Context:
A rhombus with a 60° angle is rotated about its center on a plane until it coincides with its original position. Find the minimum degree of rotation required for this to happen. Express your answer in degrees as a whole number.
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Context:
There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number?
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Context:
Determine the order of the group presented by $$\langle a, b\mid a^3, b^3, [a, [b, a^{-1}]], [b, [b, a^{-1}]]\rangle.$$
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Context:
Find the sum of the two smallest possible values of  $x^\circ$  (in degrees) that satisfy the following equation if  $x$  is greater than  $2017^\circ$ :   $$ \cos^59x+\cos^5x=32\cos^55x\cos^54x+5\cos^29x\cos^2x\left(\cos9x+\cos x\right). $$  *2017 CCA Math Bonanza Individual Round #10*
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Context:
How many ways can one color the squares of a \(6 \times 6\) grid red and blue such that the number of red squares in each row and column is exactly 2?
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Context:
Find an estimate for the sum $\sum_{1\leq x, y\leq n, x\neq y}\Lambda(x)^{2}\cdot \Lambda(y)^{2}$, where $\Lambda$ is the von-Mangoldt function.
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Context:
Sue borrowed 15 books and 6 movies. She returned 8 books. She then returned a third of the movies to the library and checked out 9 more books. How many movies and books does Sue now have?
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Context:
In how many ways can four married couples sit around a circular table such that no man sits next to his wife?
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Context:
Suppose that $X_1,X_2,\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S= \sum _{i= 1}^{k}X_{i}/ 2^{i}$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.  The original answer is in $ae^{b / c}-d$ format, please provide the value of a+b+c+d.
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Context:
Given triangle \( ABC \). On the side \( AC \), which is the largest in the triangle, points \( M \) and \( N \) are marked such that \( AM = AB \) and \( CN = CB \). It turns out that angle \( NBM \) is three times smaller than angle \( ABC \). Find \( \angle ABC \).
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Context:
what is the least number . which should be added to 0.0320 to make it a perfect square ?
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Context:
A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.
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Context:
The sequence $\{a_{n}\}_{n\geq 0}$ is defined by:

- $a_{0} = 20$
- $a_{1} = 100$
- $a_{n+2} = 4a_{n+1} + 5a_{n} + 20$ for $n = 0, 1, 2, \ldots$

Find the smallest positive integer $h$ satisfying $1998 \mid (a_{n+h} - a_{n})$ for all $n = 0, 1, 2, \ldots$
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Context:
mary passed a certain gas station on a highway while traveling west at a constant speed of 50 miles per hour . then , 15 minutes later , paul passed the same gas station while traveling west at a constant speed of 80 miles per hour . if both drivers maintained their speeds and both remained on the highway for at least 2 hours , how long after he passed the gas station did paul catch up with mary ?
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Context:
Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e., it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
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Context:
The number  $201212200619$  has a factor  $m$  such that  $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find  $m$ .
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Context:
数列 $\left\{a_{n}\right\}$ 满足: $a_{1}=a_{2}=a_{3}=1$ 。令 $$ b_{n}=a_{n}+a_{n+1}+a_{n+2}\left(n \in \mathbf{N}^{*}\right)。 $$ 若 $\left\{b_{n}\right\}$ 是公比为 3 的等比数列, 求 $a_{100}$ 的值。原始答案是 $a_{100}=\frac{m^{n}+a}{b}$的形式，请给出 m + n +a+b的值。
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Context:
A man has a certain number of small boxes to pack into parcels. If he packs 3 or 4 in a parcel, he is left with one over; if he packs some other number of boxes in a parcel, he is also left with one over; if he packs 7 in a parcel, none is left over. The number of boxes he may have to pack is 301. What is the third number of boxes that leaves one over when packed in a parcel?
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Context:
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Then \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\) equals what?
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Context:
在平面直角坐标系中，函数 $y=\frac{x+1}{|x|+1}$ 的图像上有三个不同的点位于直线 $l$ 上，且这三点的横坐标之和为 0。求 $l$ 的斜率的取值范围。原始答案格式为 $0<k<\frac{m}{n}$，请给出 m + n 的值。
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Context:
Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least 2 pairs $(x,y)$ of positive integers.
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Context:
The tax on a commodity is diminished by 19%, and its consumption increased by 15%. What is the effect on revenue in percentage?
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Context:
Find the largest number of distinct natural numbers, each less than 50, such that any two of them are coprime.
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Context:
Given positive numbers \(a, b, c, d\) such that \(a + b + c + d = 1\), find the minimum value of \(M = \sqrt{a^{2}+\frac{1}{8a}}+\sqrt{b^{2}+\frac{1}{8b}}+\sqrt{c^{2}+\frac{1}{8c}}+\sqrt{d^{2}+\frac{1}{8d}}\).
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Context:
Given \( n \) points on a plane (where \( n > 2 \)), no three of which are collinear, in how many different ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect?
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Context:
Find the value of the product \(\left(1+2 \cos \frac{2 \pi}{7}\right)\left(1+2 \cos \frac{4 \pi}{7}\right) \cdots\left(1+2 \cos \frac{12 \pi}{7}\right) \).
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Context:
Given the function y = 2^x and the domain P = {1, 2, 3}, calculate the range of the function. Express your answer as a set of values.
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Context:
What least number should be added to 29989, so that the sum is completely divisible by 73?
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Context:
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which
  $a_1 > a_2 > a_3 > a_4 > a_5 > a_6$ and $a_6 < a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12}.$
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations.
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Context:
Let \( a, b, c, d, e \) be natural numbers with \( a < b < c < d < e \), and \( a + 2b + 3c + 4d + 5e = 300 \). Determine the maximum value of \( a + b \).
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Context:
$P Q R S$ is a square with side length $8$. $X$ is the midpoint of side $P Q$, and $Y$ and $Z$ are the midpoints of $X S$ and $X R$, respectively. Calculate the area of trapezoid $Y Z R S$. Express your answer in square units.
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Context:
A set  $D$  of positive integers is called *indifferent* if there are at least two integers in the set, and for any two distinct elements  $x,y\in D$ , their positive difference  $|x-y|$  is also in  $D$ . Let  $M(x)$  be the smallest size of an indifferent set whose largest element is  $x$ . Compute the sum  $M(2)+M(3)+\dots+M(100)$ .

