99832013438925 seconds. What is the length of the second train?
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Context:
A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of $n$.
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Context:
Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today's handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor's. If the freshmen are distinguishable but the handouts are not, how many ways are there to distribute the six handouts subject to the above conditions?
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Context:
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. Find the digit $A$.
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Context:
Evaluate the following sum:

$$ \frac{1}{\log_2\left(\frac{1}{7}\right)} + \frac{1}{\log_3\left(\frac{1}{7}\right)} + \frac{1}{\log_4\left(\frac{1}{7}\right)} + \frac{1}{\log_5\left(\frac{1}{7}\right)} + \frac{1}{\log_6\left(\frac{1}{7}\right)} - \frac{1}{\log_7\left(\frac{1}{7}\right)} - \frac{1}{\log_8\left(\frac{1}{7}\right)} - \frac{1}{\log_9\left(\frac{1}{7}\right)} - \frac{1}{\log_{10}\left(\frac{1}{7}\right)} $$
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Context:
Kevin collected various toys to use as prizes at the fair. He collected x stuffed animals, y frisbees, z yo-yos, and w puzzles. The total number of stuffed animals, frisbees, and puzzles was 80.

Additionally, he discovered that 3/5 of the toys he collected were toy cars, while 1/5 of the total collected toys were toy robots. The number of toy cars and toy robots adds up to 15 more than the total number of other toys (stuffed animals, frisbees, yo-yos, and puzzles).

If Kevin collected a total of 300 prizes, determine how many yo-yos (z) did Kevin collect?
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Context:
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
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Context:
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
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Context:
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB$, $\angle BFC$, $\angle CGD$, $\angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
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Context:
A shop owner sells three different types of cloth with varying selling prices per meter, which are also subject to fluctuating market prices. The base prices are as follows: Cloth A is sold at $15 per meter, Cloth B at $20 per meter, and Cloth C at $25 per meter. On a given day, the market price for Cloth A increases by 8%, Cloth B decreases by 6%, and Cloth C remains unchanged. The owner sells 10 meters of Cloth A, 15 meters of Cloth B, and 5 meters of Cloth C. In addition, a 5% sales tax is applied to all types of cloth, and a varying discount rate is offered on the total purchase of Cloth B based on the quantity purchased: 5% for up to 10 meters, and 15% for more than 10 meters. Consider the cost prices of the cloths as $10 per meter for Cloth A, $15 per meter for Cloth B, and $20 per meter for Cloth C. Calculate the total profit percent earned by the shop owner on the given day.
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Context:
The base of a right triangular prism is an isosceles triangle \( ABC \) with \( AB = BC = a \) and \(\angle BAC = \alpha\). A plane is drawn through the side \( AC \) at an angle \(\varphi(\varphi<\pi / 2)\) to the base. Find the area of the cross-section, given that the cross-section is a triangle.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{a^2 \sin 2\alpha}{2 \cos \varphi}}


Context:
Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist \( 1 \leq i < j < k \leq 10 \) such that \( a_i < a_k < a_j \).

Find the number of such sequences \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer solution to the equation $2^{2^k} \equiv k \pmod{29}$.
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Context:
Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number.
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Context:
Shelly writes down a vector \( v = (a, b, c, d) \), where \( 0 < a < b < c < d \) are integers. Let \( \sigma(v) \) denote the set of 24 vectors whose coordinates are \( a, b, c \), and \( d \) in some order. For instance, \( \sigma(v) \) contains \( (b, c, d, a) \). Shelly notes that there are 3 vectors in \( \sigma(v) \) whose sum is of the form \( (s, s, s, s) \) for some \( s \). What is the smallest possible value of \( d \) ?
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Context:
In a certain high school, student A's math scores in 10 mock exams in the third year of high school are arranged in ascending order as $94$, $96$, $98$, $98$, $100$, $101$, $101$, $102$, $102$, $103$. The 40th percentile of student A's math scores in these 10 mock exams is ______.
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Context:
What is the maximum number of digits that a natural number can have, where all the digits are different, and the number is divisible by each of its digits?
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Context:
Let  $\mathbb{Z}_{\geq 0}$  be the set of nonnegative integers. Let  $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$  be a function such that, for all  $a,b \in \mathbb{Z}_{\geq0}$ : \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\]
Furthermore, suppose there exists  $n \in \mathbb{Z}_{\geq0}$  such that  $f(n)=577$ . Let  $S$  be the sum of all possible values of  $f(2017)$ . Find the remainder when  $S$  is divided by  $2017$ .
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Context:
Calculate the following expression:

