Context:
A lead ball with a radius of $R$ falls into a sandpit, leaving a hole with an external diameter of 24 cm and a depth of 8 cm. The radius of the ball is
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Context:
15 men are equal to as many women as are equal to 12 boys. All of them together earn Rs. 432. What are the men's wages?
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Context:
Six people, a mix of knights (who always tell the truth) and liars (who always lie), sat around a round table. Each of them was given a coin. Then, each person passed their coin to one of their two neighbors. After that, 3 people said, "I have one coin," and the other 3 said, "I have no coins." What is the maximum number of knights that could have been sitting at the table?
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Context:
Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
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Context:
若存在实数 $a$ 及正整数 $n$ 使得 $f(x)=\cos 2 x-a \sin x$ 在 $(0, n \pi)$ 内恰有 $2022$ 个零点, 则满足条件的正整数 $n$ 的值有 $\qquad$个.
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Context:
Evaluate the integral: $$\int_0^1 \frac{x\ln x}{(1+x^2)^2} \, dx$$
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Context:
双曲线 $C$ 的方程为 $x^{2}-\frac{y^{2}}{3}=1$, 左、右焦点分别为 $F_{1}, F_{2}$, 过点 $F_{2}$ 作一直线与双曲线 $C$ 的右半支交于点 $P, Q$, 使得 $\angle F_{1} P Q=90^{\circ}$, 则 $\triangle F_{1} P Q$ 的内切圆半径的原始答案为 $\sqrt{m} - n$ 的形式，请给出 $m + n$ 的值。
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Context:
## Task Condition

Find the derivative of the specified order.

$$
y=\left(2 x^{3}+1\right) \cos x, y^{V}=?
$$
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Context:
Evaluate the sum $\sum\limits_{n=1}^\infty \left(\frac{1}{2}\right)^n \prod\limits_{k=0}^{n-1}\frac{j+\frac{1}{2}}{j+1}$.
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Context:
Let $M=\left\{1,2, \cdots, 2^{m} n\right\}$ (where $m$ and $n$ are positive integers) be a set consisting of continuous $2^{m}n$ positive integers. Find the minimum value of the positive integer $k$ such that any $k$-element subset of $M$ contains $m+1$ numbers $a_1, a_2, \cdots, a_{m+1}$ that satisfy $a_i$ divides $a_{i+1}$ for $i=1,2, \cdots, m$.
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Context:
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?
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Context:
A company had a sales revenue of 10 million yuan last year, with annual costs consisting of 5 million yuan in production costs and 2 million yuan in advertising costs. If the annual profit must be taxed at a rate of p%, and the portion of the annual advertising expense that exceeds 2% of the annual sales revenue is also taxed at p%, with no other taxes applied. It is known that the company paid a total of 1.2 million yuan in taxes last year. Calculate the tax rate p%. Express your answer as a percentage.
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Context:
Evaluate the integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$.
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Context:
A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home by bus in the afternoon $15$ times, and commuted by train (either morning or afternoon) $9$ times. Find $x$.
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Context:
There were 12 seats prepared for the jury at a round table, each with a name indicated on it. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his designated seat but in the next one clockwise. Each jury member arriving at the table afterward would either sit in their designated seat, or, if it was already occupied, they would walk around the table clockwise and sit in the next available seat. The resulting seating arrangement of the jury members depends on the order in which they approached the table. How many different ways can the jury be seated?
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Context:
11.3. The teacher is going to give the children a problem of the following type. He will inform them that he has thought of a polynomial $P(x)$ of degree 2017 with integer coefficients, the leading coefficient of which is 1. Then he will tell them $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$, and separately inform them of the value of the expression $P\left(n_{1}\right) \cdot P\left(n_{2}\right) \cdot \ldots \cdot P\left(n_{k}\right)$. Based on this data, the children are supposed to find the polynomial that the teacher could have thought of. For what smallest $k$ can the teacher formulate such a problem so that the polynomial found by the children necessarily coincides with the one thought of?