*Proposed by Yannick Yao*
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Context:
Given $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(x,4)$, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, then the range of real number $x$ is ____.
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Context:
In a group of 10 basketball teams which includes 2 strong teams, the teams are randomly split into two equal groups for a competition. What is the probability that the 2 strong teams do not end up in the same group?
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Context:
We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into  $n$  sets of equal weight. What is the maximal  $n$  for which this is possible?
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Context:
Given the cryptarithm: ЖАЛО + ЛОЖА = ОСЕНЬ. Identical letters represent identical digits, and different letters represent different digits. Determine the value of the letter А.
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Context:
Given the quadratic function f(x) = ax^2 + bx + c satisfying 2a + c/2 > b and c < 0, determine an interval containing a zero of the function f(x). Express your answer as an interval (lower bound, upper bound) using parentheses.
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Context:
Donggyu wants to complete the diagram so that each box contains a positive integer and each box in the top two rows contains the product of the integers in the two boxes below it. He wants the integer in the top box to be 720. How many different values can the integer \( n \) take? Express your answer as a single integer count.
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Context:
A sphere is placed inside a dihedral angle of 60°, and the sphere touches the two half-planes of the dihedral angle at points A and B, respectively. The spherical distance between points A and B is $2\pi$ cm. Find the radius of the sphere.
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Context:
6. The password setting for a certain lock involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex either red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How many different password settings are there for this lock?
(2010, National High School Mathematics League Competition)
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Context:
记 $S=\left\{(x, y, z, w) \in \mathbb{Z}^{4} \mid 0 \leq x, y, z, w \leq 18, x^{2}+y^{2} \equiv z^{2}+w^{2}(\bmod 19)\right\}$,则 S 中有 $\qquad$个元素。
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Context:
Find all $a\in\mathbb{C}$ such that $f=X^4 −(a+4)X^3 +(4a+5)X^2 −(5a+2)X +2a$ has $a$ as a double root.
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Context:
Let the curve $y = x^n + 1$ ($n \in \mathbb{N}^*$) have a tangent line at the point $(1,1)$, and let the x-coordinate of the point where this tangent line intersects the x-axis be $x_n$.
Define $a_n = \lg x_n$, then the value of $a_1 + a_2 + \ldots + a_{99}$ is \_\_\_\_\_\_\_\_.
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Context:
The milk level in a rectangular box measuring 58 feet by some feet is to be lowered by 6 inches. To do this, 5437.5 gallons of milk must be removed. What is the width of the rectangular box?
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Context:
Suppose  $a$ ,  $b$ ,  $c$ , and  $d$  are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762,  ab+cd &= 260,  ac+bd &= 365,  ad+bc &= 244. \end{aligned}\] Compute  $abcd.$
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Context:
Given the function $f(x)=2\sin x\cos x+4\cos ^{2}x-1$, if real numbers $a$, $b$, $c$ satisfy $af(x)-bf(x+c)=3$ for any real number $x$, then $3a^{2}+2b+\cos c=\_\_\_\_\_\_$.
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Context:
Let $f_{1}(x)=\sqrt{1-x}$, and for integers $n \geq 2$, let \[f_{n}(x)=f_{n-1}\left(\sqrt{n^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N,$ the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$
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Context:
Let $n$ be a positive integer. Determine the smallest positive integer $k$ such that for any colouring of the cells of a $2n\times k$ table with $n$ colours there are two rows and two columns which intersect in four squares of the same colour.
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Context:
Evaluate the integral: $$\int_{0}^{2\pi} e^{i\cos\theta}\cos\theta \, d\theta$$
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Context:
Find the simple interest on Rs. 69,000 at 16 2/3% per year for a certain period. The simple interest is Rs. 8,625. What is the duration of the investment?
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Context:
Oleg writes a real number above each column of a blank $50 \times 50$ grid (outside the grid) and a real number to the left of each row (outside the grid). It is known that all 100 real numbers he writes are distinct, and there are exactly 50 rational numbers and 50 irrational numbers among them. Then, Oleg writes the sum of the number to the left of the row and the number above the column in each cell of the grid (creating an addition table). What is the maximum number of rational numbers that can appear in this grid?
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Context:
Determine the order of the Taylor polynomial needed to approximate $f(x) = \cos x$ centered at $x = 0$ with an error of at most $10^{-8}$ for $x = \frac{\pi}{4}$. Use the remainder formula for Taylor series with $c = \frac{\pi}{4}$.
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Context:
A permutation $ (a_1, a_2, a_3, a_4, a_5)$ of $ (1, 2, 3, 4, 5)$ is heavy-tailed if $ a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
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Context:
4. In a sequence of English letter strings, the first string $a_{1}=A$, the second string $a_{2}=B$, and each subsequent string $a_{n}(n \geq 3)$ is formed by appending the reverse of $a_{n-2}$ to $a_{n-1}$. For example, $a_{3}=a_{2} \overline{a_{1}}=B A$ (we use $\overline{a_{i}}$ to denote the reverse of $a_{i}$, which is the string read from right to left, such as $\overline{A B B}=B B A$, $\overline{A A B A}=A B A A$), $a_{4}=a_{3} \bar{a}_{2}=B A B$, $a_{5}=a_{4} \overline{a_{3}}=B A B A B$, $a_{6}=a_{5} \overline{a_{4}}=B A B A B B A B$. Therefore, among the first 1000 strings in this sequence, there are $\qquad$ palindromic strings (a palindromic string is one that reads the same from left to right as from right to left, such as $A B A$, $A A B A A$).
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Context:
In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB = 5$, $ CD = 3$, $ AC = 15$. What are the sum of different integer values of possible $ BD$?
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Context:
Given the function f(x) = sin(ωx + φ) where ω > 0 and |φ| ≤ π/2, we are told that x = -π/4 is a zero of f(x), and x = π/4 is the axis of symmetry of the graph of y = f(x). Additionally, f(x) is monotonic on the interval (π/18, 5π/36). Find the maximum value of ω. Express your answer as a single integer.
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Context:
Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of independent events defined on $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{P}(A_n)<1$ for all $n$ and $\mathbb{P}(\bigcup_{n=1}^{\infty}A_n)=1.$ Find the value of $\mathbb{P}(\bigcap_{i=1}^{\infty}\bigcup_{n=i}^{\infty}A_n).$
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Context:
A convex quadrilateral \( {ABCD} \) with area2002contains a point \( P \) in its interior such that \( {PA} = {24} \) , \( {PB} = {32},{PC} = {28} \) ,
 and \( {PD} = {45} \) . FInd the perimeter of \( {ABCD} \) .
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Context:
In square \(ABCD\), an isosceles triangle \(AEF\) is inscribed such that point \(E\) lies on side \(BC\) and point \(F\) lies on side \(CD\), and \(AE = AF\). The tangent of angle \(AEF\) is 3. Find the cosine of angle \(FAD\).
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Context:
Let $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222 < ax < 2022$?
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Context:
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$
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Context:
65. Suppose you have forgotten one digit of a phone number you need and dial it at random. What is the probability that you will have to make no more than two calls?
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Context:
Given \( x \) satisfies \(\log _{5 x} 2 x = \log _{625 x} 8 x\), find the value of \(\log _{2} x\).
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Context:
Alyssa went to 11 soccer games this year but missed 12. She also watched 8 games online. Last year, she went to 13 games in person. For each in-person game, she paid $20 for a ticket. This year, she spent $120 on a streaming subscription to watch games online. Next year, she plans to go to 15 games in person and watch 10 games online. Two of her friends will join her at 5 of the games next year, and they will pay her back for their tickets. If a streaming subscription is expected to cost $150 next year, how much will Alyssa spend in total on attending and streaming soccer games over the three years, considering the amount her friends pay for their tickets?
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Context:
How many positive five-digit integers are there that have the product of their five digits equal to $900$?

(Karl Czakler)
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Context:
Let \(p\) be a prime number, and consider a group \((G, \cdot)\) with \(p^3\) elements. The center of the group, \(\operatorname{Z}(G)\), has \(p\) elements. Determine the number of distinct centralisers \(\operatorname{C}(x)\) for elements \(x \in G\).
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Context:
Find the almost sure limit of $\frac{X_t}{t}$ as $t \to +\infty$, where $X_{t} = \inf(n:\xi_{1}+\ldots+\xi_{n}> t)$ and $\{\xi_{i}\}$ are i.i.d. random variables with $\xi_{i}\sim \mathrm{Exp}(\lambda)$. What is this limit?
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Context:
Find a positive real number $R$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.
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Context:
Given that in the expansion of $(1-2x)^{n} (n \in \mathbb{N^*})$, the coefficient of $x^{3}$ is $-80$, find the sum of all the binomial coefficients in the expansion.
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Context:
设实数 $k, l, m$ 满足：函数 $y=(x+1)\left(x^{2}+k x+l\right)$ 的图像有对称中心 $(1,0)$ ，且与函数 $y=x^{3}+m$ 的图像有公共点, 则 $k+l+m$ 的取值范围是$\left(-\infty, \frac{a}{b}\right]$，求$a+b$的值。
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Context:
Evaluate the limit \( \lim_{x\to 3} \left(\dfrac{\sqrt{2x+3}-x}{\sqrt{x+1}-x+1} \right)^{ \left(\dfrac{x-1-\sqrt{x^2-5}}{x^2-5x+6}\right)} \) and express it in the form \(\dfrac{a\sqrt{b}}{c}\), where \(a,b,c \in \mathbb{N}\). Find the least value of \(a^2+b^2+c^2\).
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Context:
A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle.
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Context:
Consider  $\triangle \natural\flat\sharp$ .  Let  $\flat\sharp$ ,  $\sharp\natural$  and  $\natural\flat$  be the answers to problems  $4$ ,  $5$ , and  $6$ , respectively.  If the incircle of  $\triangle \natural\flat\sharp$  touches  $\natural\flat$  at  $\odot$ , find  $\flat\odot$ .
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Context:
Let $\mathcal{P}$ be a regular 17 -gon; we draw in the $\left(\begin{array}{c}17 \\ 2\end{array}\right)$ diagonals and sides of $\mathcal{P}$ and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of $\mathcal{P}$ ) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?
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Context:
Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the small one. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?
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Context:
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $29x + y = 2035$. Find the number of such distinct triangles whose area is a positive integer.
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Context:
A rook has toured a chessboard, visiting each square at least once. What is the minimum number of turns it could have made in the process?
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Context:
Compute the double integral \( \iint_{M} y \, dx \, dy \) where \( M \) is the region defined by \( y \geq 0 \), \( x^2 + y^2 \leq 1 \), and \( (x - 1)^2 + y^2 \geq 1 \).
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Context:
Determine the range of values for $a$ such that the polynomial \( p(x) = x^4 + ax^3 + bx^2 + ax + 1 \) has complex (non-real) roots, given that \( x = 1 \) is a root and \( b = -2(a+1) \).
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Context:
Given $A(-2,2)$, $B$ is a point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, and $F$ is the left focus of the ellipse. Find the coordinates of $B$ when $|AB|+\frac{5}{3}|BF|$ reaches its minimum value.
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\boxed{ \left( -\frac{5}{2} \sqrt{3}, 2 \right) }