$$
\operatorname{tg} \frac{\pi}{43} \cdot \operatorname{tg} \frac{2 \pi}{43}+\operatorname{tg} \frac{2 \pi}{43} \cdot \operatorname{tg} \frac{3 \pi}{43}+\ldots+\operatorname{tg} \frac{k \pi}{43} \cdot \operatorname{tg} \frac{(k+1) \pi}{43}+\ldots+\operatorname{tg} \frac{2021 \pi}{43} \cdot \operatorname{tg} \frac{2022 \pi}{43}
$$
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Context:
A list of five positive integers has a mean of $12$ and a range of $18$. The mode and median are both $8$. Find how many different values are possible for the second largest element of the list.
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Context:
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB = 1$. If the radius of the circle is $r$, find the value of $100r$ to the nearest integer.
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Context:
Dany owns a farm, in his farm he has 4 cows and 3 sheep that eat 2 bushels a day. He also has 7 chickens that eat a certain amount of bushels a day. He should have 35 bushels to suffice the animals for a day. How many bushels do the chickens eat per day?
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Context:
A hurricane system consisting of three storms, A, B, and C, passed over a location. Storm A dropped 5 inches of rain in the first 30 minutes and then dropped half that amount of rain in the next 30 minutes. Storm B dropped 3 inches of rain in the first 45 minutes, and in the next hour, the rainfall increased by 50%. Storm C consisted of light rainfall of 1/2 inch per hour for 3 hours. Calculate the overall average rainfall amount for the entire hurricane system.
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Context:
In triangle \( ABC \), angle \( A \) is equal to \( 60^{\circ} \). The distances from vertices \( B \) and \( C \) to the center of the inscribed circle of triangle \( ABC \) are 3 and 4, respectively. Find the radius of the circumscribed circle around triangle \( ABC \).
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Context:
A circle inscribed in triangle \(ABC\) touches sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. Point \(I_A\) is the center of the excircle opposite side \(BC\) of triangle \(ABC\). Points \(K\) and \(L\) are the midpoints of segments \(DI_A\) and \(EI_A\) respectively. Lines \(BK\) and \(CL\) intersect at point \(F\) located inside angle \(BAC\). Find \(\angle BFC\), given that \(\angle BAC = 50^\circ\). (The excircle touches side \(BC\) and the extensions of sides \(AB\) and \(AC\) beyond points \(B\) and \(C\) respectively.)
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Context:
In triangle $XYZ$, $XY=15$, $YZ=18$, and $ZX=21$. Point $G$ is on $\overline{XY}$, $H$ is on $\overline{YZ}$, and $I$ is on $\overline{ZX}$. Let $XG = p \cdot XY$, $YH = q \cdot YZ$, and $ZI = r \cdot ZX$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $GHI$ to the area of triangle $XYZ$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
A \(20 \times 19\) rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?
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Context:
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have participated in the tournament?
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Context:
10.4. In triangle $A B C$, points $D$ and $F$ are marked on sides $A B$ and $A C$ respectively such that lines $D C$ and $B F$ are perpendicular to each other and intersect at point $E$ inside triangle $A B C$. It turns out that $A D=D C$ and $D E \cdot E C=F E \cdot E B$. What degree measure can angle $B A C$ have? (7 points)
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Context:
Given the function f(x) = ax^3 + b*sin(x) + 4, where a, b are real numbers, and f(lg(log_2 10)) = 5, determine the value of f(lg(lg2)). Express your answer as a single integer.
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Context:
A positive integer, which when added to a certain number, gives a sum which is greater than when it is multiplied by that number. This positive integer is 1. What is the certain number?
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Context:
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.
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Context:
In how many ways can \(a, b, c\), and \(d\) be chosen from the set \(\{0,1,2, \ldots, 9\}\) so that \(a<b<c<d\) and \(a+b+c+d\) is a multiple of three?
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Context:
What is the minimum number of L-shaped figures made of 3 cells that need to be painted in a $5 \times 5$ grid so that no more L-shaped figures can be painted? (The painted L-shaped figures must not overlap.)
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Context:
Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?
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Context:
Quadrilateral $A B C D$ has $\angle B C D=\angle D A B=90^{\circ}$. The perimeter of $A B C D$ is 224 and its area is 2205. One side of $A B C D$ has length 7. The remaining three sides have integer lengths. The sum of the squares of the side lengths of $A B C D$ is $S$. What is the integer formed by the rightmost two digits of $S$ ?
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Context:
Given a rectangular coordinate system with origin point $O$, vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfy $|\overrightarrow {a}|=|\overrightarrow {b}|=1$ and $\overrightarrow {a}\cdot \overrightarrow {b}=\frac {1}{2}$. Let $\overrightarrow {c} = (m, 1-m)$ and $\overrightarrow {d} = (n, 1-n)$. For any real numbers $m$ and $n$, the inequality $|\overrightarrow {a} - \overrightarrow {c}| + |\overrightarrow {b} - \overrightarrow {d}| \geq T$ holds. Determine the range of possible values for the real number $T$.
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Context:
Find the maximum value of the function
$$
f(x) = \sqrt{3} \sin 2x + 2 \sin x + 4 \sqrt{3} \cos x.
$$
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Context:
In triangle $ABC$, angle $A$ satisfies $\sin A + \cos A > 0$ and $\tan A < \sin A$. Determine the range of values for angle $A$.
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Context:
$S$ is the sum of the first 6 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{10} a_{16} > S + 39$ and $a_{11} a_{15} < S + 55$. Specify all possible values of $a_{1}$.
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Context:
A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the large rectangle. The sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine the minimum possible value of the area of the large rectangle.
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Context:
The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$.  A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$.  Determine the value of $P(1)/P(-1)$.
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Context:
Find the $2016$th smallest positive integer that satisfies the congruence $x^x \equiv x \pmod{5}$.
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Context:
A school offers 10 elective courses, among which courses A, B, and C have the same class time. According to the school's rules, each student must choose three courses. Find the number of different elective schemes available for each student. Express your answer as a single integer.
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Context:
Find the number of triples of sets $(A, B, C)$ such that:

(a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$.
(b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$.
(c) $|A| = |B| = |C| = 4$.

Here, $|S|$ denotes the number of elements in the set $S$.
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Context:
Let \(ABC\) be a triangle with \(AB = 13\), \(BC = 14\), \(CA = 15\). Let \(H\) be the orthocenter of \(ABC\). Find the distance between the circumcenters of triangles \(AHB\) and \(AHC\).
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Context:
Compute: $\int_{\lvert z \rvert = \frac{1}{2}} \frac{dz}{z \sin\left( \frac{1}{z} \right)}$
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Context:
Two sides of a regular polygon with $n$ sides, when extended, meet at an angle of $28$ degrees. What is the smallest possible value of $n$?
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Context:
7 For non-negative real numbers $x_{i}(i=1,2, \cdots, n)$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$.
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Context:
Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of $\angle BAD$?
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Context:
In how many ways can 2 black, 3 white, and 4 red balls be arranged in a row such that no black ball is next to a white ball?
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Context:
Sixty points are marked on a circle, divided as follows: thirty are red, twenty are blue, and ten are green. These points divide the circle into sixty arcs. Each arc is assigned a number based on the colors of its endpoints:

- An arc between a red and a green point is assigned the number $1$.
- An arc between a red and a blue point is assigned the number $2$.
- An arc between a blue and a green point is assigned the number $3$.
- An arc between two points of the same color is assigned the number $0$.