(G. Zhukov)
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Context:
Nine identical books cost 11 rubles and some kopecks, and thirteen identical books cost 15 rubles and some kopecks. Determine the exact cost of one book.
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\boxed{1 \text{ ruble } 23 \text{ kopeks}}


Context:
In the figure, the center of square \(ABCD\) is \(O\), and its area is \(1989 \text{ cm}^2\). Point \(P\) is a point inside the square such that \(\angle OPB = 45^\circ\) and \(PA : PB = 5 : 14\). Find \(PB\) in centimeters.
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Context:
Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this operation necessary to sort $A$?
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Context:
For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed? Express your answer as a whole number.
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Context:
Find
$$
\lim _{n \rightarrow \infty}\left[\frac{1}{n^{5}} \sum_{h=1}^{n} \sum_{k=1}^{n}\left(5 h^{4}-18 h^{2} k^{2}+5 k^{4}\right)\right].
$$
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Context:
A clothing store found in its sales that a certain brand of clothing can sell an average of 20 pieces per day, with a profit of $40 per piece. In order to prepare for "Double Eleven," the mall decided to take appropriate price reduction measures to increase profits and reduce inventory. According to market research, if the price of each piece of clothing is reduced by $4, then an average of 8 more pieces can be sold per day. If you want to make a profit of $1200 per day on average by selling this type of clothing, how much should the price of each piece of clothing be reduced by?
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Context:
Three male students and three female students, a total of six students, stand in a row. If female students do not stand at the end of the row, and female students A and B are not adjacent to female student C, then how many different arrangements are possible? Express your answer as a single integer.
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Context:
Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \cdot a_5 = -15$, find the value of $a_{12}$.
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Context:
Consider the set  $A=\{1,2,3\ldots ,2^n\}, n\ge 2$ . Find the number of subsets  $B$  of  $A$  such that for any two elements of  $A$  whose sum is a power of  $2$  exactly one of them is in  $B$ .
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Context:
Consider the set \( S = \{1, 2, 3, \ldots, 98\} \). Determine the smallest natural number \( n \) such that in any subset of \( S \) with \( n \) elements, it is possible to select 10 numbers where, no matter how these 10 numbers are divided into two groups of 5, one group will contain a number that is relatively prime to the other four numbers, and the other group will contain a number that is not relatively prime to the other four numbers.
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Context:
Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

2. No digit occurs more than once in the same integer.

3. The digits in each integer are in increasing order.

4. Any two integers have at least one digit in common (possibly at different positions).

5. There is no digit which appears in all the integers.
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Context:
The base of a right rectangular prism is a rhombus with a side length of 2 and an angle of $30^{\circ}$. The height of the prism is 1. A cross-section is made through the base edge at an angle of $60^{\circ}$ to the base. Find the area of this cross-section.
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Context:
Find the largest value of the non-negative integer (a) for which:
$$ \displaystyle \lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4 $$
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Context:
Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$, and $(5,8,12)$ as shown in the figures below. He can join two pieces by sticking exactly the sides of the same length together. For example, he can join the side 10 of the first piece with the side 10 of the second piece, but he cannot join the side 10 of the first piece with the side 8 of the third piece because they do not have the same length. What is the greatest perimeter Pedrinho can obtain by joining the three pieces?