Context:
Given an isosceles triangle \(XYZ\) with \(XY = YZ\) and an angle at the vertex equal to \(96^{\circ}\). Point \(O\) is located inside triangle \(XYZ\) such that \(\angle OZX = 30^{\circ}\) and \(\angle OXZ = 18^{\circ}\). Find the measure of angle \(\angle YOX\).
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Context:
On a calculator's keyboard, there are digits from 0 to 9 and signs for two operations. Initially, the display shows the number 0. You can press any keys. The calculator performs actions in the sequence of key presses. If an operation sign is pressed several times in a row, the calculator remembers only the last press. A distracted Scientist pressed a lot of buttons in a random sequence. Approximately find the probability that the result of the resulting sequence of actions is an odd number?
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Context:
A cobbler can mend 3 pairs of shoes in an hour on Mondays, Tuesdays, and Wednesdays. On Thursdays, his work pace slows down, and he can only mend 2 pairs of shoes per hour. He takes a 30-minute break each day from Monday to Thursday. On Friday, he works from 8am to 11am, during which he mends shoes at a rate of 4 pairs per hour. How many pairs of shoes can the cobbler mend in a week?
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Context:
1. When $x^{2}$ was added to the quadratic trinomial $f(x)$, its maximum value increased by $\frac{27}{2}$, and when $4 x^{2}$ was subtracted from it, its maximum value decreased by 9. How will the maximum value of $f(x)$ change if $2 x^{2}$ is subtracted from it?
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Context:
We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday.  On the first Monday the number we record is ten and a half.  On every Tuesday and every Thursday the number we record is one third of what it was on the previous workday.  On every Monday, Wednesday, and Friday the number we record is double what it was on the previous workday.  This will go on forever. What is the sum of all of the numbers we will record?
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Context:
Find the smallest natural number \( k \) such that for some natural number \( a \), greater than 500,000, and some natural number \( b \), the equation \(\frac{1}{a} + \frac{1}{a+k} = \frac{1}{b}\) holds.
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Context:
Find the product of all values of $d$ such that $x^{3} + 2x^{2} + 3x + 4 = 0$ and $x^{2} + dx + 3 = 0$ have a common root.
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Context:
抛物线 $y^{2}=2 p\left(x-\frac{p}{2}\right)(p>0)$ 上动点 $A$ 到点 $B(3,0)$ 的距离的最小值记为 $d(p)$ ，满足 $d(p)=2$ 的所有实数 $p$ 的和为 $\qquad$.
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Context:
Determine the largest constant $C>0$ satisfying the following property:

For any 2023-tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$ of strictly positive real numbers that are pairwise distinct, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$
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Context:
Find the smallest positive integer $n$ such that $$\underbrace{2^{2^{2^{2}}}}_{n 2^{\prime} s}>\underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text { factorials }}$$
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Context:
A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second best ranked team is eliminated in the last round. Compute $\lfloor 2021 p \rfloor$.
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Context:
Five consecutive integers are represented by p, q, r, s, and t. The two integers in the leftmost circle add to 63, and the two integers in the rightmost circle add to 57. Find the value of r. Express your answer as an integer.
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Context:
A person crosses a 3,500 meters long street in 9.5 minutes. What is their speed in miles per hour?
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Context:
9. Given that $\triangle A B C$ is an acute triangle, $b=2 c$, and $\sin B-\sin (A+B)=2 \sin C \cos A$. Then the range of $(\cos B+\sin B)^{2}+$ $\sin 2 C$ is $\qquad$ .
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Context:
求满足下列条件的大于 5 的最小奇数 a：存在正整数 m_1, n_1, m_2, n_2，使得 a=m_1^2+n_1^2，a^2=m_2^2+n_2^2，且 m_1-n_1=m_2-n_2.
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Context:
Let  $ABC$  be a triangle with  $AB=5$ ,  $BC=6$ ,  $CA=7$ . Let  $D$  be a point on ray  $AB$  beyond  $B$  such that  $BD=7$ ,  $E$  be a point on ray  $BC$  beyond  $C$  such that  $CE=5$ , and  $F$  be a point on ray  $CA$  beyond  $A$  such that  $AF=6$ . Compute the area of the circumcircle of  $DEF$ .

*Proposed by James Lin.*
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Context:
Given that (1+ex)2019=a0+a1x+a2x2+……+a2019x2019, find the value of:
- $$\frac {a_{1}}{e}$$+ $$\frac {a_{2}}{e^{2}}$$\- $$\frac {a_{3}}{e^{3}}$$+ $$\frac {a_{4}}{e^{4}}$$\-……- $$\frac {a_{2019}}{e^{2019}}$$
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Context:
What is the maximum number of cells on an $8 \times 8$ chessboard that can be cut by a single straight line?
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Context:
If the line $y = -2x + k$ forms a triangle with the coordinate axes and the area of this triangle is 4, then the value of $k$ is.
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Context:
Determine the number of invertible elements in the quotient ring \( \mathbb{Z}[i]/(220+55i)\mathbb{Z}[i] \).
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Context:
Given the equations \(xy + x + y = 44\) and \(x^2y + xy^2 = 448\), find the value of \(x^2 + y^2\).
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Context:
A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the $9^\text{th}$ day?
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Context:
Given the function \(f_X(x) = \begin{cases}4x^3 & 0<x<1\\ 0 & \text{otherwise}\end{cases}\) and the transformation \(y = -\log(x)\), find the cumulative distribution function (CDF) \(F_Y(y)\).
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Context:
Let $A$ be a regular $12$-sided polygon. A new $12$-gon $B$ is constructed by connecting the midpoints of the sides of $A$. The ratio of the area of $B$ to the area of $A$ can be written in simplest form as $\frac{a + \sqrt{b}}{c}$, where $a$, $b$, and $c$ are integers. Find $a + b + c$.
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Context:
Inside an unequal-sided triangle \( ABC \), there is a point \( O \) such that \(\angle OBC = \angle OCB = 20^\circ\). Additionally, \( \angle BAO + \angle OCA = 70^\circ \). Find angle \( A \).
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Context:
Evaluate the integral:

$$ \frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac{x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$

where $N > 0$. Provide the result of the integral.
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Context:
Let  $\sigma(n)$  be the number of positive divisors of  $n$ , and let  $\operatorname{rad} n$  be the product of the distinct prime divisors of  $n$ . By convention,  $\operatorname{rad} 1 = 1$ .  Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural*
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Context:
Given that the even function $f(x)$ is increasing on the interval $[0, +\infty)$, determine the relationship between $f(-1)$ and $f(a^2-2a+3)$. Express your answer as an inequality using standard notation.
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Context:
In city \( \mathrm{N} \), there are only blondes and brunettes. Blondes always lie, and brunettes always tell the truth. Every day, the residents dye their hair to the opposite color. One Monday in October, all people born in autumn were asked, "Were you born this month?" - 200 residents answered "yes", and no one answered "no". On Friday of the same week, these same people were asked again, "Were you born this month?" - and now only 50 answered affirmatively. How many "yes" answers were given to the same question on the last Monday of September, given that there are exactly four Mondays in October and no one in city \( \mathrm{N} \) was born in November?
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Context:
The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle.
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Context:
Petya and Vasya came up with ten quadratic trinomials. Then Vasya sequentially named consecutive natural numbers (starting from some number), and Petya substituted each named number into one of the trinomials of his choice and wrote the obtained values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in that specific order).