What is the greatest possible sum of all the numbers assigned to the arcs?
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Context:
Given that the graph of the inverse function of \( y = -\frac{x-a}{x-a-1} \) is centrally symmetric with respect to the point \((-1,3)\), determine the real number \( a \). Express your answer as a single real number.
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Context:
174. \(n^{2}\) integers from 1 to \(n^{2}\) are written in a square table of size \(n \times n\): number 1 is in any position in the table; number 2 belongs to the row whose number is equal to the column number containing 1; number 3 belongs to the row whose number matches the column number containing 2, and so on. By how much does the sum of the numbers in the row containing number 1 differ from the sum of the numbers in the column containing the number \(n^{2}\)?
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Context:
Given that point \( F \) is the right focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), and the eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\), a line \( l \) passing through point \( F \) intersects the ellipse at points \( A \) and \( B \) (point \( A \) is above the \( x \)-axis), and \(\overrightarrow{A F} = 3 \overrightarrow{F B}\). Find the slope of the line \( l \).
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Context:
Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)
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Context:
You are given a positive integer  $k$  and not necessarily distinct positive integers  $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from  $1$  to  $2021$  in one of the  $k$  colors so that there are exactly  $a_1$  numbers of the first color,  $a_2$  numbers of the second color,  $\ldots$ , and  $a_k$  numbers of the  $k$ -th color, there is always a number  $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as  $x$  is exactly  $x$ . What are the possible values of  $k$ ?
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Context:
Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEFB] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
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Context:
In a box, there are 100 balls of different colors: 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls. How many balls must be drawn randomly from the box to ensure that at least 15 of them are of the same color?
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Context:
设A（2,0）为平面上一定点，$P ( \sin ( 2 t - 6 0 ^ { \circ } ) , \cos ( 2 t - 6 0 ^ { \circ } )$为动点，则当t由15°变到45°时，线段AP所扫过的图形的面积是
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Context:
Given the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 (a > 0, b > 0)$ with its right focus at $F$. If a line passing through point $F$ with a slope angle of $60^\circ$ intersects the right branch of the hyperbola at exactly one point, then the range of the eccentricity of this hyperbola is ______.
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Context:
The graph of the function $f(x) = \log_a x + 1$ must pass through a fixed point. Determine the coordinates of this fixed point. Express your answer as an ordered pair $(x, y)$.
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Context:
The complex number $z=(a-2)+(a+1)i$, where $a\in\mathbb{R}$, corresponds to a point located in the second quadrant. The range of $|z|$ is __________.
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Context:
Let \( x_{1}, x_{2} \) be the roots of the equation \( x^{2} - x - 3 = 0 \). Find \(\left(x_{1}^{5} - 20\right) \cdot \left(3 x_{2}^{4} - 2 x_{2} - 35\right)\).
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Context:
Given a constant $a \gt 0$, the graph of the function $f(x)=\frac{2^x}{{{2^x}+ax}}$ passes through the points $P(p,\frac{6}{5})$ and $Q(q,-\frac{1}{5})$. If $2^{p+q}=16pq$, then $a=$____
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Context:
pipe a that can fill a tank in an hour and pipe b that can fill the tank in 15 are opened simultaneously when the tank is empty . pipe b is shut 15 minutes before the tank overflows . when will the tank overflow ?
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Context:
How many subsets with $3$ elements of the set $S = \{1, 2, 3, \ldots, 19, 20\}$ exist, such that their product is divisible by $4$?
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Context:
Two trains 130 m and another train of unknown length run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 10.439164866810657 seconds. What is the length of the second train?
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Context:
Suppose a society has the following rule: every couple has exactly two children, with each child independently having a 50% chance of being male or female. If a couple has one male and one female child, what is the probability that the male line will eventually become extinct?
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Context:
An event is a hit or a miss. The first event is a hit, the second is a miss. Thereafter, the probability of a hit equals the proportion of hits in the previous trials. For example, the probability of a hit in the third trial is 1/2. What is the probability of exactly 50 hits in the first 100 trials?
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Context:
若在 1,2,3,\cdots,18 的任意一个排列中，总能找到连续 6 个数之和不小于 m，则实数 m 的最大值为__________.
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Context:
Given that a positive number is singular if its representation as a product of powers of distinct prime numbers contains no even powers other than $0$, determine the maximum number of consecutive singular numbers.
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Context:
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
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Context:
Determine the pressure force of water on a vertical parabolic segment, whose base is $4 \mathrm{m}$ and is located at the water surface, and its vertex is at a depth of $4 \mathrm{м}$.
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Context:
Find $\sum_{k=0}^{\infty}\left\lfloor\frac{1+\sqrt{\frac{2000000}{4^{k}}}}{2}\right\rfloor$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
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Context:
Find the angle of inclination of the tangent line to the curve $y=\frac{1}{3}x^3-5$ at the point $(1,-\frac{3}{2})$.
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Context:
The poetry lecture lasted 2 hours and $m$ minutes. The positions of the hour and minute hands on the clock at the end of the lecture are exactly swapped from their positions at the beginning of the lecture. If $[x]$ denotes the integer part of the decimal number $x$, find $[m]=$ $\qquad$ .
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Context:
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25\%$ have sent a message to $A$. How many possible two-digit values of $n$ are there?
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Context:
In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is $\frac{1}{4}$ length from $R$. Given $LM = 4$, and $LO = MO = 3$. Find the area of quadrilateral $PRSQ$.
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Context:
Three rays emanate from a single point and form pairs of angles of $60^{\circ}$. A sphere with a radius of one unit touches all three rays. Calculate the distance from the center of the sphere to the initial point of the rays.
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Context:
Determine the number of integers \( D \) such that whenever \( a \) and \( b \) are both real numbers with \( -\frac{1}{4} < a, b < \frac{1}{4} \), then \( \left|a^{2} - D b^{2}\right| < 1 \).
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Context:
Let $\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles.

Suppose that the edges of $WXYZ$ have lengths of either $24$ or $49$, and no face of $WXYZ$ is equilateral. Determine the surface area of the pyramid $WXYZ$.
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Context:
Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
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Context:
In the polar coordinate system, randomly select two curves from the four curves $C\_1$: $ρ=1$, $C\_2$: $θ= \frac {π}{3}(ρ>=0)$, $C\_3$: $ρ=\cos θ$, $C\_4$: $ρ\sin θ=1$. Let the number of intersection points of these two curves be the random variable $ξ$. Find the mathematical expectation $Eξ=$ \_\_\_\_\_\_.
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Context:
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
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Context:
Let \(ABCDA'B'C'D'\) be a rectangular prism with \(|AB|=2|BC|\). \(E\) is a point on the edge \([BB']\) satisfying \(|EB'|=6|EB|\). Let \(F\) and \(F'\) be the feet of the perpendiculars from \(E\) at \(\triangle AEC\) and \(\triangle A'EC'\), respectively. If \(m(\widehat{FEF'})=60^{\circ}\), find the value of \(|BC|/|BE|\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). Please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.