(Note: The figures provided in the original problem have been omitted for text-based translation purposes.)
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Context:
There are  $ 2010 $  people sitting around a round table. First, we give one person  $ x $  a candy. Next, we give candies to  $1$  st person,  $1+2$  th person,  $ 1+2+3$  th person,  $\cdots$  , and  $1+2+\cdots + 2009 $  th person clockwise from  $ x $ . Find the number of people who get at least one candy.
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Context:
Given two lines $l_{1}$: $mx+y+2m-3=0$ and $l_{2}$: $mx+y-m+1=0$, the maximum value of the distance between line $l_{1}$ and $l_{2}$ is ____.
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Context:
在 $\triangle A B C$ 中, $A B=A C=2, B C$ 边有 100 个不同的点 $P_1, P_2, \cdots, P_{100}$, 记 $m_i=A P_2^2+$ $B P_i \cdot P_i C(i=1,2, \cdots, 100)$, 求 $m_1+m_2+\cdots+m_{10}$ 的值.
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Context:
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^\circ$, $\angle B = 105^\circ$, $\angle C = 90^\circ$ and $AB = 2$, $BC = CD = DE = \sqrt{2}$. If the length of $AE$ is $\sqrt{a} - b$ where $a$ and $b$ are integers, what is the value of $a + b$?
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Context:
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ $\mathbf{d}$ be four distinct unit vectors in space such that
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{d} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{7}.\]
Find $\mathbf{a} \cdot \mathbf{d}$.
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Context:
An n-pointed star is formed by a series of equal-length segments, with each segment measuring 2n. The angles A1, A2,..., An are equal, and the angles B1, B2,..., Bn are equal. If angle A1 = angle B1 - 10 degrees, find the value of n. Express your answer as a whole number.
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Context:
Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can be assembled into $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are identical after rotation are considered the same).
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Context:
Find all natural numbers n that satisfy the equation 1*2 + 2*3 + 3*4 +... + n*(n+1) = 3n^2 - 3n + 2. Express your answer as a list of natural numbers.
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Context:
Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each subsequent placement must share at least one edge with a piece placed in the previous placement, and apart from sharing edges, there should be no other overlaps (see diagram). After the 20th placement, the total number of equilateral triangle pieces used is:
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Context:
In the sequence below, each △ represents a number, and starting from the third term, each number is the sum of the two preceding numbers. Calculate the sum of all 9 numbers listed: △, △, △, △, 7, △, △, △, 47. Express your answer as a single integer.
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Context:
The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150.
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Context:
在由实数构成的 $n \times n$ ($n \geq 2$) 数表中，第 $i$ 行 $n$ 个数之和为 $s_i$，第 $j$ 列 $n$ 个数之和为 $t_j$，记 $a_{ij} = s_i - t_j$，$i,j = 1, 2, \dots, n$。求 $n^2$ 个数 $a_{ij}$ ($i,j = 1, 2, \dots, n$) 中正数个数的最大值。
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Context:
A conical flask contains some water. When the flask is oriented so that its base is horizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep. When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2 inches deep. What is the height of the cone?
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Context:
Assign each letter a digit from 1, 2, 3, 4, 5 such that all the inequalities are satisfied.

$$
3>\mathrm{A}>\mathrm{M}<\mathrm{E}<\mathrm{H}<\mathrm{A}
$$

Different letters must correspond to different digits. Write the number ZAMENA as the answer.
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Context:
5. Ms. Olga Ivanovna, the class teacher of 5B, is staging a "Mathematical Ballet." She wants to arrange the boys and girls so that there are exactly 2 boys 5 meters away from each girl. What is the maximum number of girls that can participate in the ballet, given that 5 boys are participating?
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Context:
Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles?
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Context:
The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
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Context:
设实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 5 } \in [ - 1 , 1 ]$，求下列表达式的最小值