What is the maximum number of numbers Vasya could have named?
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Context:
Find the radius of convergence of the series \(\sum_{k=1}^{\infty} \frac{3^n}{z^n+z^{-n}}\) using the ratio test.
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Context:
A frog sitting at the point \((1, 2)\) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length \(1\), and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices \((0,0), (0,4), (4,4),\) and \((4,0)\). The probability that the sequence of jumps ends on a vertical side of the square is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
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Context:
Determine the degree of the minimal polynomial of $\cos\left(\frac{\pi}{13}\right)$ over $\mathbb{Q}$.
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Context:
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. Find how many pairs of integers $(m,n)$ there are such that $1\leq m\leq 2012$ and
$$5^n<2^m<2^{m+2}<5^{n+1}.$$
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Context:
7. Given the hyperbola $C: x^{2}-\frac{y}{3}=1$, with left and right foci $F_{1}$ and $F_{2}$, respectively. A line is drawn through point $F_{2}$ intersecting the right branch of the hyperbola $C$ at points $P$ and $Q$, such that $\angle F_{1} P Q=90^{\circ}$. Then the inradius of $\triangle F_{1} P Q$ is $\qquad$
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Context:
Determine the largest interval where the initial value problem \( t\dfrac{dy}{dt} = t + \left|y\right| \) with the initial condition \( y(-1) = 1 \) has a unique solution.
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Context:
What is the probability of finding a trinomial multiple of a given polynomial \( p(x) \) in \( GF(2)[x] \) that satisfies the conditions \( p(1) = 1 \) and \( p(x) \) has a constant term of 1?
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Context:
On an infinite chessboard, some bishops and knights are placed such that:

- Each bishop has a knight on the same diagonal (there may be other pieces between them).
- Each knight has a bishop exactly \( \sqrt{5} \) units away.
- Removing any piece causes at least one of these conditions to fail.

Determine all possible values of \( n \), the total number of pieces on the board.
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Context:
Evaluate the integral \( \int_0^\infty \left(x^3-\frac{1}{x^3}\right)e^{-x^2/2}\,\mathrm{d}x. \)
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Context:
Find the derivative of the convolution operation \( z(t) = (h \ast x)(t) \) with respect to \( x(t) \).
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Context:
A non-zero vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by vectors \( \hat{i} \) and \( \hat{i}+\hat{j} \) and the plane determined by vectors \( \hat{i}-\hat{j} \) and \( \hat{i}+\hat{k} \). Find the angle between \( \vec{a} \) and the vector \( \hat{i}-2\hat{j}+2\hat{k} \).
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Context:
In a convex decagon \(A_{1} A_{2} \ldots A_{10}\), all the sides and diagonals connecting vertices at intervals of one (e.g., \(A_{1} A_{3}, A_{2} A_{4}\), etc.) are drawn, except for the side \(A_{1} A_{10}\) and diagonals \(A_{1} A_{9}, A_{2} A_{10}\). A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-intersecting broken line with endpoints \(A_{1}\) and \(A_{10}\), where each segment coincides with one of the drawn sides or diagonals. Determine the number of such paths.
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Context:
Determine the spectrum of the operator $T: C[0,1] \to C[0,1]$ defined by $Tf(x) = \int_0^x f(t) \, dt$.
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Context:
Find all polynomials \( f(x) \) with integer coefficients such that for any two distinct primes \( p \) and \( q \), the values \( f(p) \) and \( f(q) \) are relatively prime.
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Context:
Evaluate the limit: $$\lim_{x\to 0}\frac{\sin 5x - 5\sin 3x + 10\sin x}{\sin(\sin x) + \tan x - 2x}$$
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Context:
Determine the cardinality of the set of injective functions from the set of natural numbers \( \mathbb{N} \) to the power set of natural numbers \( \mathcal{P}(\mathbb{N}) \).
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Context:
Calculate the limit: \[ \lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1} \left(\frac{1}{5}\right)^{n} \left(\frac{4}{5}\right)^{k-n} \]
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Context:
A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset".

The probability that 10 minutes pass without either baby crying can be expressed as $\frac{p}{q}$. Compute $p+q$.
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Context:
A 2D random walker starts at the origin \((0,0)\) and takes one step randomly per second. There is a square absorbing boundary defined by the points \((2,2)\), \((2,-2)\), \((-2,2)\), and \((-2,-2)\). Determine the average time it takes for the random walker to reach the absorbing boundary.
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Context:
Calculate the isothermal compressibility \( k_T \) at the critical temperature \( T = T_c \) and pressure \( p \) using the Van der Waals equation:

\[ p = \frac{R T}{v-b} - \frac{a}{v^2} \]

Given:
- \( p_c = \frac{a}{27b^2} \)
- \( v_c = 3b \)
- \( T_c = \frac{8a}{27bR} \)

The partial derivative of pressure with respect to volume at constant temperature is:

\[ \left(\frac{\partial p}{\partial v}\right)_T = -\frac{R T}{(v-b)^2} + 2\frac{a}{v^3} \]

At \( v = v_c \):

\[ -\frac{R T}{(v-b)^2} + 2\frac{a}{v^3} = -\frac{R(T-T_c)}{4b^2} \]

Hence, the isothermal compressibility is:

\[ k_T(T, v = v_c) = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T \approx C(T-T_c)^{-\gamma} \]

Determine \( k_T(T = T_c, p) \).
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Context:
Find the condition on positive integers \(m\) and \(n\) such that the quotient rings \(\mathbb{Z}[x,y]/(x^2-y^n)\) and \(\mathbb{Z}[x,y]/(x^2-y^m)\) are isomorphic.
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Context:
Given the function \( u = x^2y^2 + yz - z^3 \) and the constraint \( x^2 + y^2 + z^2 = 11 \), find the value of \( \left( \frac{\partial u}{\partial y} \right)_{x} \) at the point \((u, x, y, z) = (5, 1, -3, 1)\).
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Context:
Find the value of \( l \) for which the integral \( I(l) = \int_{0}^{\infty} \frac{x^l}{2x^6 + 4x^5 + 3x^4 + 5x^3 + 3x^2 + 4x + 2} \, dx \) is minimized, where \( 0 < l < 5 \).
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Context:
Find the greatest positive integer  $k$  such that the following inequality holds for all  $a,b,c\in\mathbb{R}^+$  satisfying  $abc=1$  \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
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Context:
XLIII OM - II - Problem 5

Determine the supremum of the volumes of spheres contained in tetrahedra with all altitudes no longer than $1$.
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Context:
What is the maximum number of rooks that can be placed on a $300 \times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.)
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Context:
For how many values of $x$ in $[0,\pi]$ is $\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)$?
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Context:
What values of \( d \), where \( 0 < d < 1 \), result in a generalized Cantor set with Lebesgue measure zero, when at each stage intervals of size \( 1 - 2d \) are retained at both ends and intervals of size \( 1 - 2d^n \) are removed, with \( 2^{n-1} \) such intervals at the \( n^{th} \) stage?
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Context:
How many numbers of the form $\overline{a b c d a b c d}$ are divisible by 18769?
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Context:
Krishna is preparing a pizza with 8 slices and has 10 different toppings available. Each slice can have one topping, and the same topping cannot be used on adjacent slices. How many different ways can Krishna arrange the toppings on the pizza slices?
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Context:
已知多项式
$$
f(z)=z^{3}+a z^{2}+b z+c(a, b. c \in \mathbf{Z})
$$
的所有根的模都是 20 或 15. 则这样的多
项式有 $\qquad$个。
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Context:
Evaluate the surface integral \( \int_{S} \vec{F} \cdot d\vec{S} \) where \( \vec{F} = y\hat{i} + 2x\hat{j} - z\hat{k} \) and \( S \) is the surface of the plane \( 2x + y = 4 \) in the first octant, cut off by the plane \( z = 4 \).
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Context:
In a 100x100 grid, each unit square contains a natural number. A rectangle with sides parallel to the grid is called "good" if the sum of the numbers inside the rectangle is divisible by 17. We can paint all unit squares in a "good" rectangle, and no unit square can be painted more than once. Determine the maximum number of unit squares, denoted as \(d\), that can be guaranteed to be painted.
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Context:
In a fair coin game, you win $1 if a head shows up and lose $1 if a tail shows up. What is the expected number of rounds you will play before you lose the first dollar from your own pocket?
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Context:
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac{n}{4} \right\rfloor$, $\left\lfloor \frac{n}{5} \right\rfloor$, and $\left\lfloor \frac{n}{6} \right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.
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Context:
Given that $f(z)$ satisfies the condition $\Im (f'(z)) = 6x(2y-1)$, with $f(0) = 3-2i$ and $f(1) = 6-5i$, find the value of $f(1+i)$. Provide your answer in the form $a + bi$.
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Context:
Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3} + 40n - 27$ for any positive integer $n$.
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Context:
Determine the tightest possible constant \( \epsilon_1 \) such that the inequality \( \left| 1 - \frac{1}{1+E[x]} - E \left[ 1 - \frac{1}{1+x} \right] \right| \leq \epsilon_1 \cdot \text{Var}(x) \) holds for a real-valued random variable \( X \geq 0 \).
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Context:
At Jefferson High School, there are 500 students enrolled. One hundred twenty students are in the orchestra, 190 are in band, and 220 are in chorus. If only 400 students are in orchestra, band, and/or chorus, how many students are in exactly two of these groups?
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Context:
? % of 480 = 193.2
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Context:
Suppose we have $10$ balls and $10$ colors. For each ball, we independently color it with one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal.
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Context:
How many ways can you line up the players of two soccer teams in a row so that no two players from the same team stand next to each other?
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Context:
Andy is mixing blue, green, white, and red paint in a 1 : 2 : 5 : 3 ratio. If he uses 6 liters of green paint and needs to convert the total amount of paint used into gallons, how many gallons of paint does he use total? (Note: 1 liter = 0.264172 gallons)
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Context:
Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \).
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Context:
Given the sequence ${a_n}$ that satisfies the equation $a_{n+1}+(-1)^{n}a_{n}=3n-1,(n∈N^{*})$, determine the sum of the first 40 terms of the sequence ${a_n}$.
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Context:
Determine the values of \( a > 0 \) for which the improper integral \( \int_{0}^{\infty}\frac{\sin x}{x^{a}} \, dx \) converges.
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Context:
Determine the integer $d$ such that the homotopy equivalence $\mu: S^m \wedge S^n \rightarrow S^{m+n}$ satisfies $\mu^*(\sigma^{m+n}1) = d \cdot \sigma^m1 \wedge \sigma^n1$, where $\mu^*: H^{m+n}(S^{m+n}) \rightarrow H^{m+n}(S^m \wedge S^n)$ is the induced homomorphism. Here, $\sigma^n 1$ is the image of the unit $1 \in H^0(S^0)$ under the $n$-fold suspension isomorphism $\sigma^n: H^0(S^0) \rightarrow H^n(S^n)$. Find the value of $d$.
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Context:
Given that the domain of function $f(x)$ is $\mathbf{R}$, $f''(x)$ is the derivative of $f(x)$, and $f''(x) - 2f(x) > 0$, with $f(\frac{1}{2}) = e$, where $e$ is the base of the natural logarithm, determine the solution set for the inequality $f(\frac{1}{2}\ln{x}) < x$. Express your answer in interval notation.
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Context:
Determine the greatest positive integer \( n \) for which there exist \( n \) nonnegative integers \( x_1, x_2, \ldots, x_n \), not all zero, such that for any choice of \( \varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n \) from the set \( \{-1, 0, 1\} \), not all zero, the sum \( \varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n \) is not divisible by \( n^3 \).
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Context:
Find the number of matrices \( A \) in \( F_4^{5 \times 3} \) such that \( \text{rank}_{F_4}(A) = 3 \).
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Context:
Consider numbers of the form \( {1a1} \) , where \( a \) is a digit. How many pairs of such numbers are there such that their sum is also a
 palindrome?
 Note: A palindrome is a number which reads the same from left to right and from right to left. Examples:353,91719.
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Context:
Let $x, y, a \in \mathbb{R}^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Determine the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.
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Context:
Given the function \(f(x) = \lim_{n\to \infty}\dfrac{\left(1- \cos \left(1- \tan \left(\dfrac {\pi}{4}-x\right)\right)\right)(x+1)^n+ \lambda\sin((n - \sqrt{n^2 -8n})x)}{x^2(x+1)^n+x}\), where \(x \neq 0\), determine the value of \(f(0) + 2\lambda\) if \(f(x)\) is continuous at \(x=0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the automorphism $\psi$ of the cyclic group $\Bbb Z_{30}$ such that $\psi(7) = 11$. What is the value of $\psi(x)$ for any $x \in \Bbb Z_{30}$?
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Context:
In triangle $ABC$, points $D$ and $E$ are on side $BC$. Perpendiculars from $D$ and $E$ to $BC$ intersect $BC$ at points $K$ and $L$, respectively. Given that $BK = R$ and $KL = 2R$, and that $DK$ and $EL$ are tangent to a circle centered at $B$, find the length of $BC$. Express your answer in terms of $R$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface area of the portion of the cylinder defined by \(y^2+z^2=2z\) that is intersected by the cone \(x^2=y^2+z^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among all fractions of the form $\frac{m}{n}$, where $m$ and $n$ are four-digit numbers with the same sum of digits, the greatest one was selected. Find it. Record your answer as an improper fraction (do not round or simplify!).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), the sides opposite to the angles \( \angle A \), \( \angle B \), and \( \angle C \) are denoted as \( a \), \( b \), and \( c \) respectively. If \( b^{2}=a^{2}+c^{2}-ac \), and \( c-a \) is equal to the height \( h \) from vertex \( A \) to side \( AC \), then find \( \sin \frac{C-A}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x,y,z,u)=\frac{24}{(1+x+y+z+u)^5}, x>0,y>0,z>0,u>0$ be a density function. Find $P(X+Y+Z+U \ge 1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An article is sold for Rs. 350, and 5% more is gained than by selling it for another price. If the cost of the article is Rs. 200, what is the lower selling price?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence of 2020 natural numbers is written in a row. Each of them, starting from the third number, is divisible by the previous one and by the sum of the two preceding ones.

What is the smallest possible value for the last number in the sequence?
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Context:
In a course conducted by Professor Jones, each student is on average absent for one day out of a 40-day course. What is the probability that out of any two randomly selected students, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all positive divisors of $40081$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. In how many ways can the set $\{1,2, \ldots, 12\}$ be divided into six disjoint two-element subsets such that each of them contains coprime numbers (i.e., numbers that do not have a common divisor greater than one)?
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Context:
Find the value of the limit:

$$\lim_{x\to0}\frac{\log_{\sin^2x}\cos x}{\log_{\sin^2(x/2)}\cos(x/2)}$$
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Context:
Nylah has her living room lights on a timer. Each evening, the timer switches the lights on at exactly 7:00 p.m., 7:30 p.m., 8:00 p.m., 8:30 p.m., or 9:00 p.m. Later in the evening, the timer switches the lights off at any random time between 11 p.m. and 1 a.m. What is the probability that the lights are on for 4 < t < 5 hours? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
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Context:
已知a、b、c、d取某些实数值时，方程$x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c x + d = 0$有4个非实数根，其中2个根的积为13+i,，另2个根的和为3+4i，这里i为虚数单位，求b.
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Context:
There is a strip with a length of 100, and each cell of the strip contains a chip. You can swap any two adjacent chips for 1 ruble, or you can swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the chips in reverse order?
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Context:
An event has some adults and 20 children. The event planner wants to make each table identical, with the same combination of adults and children and no people left over. The greatest number of tables the planner can set up is 4. How many adults are at the event?
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Context:
For which values of \( p \) does the triangle inequality \( ||x + y||_p \leq ||x||_p + ||y||_p \) hold true for the \( \ell^p \) norm in \( \mathbb{R}^n \)? Here, \( ||x||_p = \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} \) for \( p \in (0, \infty) \) and \( ||x||_\infty = \sup |x_k| \) for \( 1 \leq k \leq n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts wins the district; finally, the giraffe that wins in the majority of the districts is declared the winner of the final. The giraffe Tall won. What is the minimum number of voters who could have voted for Tall?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{\frac{\pi}{4}} Re^{-R^2 \sin x} \, dx \) as \( R \to \infty \) using a suitable substitution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define the sequence $a_1, a_2, a_3, \cdots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the $100$th term for which $a_n < 0$.
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Context:
In a rectangle \(ABCD\) with an area of 1 (including the boundary), there are 5 points, none of which are collinear. Find the minimum number of triangles, formed by these 5 points as vertices, that have an area not exceeding \(\frac{1}{4}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
When \( 2x^3 - 5x^2 - 12x + 7 \) is divided by \( 2x + 3 \), the quotient is \( x^2 - 4x + 2 \). What is the remainder?
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Context:
John climbs up some flights of stairs. Each flight is 10 feet. He then climbs a rope that is half that height. Finally, he climbs a ladder that is 10 feet longer than the rope. He went up 70 feet high. How many flights of stairs did he climb?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose we have an infinite number of counters (numbered $0,1,2,\dots$) which fire (i.e. increment by $1$) independently according to a Poisson process of rate $1$. All counters are initially switched off. At time $0$ we turn on counter $0$. Each time counter $0$ fires, the next counter which is still off turns on (e.g. the first time counter $0$ fires, counter $1$ turns on, the second time counter $0$ fires, counter $2$ turns on, etc.). What is the expected number of fires over all counters in the interval $[1,2]$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\begin{aligned} & \mathrm{Let~}ABCDE\text{ be a convex pentagon such that }AB=AE=CD=1,\angle ABC=\angle DEA=90^{\circ}\mathrm{~and~}BC+DE=1 \\ & \text{Compute the area of the pentagon.} \end{aligned}$
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Context:
Determine all real values of \( \alpha \) for which the improper integral \( \int_{0}^{\infty} x(x^2/2 + 1)^{\alpha} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice and Bob are on a 1 km long closed and dark train. They start at random positions and can only see their own and each other's information on their iPhone 133, which includes their facing direction, total walking distance, and whether they are at the front or end of the train. They can discuss their strategy before the game starts. Determine the minimum total walking distance, \( x \), that guarantees they will meet, with \( x \leq 1 \) km.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the highest dimension for which the classification of nilpotent Lie algebras is currently unknown.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The distance between Luga and Volkhov is 194 km, between Volkhov and Lodeynoe Pole is 116 km, between Lodeynoe Pole and Pskov is 451 km, and between Pskov and Luga is 141 km. What is the distance between Pskov and Volkhov?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate: $$\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$$
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Context:
A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor removes N marbles all at once. If he is to be sure that, no matter which choice of N marbles he removes, there are at least four marbles of one color and at least three marbles of another color left in the bag, what is the maximum possible value of N? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John has 20% more boxes than Jules. Jules has 5 boxes more than Joseph. Joseph has 80% fewer boxes than Stan. Stan has twice as many boxes as Lily. Martin has 150% of the boxes Jules has. Alice has 3 boxes less than half the boxes Martin has. Gabriel has the average number of boxes between Martin and Alice. Emma has the median number of boxes among John, Jules, Joseph, Stan, Lily, Martin, Alice and Gabriel. If Stan has 150 boxes, how many boxes does Emma have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \int_{-5}^5 \int_{-5}^5 -\frac{3}{2}|x+y| - \frac{3}{2}|x-y| + 15 \,\mathrm{d}x\,\mathrm{d}y \).
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Context:
Given the line $y=kx$ is tangent to the circle $C$: $(x-4)^{2}+y^{2}=r^{2}$, and the surface area of the geometric solid obtained by rotating circle $C$ around the $x$-axis for one week is $S=16π$, the value of $k$ is $\_\_\_\_\_\_$.
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Context:
定义 $x * y=\frac{x+y}{1+x y}$, 则 $(\cdots((2 * 3) * 4) \cdots) * 21=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
A product of five primes is of the form $A B C, A B C$, where $A, B$, and $C$ represent digits. If one of the primes is 491, find the product $A B C, A B C$.
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Context:
Each segment with endpoints at the vertices of a regular 100-gon is painted red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the products of the numbers at the endpoints are placed on the segments. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the largest possible result?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA=90^{\circ}$. Given that $AI=97$ and $BC=144$, compute the area of $\triangle ABC$.
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Context:
Consider a symmetric random walk on a graph constructed from two cycles, $C_{m}$ and $C_{n}$, which share exactly two consecutive common vertices. Let $S$ and $S'$ be the common vertices, $A$ be a vertex of $C_{m}$ not in $C_{n}$, and $B$ be a vertex of $C_{n}$ not in $C_{m}$. Given $m = 2 \times 10^{6} + 1$ with the $SS'A$-arc being a path of $10^{6} + 1$ edges and the other $SA$-arc being a path of $10^{6}$ edges, and $n = 4 \times 10^{6} + 1$ with the $SS'B$-arc being a path of $2 \times 10^{6} + 1$ edges and the other $SB$-arc being a path of $2 \times 10^{6}$ edges, calculate the probability that a symmetric random walk starting from $S$ ends at vertex $A$. Assume the walk ends immediately upon reaching $A$. What is this probability?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Gini index for the probability density function \( f(x) \) defined as \( f(x) = 1 \) for \( 0 \leq x \leq 1 \) and \( f(x) = 0 \) elsewhere.
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Context:
For every positive integeer  $n>1$ , let  $k(n)$  the largest positive integer  $k$  such that there exists a positive integer  $m$  such that  $n = m^k$ .

Find   $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$
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Context:
Find the period of the function: $$f(x) = \left\lfloor x + 2 \right\rfloor -
         \left\lfloor x - 1 \right\rfloor -
        3\left\lfloor -x \right\rfloor$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the Horner's Rule to calculate the value of the polynomial: $f(x) = 1 + x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 7x^7$ at $x = 2$, the value of $v_3$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a positive integer \( n \) as a "factorial tail" if there exists a positive integer \( m \) such that the decimal representation of \( m! \) ends in exactly \( n \) zeros. How many positive integers less than 1992 are not "factorial tails"?
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Context:
How many values of $\theta$ in the interval $0 < \theta \le 2\pi$ satisfy $1 - 3\sin\theta + 5\cos3\theta = 0$? Provide your answer as an integer.
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Context:
For any real number x, find the range of values for the real number k such that the inequality |x+2|+|x+1|>k is always true. Express your answer as an inequality using standard notation (e.g., <, >, ≤, ≥).
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Context:
The radius of a circle is increased by some percentage. Its area increases by 2.01%. By what percentage was the radius increased?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of ordered pairs of complex numbers \((u, v)\) such that \(uv = 10\) and such that the real and imaginary parts of \(u\) and \(v\) are integers.
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Context:
David and Evan each repeatedly flip a fair coin. David will stop when he flips a tail, and Evan will stop once he flips 2 consecutive tails. Find the probability that David flips more total heads than Evan.
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Context:
Calculate the integral $$\int_{C}{xy^{2}dy-yx^{2}dx}$$ where C is the circle $x^{2}+(y-1)^{2}=1$ using polar coordinates and Green's theorem.
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Context:
The vertices of a regular nonagon are colored such that:
1. Adjacent vertices are different colors.
2. If three vertices form an equilateral triangle, they are all different colors.

Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In rectangle PQRS, the ratio of ∠PSQ to ∠PQS is 1:5. Find the size of ∠QSR. Express your answer in degrees as a whole number.
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Context:
Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n$. Compute the number of positive integers $n$ at most $10^{4}$ that satisfy $$s(11 n)=2 s(n)$$
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Context:
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), \( f(2x) + f(2y) = f(x+y) f(x-y) \). Additionally, \( f(\pi) = 0 \) and \( f(x) \) is not identically zero. What is the period of \( f(x) \)?
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Context:
23. Using the digits $1,2,3,4$ (digits can be repeated), how many $n(n \geqslant 2)$-digit numbers can be formed that contain an odd number of 1s, an even number of 2s, and at least one 3?
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{\ln(n+1)^{n+1}}{\ln n^n} \right)^n. \]
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Context:
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by 2 and add 1 or he may multiply the stored value by 4 and add 3. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at 1 and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
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Context:
设 m 是正整数，n=2^m-1，P_n=\{1,2,\cdots,n\} 是数轴上 n 个点构成的集合. 一只蚱蜢在这些点上跳跃，每步从一个点跳到与之相邻的点. 求 m 的最大值，使得对任意 x,y\in P_n，从点 x 跳 2012 步到点 y 的跳法种数为偶数（允许中途经过点 x,y）.
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Context:
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. Calculate the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out). Express your answer as a simplified ratio.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a party with \( n + 1 \) couples, where handshakes between spouses are forbidden, a total of \( 2n + 1 \) guests (excluding the host and hostess) report \( 2n + 1 \) unique numbers of handshakes. Determine the number of handshakes the host made.
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Context:
A company sells a brand of cars in locations A and B, with profits (in units of ten thousand yuan) given by $L_1 = 5.06x - 0.15x^2$ and $L_2 = 2x$ respectively, where $x$ is the sales volume (in units of vehicles). If the company sells a total of 15 cars in both locations, calculate the maximum profit attainable. Express your answer as a single number in units of ten thousand yuan.
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Context:
Consider the Hilbert space \(\mathcal H=L^2(\mathbb R_{+})\) and the operator \(Af(x)=-f''(x)\) with domain \(D(A)=\{f\in H_2(\mathbb R_{+})\;\;| \;\;f'(0)+\beta f(0)=0\}\) for some \(\beta \in \mathbb R\). Determine the conditions on \(\beta\) for which the operator \(A\) has a point spectrum.
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Context:
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. A plane is tangent to these two spheres and intersects the cylinder to form an ellipse. Find the sum of the lengths of the major and minor axes of this ellipse.
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Context:
An up-right path from lattice points $P$ and $Q$ on the $xy$-plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from $(0,0)$ to $(10,3)$ does not intersect the graph of $y=x^2+0.5$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Among the four numbers $-\frac{1}{2}$, $-\frac{2}{3}$, $4$, $-5$, find the number with the smallest absolute value. Express your answer as a single number.
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Context:
Given three numbers \(x, y=x^{x}, \quad z=x^{x^{x}}\), where \(0.9<x<1.0\), determine the order of them in increasing sequence.
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Context:
Side \(AB\) of triangle \(ABC\) has length \(8\) inches. Line \(DEF\) is drawn parallel to \(AB\) so that \(D\) is on segment \(AC\), and \(E\) is on segment \(BC\). Line \(AE\) extended bisects angle \(FEC\). If \(DE\) has length \(5\) inches, then find the length of \(CE\) in inches. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
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Context:
If it is assumed that 60 percent of those who receive a questionnaire by mail will respond initially, and the response rate declines exponentially by 20% for each of the 2 follow-up attempts. If 750 responses are needed, and there is a deadline of 30 days to receive all responses, what is the minimum number of questionnaires that should be mailed initially to achieve the desired number of responses?
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Context:
A train with varying speeds crosses three poles in the following way:
- At the first pole, the train is moving at a speed of 50 km/h and takes 18 seconds to cross it.
- At the second pole, the speed increases to 60 km/h, and it takes 20 seconds to cross it.
- At the third pole, the speed further increases to 70 km/h, and it takes 22 seconds to cross it.