$P = x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 } + \cdots + x _ { 2 0 0 4 } x _ { 2 0 0 5 } + x _ { 2 0 0 5 } x _ { 1 }$
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Context:
3. Encipher the message SURRENDER IMMEDIATELY using the affine transformation $C \equiv 11 P+18(\bmod 26)$.
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Context:
Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9) > f(16)$. The probability that $f(1) > f(16) > f(25)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1, 2, \ldots, 100\}$.
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Context:
Let \(X\), \(Y\), and \(Z\) be points on the sides \(BC\), \(AC\), and \(AB\) of \(\triangle ABC\), respectively, such that \(AX\), \(BY\), and \(CZ\) concur at point \(O\). The area of \(\triangle BOC\) is \(a\). If \(BX : XC = 2 : 3\) and \(CY : YA = 1 : 2\), what is the area of \(\triangle AOC\)?
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Context:
Find the sum of all values of  $a$  such that there are positive integers  $a$  and  $b$  satisfying  $(a - b)\sqrt{ab} = 2016$ .
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Context:
In a geometric sequence $\{a_n\}$, $a_1=1$, and the sum of the first $n$ terms is denoted by $S_n$. It is given that $S_7 - 4S_6 + 3S_5 = 0$. Find the value of $S_4$.
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Context:
Two sides of a regular $n$-gon are extended to meet at a $28^{\circ}$ angle. What is the smallest possible value for $n$?
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Context:
There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.
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Context:
A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds.  Leah picks a random three-second time interval to watch the light.  What is the probability that the color changes while she is watching?
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Context:
In tetrahedron $ABCD$, $AD > AB$, $AD \perp AB$, $AD \perp AC$, $\angle BAC = \frac{\pi}{3}$. Let the areas of triangles $ADB$, $ADC$, $ABC$, and $BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively. It is known that $S_{1} + S_{2} = S_{3} + S_{4}$. Find the value of $\frac{S_{3}}{S_{1}} + \frac{S_{3}}{S_{2}}$.
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Context:
On an $xy$ plane, how many $2n$-move paths can be made that begin and end at $(0,0)$ but pass through $(k,n-k)$ with $0\leq k\leq n$?
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Context:
Given that the solution set of the inequality $(ax-a^{2}-4)(x-4) \gt 0$ with respect to $x$ is $A$, and $A$ contains a total of $n$ integers, the minimum value of the real number $a$ is ____.
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Context:
The graph of the power function $y=f(x)$ passes through the point $(-2, -\frac{1}{8})$. Find the value of $x$ that satisfies $f(x)=27$.
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Context:
In the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\).
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Context:
A sequence $\{a_n\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\displaystyle\lim_{n \to \infty} a_n$?
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Context:
Given natural numbers \( k \) and \( n \) such that \( 1 < k \leq n \). What is the smallest \( m \) for which the following statement holds: for any placement of \( m \) rooks on an \( n \times n \) board, it is always possible to select \( k \) rooks from these \( m \) such that no two of the selected rooks can attack each other?
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Context:
At the base of the pyramid $S A B C$ lies an equilateral triangle $A B C$, and all lateral faces have equal areas. The edge $S A$ is equal to 2, and the edge $S B$ is equal to $\sqrt{2}$. A section of the pyramid is made through vertex $B$ perpendicular to the edge $S C$. Find the area of this section.
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Context:
The sequence \(\{a_n\}\) is defined as follows: \[ a_n = \sqrt{1 + \left(1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left(1 - \frac{1}{n} \right)^2}. \] Calculate the value of the expression given below: \[ \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}. \] The original answer is in the form \(\sqrt{m} - 1\). Please provide the sum of the integer part of \(\sqrt{m}\) and 1.
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Context:
In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remaining two companies' flights. What is the minimum number of airline routes in the country?
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Context:
Natural numbers $1, 2, 3, \cdots, 2014, 2015, 2016$ are arranged in a circle in clockwise order.
Starting from number 1, we perform the following operations in clockwise order.

Step 1: Cross out 1 and keep 2.

Step 2: Cross out 3 and 4, and keep 5.

Step 3: Cross out 6, 7, and 8, and keep 9.

Step 4: Cross out 10, 11, 12, and 13, and keep 14;

and so on: in the nth step, first cross out n numbers, then keep 1 number. Repeat these operations until all numbers are crossed out.

What is the last number to be crossed out?
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Context:
Given the universal set $U=\{a^{2}-2,2,1\}$ and set $A=\{a,1\}$, find the complement of $A$ with respect to $U$, denoted as $∁\_U A$.
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Context:
Find the anti-derivative of $\frac{e^{2x}-1}{e^{2x}+1}$.
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Context:
Let point \( P \) lie on the face \( ABC \) of a tetrahedron \( ABCD \) with edge length 2. The distances from \( P \) to the planes \( DAB \), \( DBC \), and \( DCA \) form an arithmetic sequence. Find the distance from \( P \) to the plane \( DBC \).
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Context:
Given a function $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\;\;∀n \geq 0 $$ I would like to find a function $g(n)$ such that $f ∈ Ө(g(n))$.
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Context:
For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which
\[n(A) + n(B) + n(C) = n(A \cup B \cup C)\]
and
\[|A| = |B| = 100,\]
then find the minimum possible value of $|A \cap B \cap C|$.
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Context:
Find the smallest positive integer \( n \) that satisfies the following condition: For any finite set of points \( A \) in the plane, if any \( n \) points in \( A \) can always be covered by two lines, then there exist two lines that can cover all the points in \( A \).
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Context:
Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$
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Context:
5. In a convex quadrilateral $A B C D$, the areas of triangles $A B D$ and $B C D$ are equal, and the area of $A C D$ is half the area of $A B D$. Find the length of the segment $C M$, where $M$ is the midpoint of side $A B$, if it is known that $A D=12$.
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Context:
The supermarket sold two types of goods, both for a total of 660 yuan. One item made a profit of 10%, while the other suffered a loss of 10%. Express the original total price of these two items using a formula.
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Context:
7.1. Two wheels of radii $r_{1}$ and $r_{2}$ roll along a straight line $l$. Find the set of points of intersection $M$ of their common internal tangents.
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Context:
Find the smallest positive integer  $n$  for which one can select  $n$  distinct real numbers such that each of them is equal to the sum of some two other selected numbers.

*Proposed by Anton Trygub*
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Context:
On Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on January 15th. On how many days in the period from Thursday, January 1, 2015, until (and including) Saturday, December 31, 2016, is it possible for Anna to arrange all her books on all her shelves such that there is an equal number of books on each shelf?
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Context:
There are 4 soap bars shaped like rectangular prisms, with lengths of 10 cm, widths of 5 cm, and heights of 3 cm. If these 4 soap bars are placed in a packaging box, what is the minimum amount of material in square centimeters needed for the box?
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Context:
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.
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Context:
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
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Context:
Find the largest positive integer solution of the equation $\left\lfloor\frac{N}{3}\right\rfloor=\left\lfloor\frac{N}{5}\right\rfloor+\left\lfloor\frac{N}{7}\right\rfloor-\left\lfloor\frac{N}{35}\right\rfloor$.
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Context:
A chemical plant plans to improve the technology of a chemical product and now decides to optimize the processing temperature. The experimental range is set to 60～81℃, with an accuracy requirement of ±1℃. If the technicians are preparing to use the fractional method for optimization, the maximum number of experiments needed to find the optimal temperature is \_\_\_\_\_\_.
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Context:
Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$.
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Context:
Embed $R^2$ in the projective plane $RP^2$ by the map $(x, y) → [1, x, y]$. Find the point of intersection in $RP^2$ of the projective lines corresponding to the parallel lines $y = mx$ and $y = mx + c$ in $R^2$ (where $c \neq 0$).
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Context:
How many Carmichael numbers are there less than 100,000?
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Context:
Mouse Jerry decided to give cat Tom a birthday pie in the shape of an $8 \times 8$ square. He put fish in three pieces marked with the letter "P", sausage in two pieces marked with the letter "K", and in one piece added both fish and sausage, but did not mark this piece (all other pieces are without filling). Jerry also told Tom that in any $6 \times 6$ square there are at least 2 pieces with fish, and in any $3 \times 3$ square at most one piece with sausage.