The distances between the poles are 500 meters from the first to the second, and 800 meters from the second to the third. Find the total length of the train.
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Context:
There are 100 silver coins, ordered by weight, and 101 gold coins, also ordered by weight. It is known that all coins have different weights. We have a two-pan balance scale that allows us to determine which of any two coins is heavier. How can we find the coin that ranks 101st in weight among all the coins with the minimum number of weighings?
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Context:
In a test, the teacher gave five problems and awarded a grade equal to the number of problems solved. All students, except Peter, solved the same number of problems, and Peter solved one more problem. Nine students solved the first problem, seven solved the second, five solved the third, three solved the fourth, and one solved the fifth.

How many grades of four and five were given on the test?
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Context:
Sets \(A, B\), and \(C\) satisfy \(|A| = 92\), \(|B| = 35\), \(|C| = 63\), \(|A \cap B| = 16\), \(|A \cap C| = 51\), and \(|B \cap C| = 19\). Compute the number of possible values of \(|A \cap B \cap C|\).
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Context:
Two cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a $\diamondsuit$ and the second card is an ace?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the volume of solid $T$ defined by the inequalities $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$.
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Context:
Given $\sin\left(\frac{\pi}{3}+\frac{\alpha}{6}\right)=-\frac{3}{5}$, $\cos\left(\frac{\pi}{12}-\frac{\beta}{2}\right)=-\frac{12}{13}$, $-5\pi < \alpha < -2\pi$, $-\frac{11\pi}{6} < \beta < \frac{\pi}{6}$,

Find the value of $\sin \left(\frac{\alpha }{6}+\frac{\beta }{2}+\frac{\pi }{4}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the following expression: $$(-(-2)+(1+\pi)^{0}-|1- \sqrt {2}|+ \sqrt {8}-\cos 45^{\circ})$$.
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Context:
John and Ingrid pay 30% and 40% tax annually, respectively. If John makes $57,000 and Ingrid makes $72,000, what is their combined tax rate?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. Let $B_{n}$ be the set of all binary strings of length $n$. For a binary string $s_{1} s_{2} \ldots s_{n}$, we define its twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_{b}$ with $1-s_{b}$. A string $a$ is said to be a descendant of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_{n}$ is called divided if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_{n}$. Please provide the value when $n = 4$. The answer should be an integer.
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Context:
Let \[P(x) = (3x^4 - 39x^3 + ax^2 + bx + c)(4x^4 - 96x^3 + dx^2 + ex + f),\] where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\{1, 2, 2, 3, 3, 4, 6\}.$ Find $P(7).$
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Context:
John makes $30 a week from his job. After six months, he earns a raise and begins to make $40 a week. Four months later, he earns another raise and starts making $50 a week. What is the total percentage increase in John's weekly earnings after ten months?
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Context:
Let $X$ be the number of dice tosses required to obtain the first even number, and $Y$ the number of tosses for the first number greater than 3. Both $X$ and $Y$ are geometric random variables with a mean of 2. Determine the probability $P(X=Y)$.
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Context:
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is
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Context:
For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \leq i<$ $j \leq n$ is called \underline{interesting} if there exists a pair ( $a_{k}, a_{l}$ ) of integers with $1 \leq k<l \leq n$ such that
$$\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2$$
For each $n \geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$. Please provide the value of this expression when $n = 5$.
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Context:
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
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Context:
设函数 f(z)（z 为复数）满足 f(f(z))=(z\overline{z}-z-\overline{z})^2，若 f(1)=0，则 |f(i)-1|=__________.
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Context:
Find the greatest constant $C$ so that for all real numbers $x_1, x_2,\ldots,x_6$ the inequality $$(x_1 + x_2 + ...+ x_6)^2 \geq C \cdot (x_1 (x_2 + x_3) + x_2 (x_3 + x_4) + \ldots+ x_6 (x_1 + x_2))$$ applies.
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Context:
Given a linear function $y=x-k$, if for any value of the independent variable $x$ in the range $x \lt 3$, the corresponding function value $y$ is less than $2k$, then the range of values for $k$ is ______.
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Context:
Between a certain natural number and 81, there are 15 consecutive natural numbers. What is this certain number if it is a natural number less than 81?
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Context:
Reggie is playing marbles with his friend. His friend arrives with some marbles. Each game, they bet ten marbles and whoever wins gets to keep all the marbles. After 9 games, Reggie has 90 marbles and lost 1 game. How many marbles did his friend arrive with?
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Context:
the regular price per can of a certain brand of soda is $ 0.40 . if the regular price per can is discounted 15 percent when the soda is purchased in 24 - can cases , what is the price of 100 cans of this brand of soda purchased in 24 - can cases ?
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Context:
Given that $c > 0$, suppose proposition $p$: the function $y=(2c-1)^{x}$ is monotonically decreasing on $\mathbb{R}$; proposition $q$: the solution set of the inequality $x+|x-2c| > 1$ is $\mathbb{R}$. If $p \vee q$ is true and $p \wedge q$ is false, find the range of values for $c$.
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Context:
Five runners ran a relay race. If the first runner had run twice as fast, they would have spent 5% less time. If the second runner had run twice as fast, they would have spent 10% less time. If the third runner had run twice as fast, they would have spent 12% less time. If the fourth runner had run twice as fast, they would have spent 15% less time. By what percentage less time would they have spent if the fifth runner had run twice as fast?
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Context:
Determine the value of the integral $\int_0^{\infty} \frac{dx}{(1+x^n)(1+x^2)}$ for any $n \neq 0$.
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Context:
Round up the smallest six-digit number that you can make by using the six number cards 0, 1, 3, 5, 7, and 9 once, to the thousandth place.
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Context:
In unit square \(ABCD\), points \(E, F, G\) are chosen on sides \(BC, CD,\) and \(DA\) respectively such that \(AE\) is perpendicular to \(EF\) and \(EF\) is perpendicular to \(FG\). Given that \(GA = \frac{404}{1331}\), find all possible values of the length of \(BE\).
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Context:
Find the limit of the function $$ \lim_{x\rightarrow 0}\left( \frac{1}{x\cos(x)\tan(x\cos(x))}-\frac{1}{x^2\cos^2(x)} \right) $$
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Context:
A circle centered at $O$ has radius 1 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle
AOB=\theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then express $OC$ in terms of $s$ and $c,$ where $s = \sin \theta$ and $c = \cos \theta.$