What is the minimum number of pieces of pie Tom needs to eat to ensure he gets a piece with both fish and sausage?

|  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | P |  |  |  |  |  |  |
|  | K |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  | P | K |  |  |
|  |  |  |  |  |  |  |  |
|  | P |  |  |  |  |  |  |
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Context:
Given $m\in R$, find the range of values for $m$ such that the maximum value of the function $f(x)=|x^{2}-4x+9-2m|+2m$ on the interval $[0,4]$ is $9$.
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Context:
Rectangle $ABCD_{}^{}$ has sides $\overline {AB}$ of length $4$ and $\overline {CB}$ of length $3.$ Divide $\overline {AB}$ into $168$ congruent segments with points $A=P_0, P_1, \cdots, P_{168}=B$ , and divide $\overline {CB}$ into $168$ congruent segments with points $C=Q_0, Q_1, \cdots, Q_{168}=B$ . For $1 \le k \le 167$ , draw the segments $\overline {P_kQ_k}$ . Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$ , and then draw the diagonal $\overline {AC}$ . Find the sum of the lengths of the $335$ parallel segments drawn.
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Context:
Evaluate the integral $\int_0^\infty J_n(bx)\,dx$, where $J_n(x)$ is the $n$th order Bessel function of the first kind.
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Context:
Given a hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with left and right foci $F_{1}$, $F_{2}$, and a point $P$ on $C$, where $\angle F_{1}PF_{2}=60^{\circ}$, find the eccentricity $e$ of $C$ when $\sin \angle PF_{2}F_{1}=3\sin \angle PF_{1}F_{2}$.
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Context:
A teacher is making packed lunches for a field trip. Each student needs 2 sandwiches, and will travel in a group with 5 other students. There are a total of 5 groups. How many pieces of bread will the teacher need to make enough sandwiches for all the students?
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Context:
Given that the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfy ${b_{n+1}}{a_n}+{b_n}{a_{n+1}}={({-2})^n}+1,{b_n}=\frac{{3+{{({-1})}^{n-1}}}}{2}({n∈{N^*}})$, and $a_{1}=2$, find $a_{2n}$.
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Context:
In the diagram, $\triangle ABC$ is right-angled. Side $AB$ is extended in each direction to points $D$ and $G$ such that $DA = AB = BG$. Similarly, $BC$ is extended to points $F$ and $K$ so that $FB = BC = CK$, and $AC$ is extended to points $E$ and $H$ so that $EA = AC = CH$. Find the ratio of the area of the hexagon $DEFGHK$ to the area of $\triangle ABC$.
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Context:
2. Let $x, y, z$ be positive real numbers, and $x y z + x + z = y$, find the maximum value of $P = \frac{2}{x^{2}+1} - \frac{2}{y^{2}+1} + \frac{3}{z^{2}+1}$. (1999 Vietnam Mathematical Olympiad)
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Context:
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?
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Context:
What is the molecular weight of a compound having 2 Al, 3 C, and 9 O
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Context:
Diana can either invest $20,\!000$ dollars for $4$ years with a simple interest rate of $6\%$ or an interest rate of $7\%$ which compounds quarterly. How many more dollars, rounded to the nearest dollar, would she get with the better interest rate than with the worse one?
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Context:
Acute-angled triangle \(ABC\) is inscribed in a circle with center at \(O\). The arcs are given as follows: \(\overarc{AB} = 120^\circ\) and \(\overarc{BC} = 72^\circ\). A point \(E\) is located on the minor arc \(AC\) such that \(OE\) is perpendicular to \(AC\). Determine the ratio of the magnitudes of angles \(\angle OBE\) and \(\angle BAC\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
The volume of the triangular pyramid P-ABC is $8 \sqrt {3}$. All the vertices of the pyramid are on the surface of the sphere O. PC is perpendicular to the plane ABC, PC=4, and $∠CAB= \frac {π}{3}$. What is the minimum surface area of sphere O?
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Context:
Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the grain in Warehouse A will be six times that in Warehouse B. What is the minimum number of bags originally stored in Warehouse A?
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Context:
Bald Qiang said, "I will take a 45-minute nap, and when I wake up, it will be just time for afternoon work." However, he slept 20 minutes longer and woke up at 1:30 PM. Calculate the time at which Bald Qiang started his nap. Express your answer in 12-hour time format, including AM/PM designation (e.g., 12:00 PM).
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Context:
求具有下述性质的最小正数 $c$ ：对任意整数 $n \geq 4$,以及集合 $A \subseteq\{1,2, \cdots, n\}$, 若 $|A|>c n$, 则存在函数 $f: A \rightarrow\{1,-1\}$, 满足
$$
\left|\sum_{a \in A} f(a) \cdot a\right| \leq 1.
$$
The original answer is in \frac{m}{n} format, please give the value of m + n.
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Context:
Let $\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Assume that $B$ and $C$ lie on the same side of $l$, and that the distance from $B$ to $l$ is twice the distance from $C$ to $l$. Suppose also that the length $BA$ is twice that of $CA$. If $\triangle ABC$ has integer side lengths and is as small as possible, what is $AB^2+BC^2+CA^2$?
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Context:
For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $\lfloor M \rfloor$.
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Context:
A circle passes through the vertices $A$ and $C$ of triangle $ABC$, intersecting side $AB$ at point $E$ and side $BC$ at point $F$. The angle $AEC$ is 5 times the angle $BAF$, and the angle $ABC$ is $72^{\circ}$. Find the radius of the circle if $AC=6$.
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Context:
Given a triangle with  $ \triangle ABC $ , with:  $ \angle C = 36^o$  and   $ \angle A = \angle B $ . Consider the points  $ D $  on  $ BC $ ,  $ E $  on  $ AD $ ,   $ F $  on  $ BE $ ,   $ G $  on  $ DF $  and   $ H $  on   $ EG $ , so that the rays  $ AD, BE, DF, EG, FH $  bisect the angles   $ A, B, D, E, F $  respectively. It is known that  $ FH = 1 $ . Calculate   $ AC$ .
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Context:
Given an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $2\vec{OC} = a_4\vec{OA} + a_8\vec{OB}$, and points A, B, C are not collinear (the line does not pass through point O), find $S_{11}$.
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Context:
Let \( a_{i} = 0,1 \) for \( i = 1, 2, \cdots, n \). Find the number of sequences \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) that satisfy \( a_{1} \leqslant a_{2}, a_{2} \geqslant a_{3}, a_{3} \leqslant a_{4}, a_{4} \geqslant a_{5}, \cdots \).
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Context:
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, and the results are a, b, c, d. What is the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} equals 4? Express your answer as a fraction.
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Context:
Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have:
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$.
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Context:
在锐角$\triangle ABC$中，三边a，b，c满足$a>b>c$，则$\sin B$的取值范围为
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Context:
Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly \(k\) chocolate chip cookies. Find the sum of the \(k\) for which Pearl is guaranteed to succeed regardless of how the cookies are arranged.
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Context:
Let \( S = \{1, 2, 3, 4\} \), and consider a sequence \( a_1, a_2, \cdots, a_n \) of \( n \) terms where \( a_i \in S \) such that for any non-empty subset \( B \) of \( S \), there exists a subsequence of \( n \) adjacent terms that forms the set \( B \). Find the minimum value of \( n \).
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Context:
Let a positive integer \( n \) be called a cubic square if there exist positive integers \( a, b \) with \( n = \gcd(a^2, b^3) \). Count the number of cubic squares between 1 and 100 inclusive.
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Context:
The distances between the points on a line are given as $2, 4, 5, 7, 8, k, 13, 15, 17, 19$. Determine the value of $k$.
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Context:
Let $f(X, Y, Z) = X^5Y - XY^5 + Y^5Z - YZ^5 + Z^5X - ZX^5$.