[asy]
pair A,B,C,O;
O=(0,0);
A=(1,0);
C=(0.6,0);
B=(1,2);
label("$\theta$",(0.1,0),NE);
label("$O$",O,S);
label("$C$",C,S);
label("$A$",A,E);
label("$B$",B,E);
draw(A--O--B--cycle,linewidth(0.7));
draw(C--B,linewidth(0.7));
draw(Circle(O,1),linewidth(0.7));
[/asy]
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Context:
Calculate the limit: $$ \lim_{n\to\infty} \frac{2^n + \left(1+\frac{1}{n^2}\right)^{n^3} + \frac{4^n}{n^4}}{\frac{4^n}{n^4} + n^3\cdot 3^n} $$
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Context:
Determine the number of distinct necklaces that can be formed using 15 identical beads and 3 different gems, taking into account rotations and reflections.
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Context:
Task 1. We have 1000 balls in 40 different colors, with exactly 25 balls of each color. Determine the smallest value of $n$ with the following property: if you randomly arrange the 1000 balls in a circle, there will always be $n$ balls next to each other where at least 20 different colors occur.
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Context:
Given the sequence $\{a\_n\}$ that satisfies: $(a\_1=-13$, $a\_6+a\_8=-2)$, and $(a_{n-1}=2a\_n-a_{n+1}\ (n\geqslant 2))$, find the sum of the first 13 terms of the sequence $\left\{\frac{1}{a\_n a_{n+1}}\right\}$.
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Context:
Solve for $x$ in the equation
\[2^{(16^x)} = 16^{(2^x)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Households A, B, and C plan to subscribe to newspapers. There are 5 different types of newspapers available. Each household subscribes to two different newspapers. It is known that each pair of households shares exactly one common newspaper. How many different subscription ways are there for the three households?
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Context:
Given the real numbers \( x_1, x_2, \ldots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left|x_k - x_{k+1}\right| = 2001 \). Let \( y_k = \frac{1}{k} \left( x_1 + x_2 + \cdots + x_k \right) \) for \( k = 1, 2, \ldots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_k - y_{k+1} \right| \).
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Context:
Find the area of the region defined by the inequality: \( |y - |x - 2| + |x|| \leq 4 \).
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Context:
Evaluate $$\lim_{x \to 0}\frac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}$$
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Context:
Determine the maximum value of the sum
$$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$
over all sequences $a_{1},a_{2},a_{3},\cdots\$ of nonnegative real numbers satisfying
$$\sum_{k=1}^{\infty}a_{k}=1.$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Let \( S \) be the set of all non-negative integers with at most \( n \) digits in decimal, and \( S_{k} \) be the subset of \( S \) consisting of elements whose digit sum is less than \( k \). Determine the value of \( n \) for which there exists a \( k \) such that \( |S| = 2|S_{k}| \).
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Context:
Let $N = 6 + 66 + 666 + \ldots + \underbrace{666\ldots66}_{100\text{ sixes}}$, where there are one hundred $6$'s in the last term in the sum. How many times does the digit $7$ occur in the number $N$?
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Context:
Allie has 18 toys in total worth $108. There are 3 types of toys: Type A, Type B, and Type C. We know that one Type A toy is worth $14, and there are 4 Type A toys. Moreover, there are 7 Type B toys and they all have the same value. Finally, there are 7 Type C toys and they also have the same value, but it is different than the value of Type B toys. If we know that the combined value of all Type B and Type C toys is $84, how much does one Type B toy cost?
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Context:
Let point \( P \) be on the ellipse \( \frac{x^{2}}{5}+y^{2}=1 \), and let \( F_{1} \) and \( F_{2} \) be the two foci of the ellipse. If the area of \( \triangle F_{1} P F_{2} \) is \( \frac{\sqrt{3}}{3} \), find \( \angle F_{1} P F_{2} \).
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Context:
Six different digits from the set
$\{ 1,2,3,4,5,6,7,8,9\}$
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is $23$ and the sum of the entries in the horizontal row is $12$.
What is the sum of the six digits used?
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Context:
定义“$\frac{{{\left| x \right|}^{n}}}{{{an}}}+\frac{{{\left| y \right|}^{n}}}{{{bn}}}=1\left( a>b>0,n\in {{N}^{*}},n\ge 3 \right)$”代表的曲线为“超椭圆”，设$a$、$b$为常数，设超椭圆的周长为${{C}_{n}}$，那么$\underset{n\to \infty }{\mathop{\lim }}\,{{Cn}}=$______.
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Context:
On a grid sheet, a $1 \times 5$ rectangle is initially shaded. Each minute, all unshaded cells that have at least one shaded neighboring cell are shaded. For example, after one minute, 17 cells will be shaded. How many cells will be shaded after 5 minutes?
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Context:
The sphere with radius 1 and center $(0,0,1)$ rests on the $xy$-plane.  A light source is at $P = (0,-1,2).$  Then the boundary of the shadow of the sphere can be expressed in the form $y = f(x),$ for some function $f(x).$  Find the function $f(x).$
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Context:
The 5th root of what integer power of 8 is closest to 100?
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Context:
Given that $\angle BAC = 90^{\circ}$ and the quadrilateral $ADEF$ is a square with side length 1, find the maximum value of $\frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$.
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Context:
Divide the plane into 1000 regions using the fewest possible number of lines.
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Context:
In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$.
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Context:
Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is chocolate if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?
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Context:
A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has.
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Context:
Consider the sum:
\[
  S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}}
\]
Determine \( \lfloor S_{4901} \rfloor \). Recall that if \( x \) is a real number, then \( \lfloor x \rfloor \) (the floor of \( x \)) is the greatest integer that is less than or equal to \( x \).
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Context:
已知集合 $A=\left\{x \mid \log _{a}(a x-1)>1\right\}$ 。若 $3 \in A$, 请给出 a 的取值范围中所有有限端点的和的6倍。
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Context:
In the convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is the angle \(BCA\)?
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Context:
A person distributed 20% of his income to his 3 children each. He deposited 30% of his income to his wife's account. He donated 5% of the remaining amount to an orphan house. Finally, he has $40,000. What is his total income?
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Context:
A total of 2021 coloured koalas are arranged in a row and are numbered from 1 to 2021. Each koala is coloured red, white, or blue. Amongst any three consecutive koalas, there are always koalas of all three colours. Sheila guesses the colours of five koalas: Koala 2 is white; Koala 20 is blue; Koala 202 is red; Koala 1002 is blue; Koala 2021 is white. Only one of her guesses is wrong. Find the number of the koala whose colour she guessed incorrectly. Express your answer as a whole number.
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Context:
The sum of the two base angles of a trapezoid is 90 degrees. The length of the upper base is 5, and the length of the lower base is 11. Find the length of the segment connecting the midpoints of the two bases. Express your answer as a whole number.
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Context:
Let  $M$  be a set of  $99$  different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of  $M$  inside in. What is the maximum number of obtuse angles formed by two rays in  $M$ ?
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Context:
Let \( M = \{1, 2, 3, \ldots, 40\} \). Find the smallest positive integer \( n \) such that \( M \) can be partitioned into \( n \) pairwise disjoint subsets, and for any 3 numbers \( a, b, c \) (not necessarily distinct) within the same subset, \( a \neq b + c \).
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Context:
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Provide your answer as a simplified fraction in the form \frac{k}{m}. Please give the value of k + m.
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Context:
求最小正整数 $n$, 使得任取 $n$ 个不同的正整数中必存在 8 个不同的正整数,其和被 8整除
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Context:
My grandson is about as many days as my son in weeks, and my grandson is as many months as I am in years. My grandson, my son, and I together are a certain number of years. If I am 72 years old, how many years are we together?
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Context:
Given a sequence of 99 terms \( (p_1, p_2, \cdots, p_{99}) \) with a Cesaro sum of 1000, find the Cesaro sum of a new 100-term sequence \( (1, p_1, p_2, \cdots, p_{99}) \). Express your answer as a single number.
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Context:
Given that the incircle of quadrilateral ABCD touches sides AB, BC, CD, and DA at points E, F, G, and H respectively, and AF, DF, and EG intersect at points M and N. If BF = CF = 5 and EG = 6, then find the length of MN. Express your answer as a whole number.
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Context:
In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$ A A, A B, A C, B B, B C, A D, C C, B D, C D, D D $$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A + A, A + B, A + C, B + B$, etc., corresponding to the order above will be in descending magnitude. Please provide the sum of A + B + C + D for the least positive integers that satisfy this condition.
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Context:
Modern widescreen television screens are rectangles and are typically measured by the length of their diagonals. The ratio of the horizontal length to the height in these widescreen televisions is $16:9$. Calculate the horizontal length (in inches) of a ``50-inch'' widescreen television screen.
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Context:
A rectangular grazing area is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 12 meters along the fence, including the two posts where the fence meets the rock wall. What is the minimum number of fence posts required to fence an area 36 m by 60 m? Express your answer as a whole number.
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Context:
Find the maximum number of $1$'s in a square matrix of size $n$ such that the matrix remains invertible, given that all the entries are either $0$ or $1$ and not all the diagonal entries are $0$.
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Context:
Given two power series, where one converges for \(|x| < 6\) and the other converges for \(|x| < 4\), determine the interval of convergence for the product of these two series.
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