Find \( \frac{f(2009, 2010, 2011) + f(2010, 2011, 2009) - f(2011, 2010, 2009)}{f(2009, 2010, 2011)} \)
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Context:
A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\sin \frac{\pi}{8}$.  Find $rt$.
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Context:
Maddie has a paper ribbon of length \(36 \text{ cm}\). She divides it into four rectangles of different lengths. She draws two lines joining the centers of two adjacent rectangles. What is the sum of the lengths of the lines that she draws?
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Context:
In the "Turn Back Difference" game, two polynomials $-m$ and $-n$ are operated on in sequence as follows: after the first operation, three polynomials $-m$, $-n$, $-n+m$ are obtained; after the second operation, four polynomials $-m$, $-n$, $-n+m$, $m$ are obtained, and so on. The operation rule is: the polynomial added in each operation is the difference between the last polynomial obtained in the previous operation and the one before it. Calculate the sum of the polynomials obtained after the $2023$rd operation of this game. Express your answer as a simplified polynomial expression.
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Context:
3. In triangle $A B C$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn, $L$ is the intersection point of segments $B_{1} C_{1}$ and $A A_{1}$, $K$ is the intersection point of segments $B_{1} A_{1}$ and $C C_{1}$. Find the ratio $L M: M K$, if $M$ is the intersection point of the angle bisector $B B_{1}$ with segment $L K$, and $A B: B C: A C=2: 3: 4$. (16 points)
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Context:
Given a positive integer \(n\), find the number of positive integer solutions \((x, y)\) to the equation
$$
\frac{1}{n}=\frac{1}{x}+\frac{1}{y} \quad(x \neq y)
$$
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Context:
find the least number must be subtracted from 5474827 so that remaining no . is divisible by 12 ?
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Context:
A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?
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Context:
Over all real numbers \( x \) and \( y \) such that
\[ x^3 = 3x + y \quad \text{and} \quad y^3 = 3y + x \]
compute the sum of all possible values of \( x^2 + y^2 \).
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Context:
Pablo has 27 solid $1 \times 1 \times 1$ cubes that he assembles in a larger $3 \times 3 \times 3$ cube. If 10 of the smaller cubes are red, 9 are blue, and 8 are yellow, what is the smallest possible surface area of the larger cube that is red?
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Context:
If $f(x)=\frac{1}{(1-x^{2011})^{\frac{1}{2011}}}$, then what is the number of solutions of the equation $\underbrace{f\circ f\circ\dots\circ f}_{2013} (x)=\{-x\}$?
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Context:
Helen the hippo and her friends are preparing for thanksgiving at Helen's house. Helen baked 519 chocolate chip cookies and 300 raisin cookies yesterday. And she baked 280 raisin cookies and some chocolate chip cookies this morning. Helen baked 20 more raisin cookies yesterday compared to today. How many chocolate chip cookies did Helen bake today?
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Context:
Find the maximum possible sum of the lengths of disjoint intervals in $(0,1)$ such that no two points in the union are at a distance of $\frac{1}{6}$.
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Context:
Let  $n$  be a positive integer, and let  $S_n = \{1, 2, \ldots, n\}$ . For a permutation  $\sigma$  of  $S_n$  and an integer  $a \in S_n$ , let  $d(a)$  be the least positive integer  $d$  for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or  $-1$  if no such integer exists). Compute the value of  $n$  for which there exists a permutation  $\sigma$  of  $S_n$  satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017,  \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\]

*Proposed by Michael Tang*
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Context:
On the health and wellness day, fruits were distributed among the students in a school so that each student would get two bananas, one apple, and an orange. On the particular day, 420 students were absent, which resulted in the remaining students receiving four extra bananas, two more apples, and an additional orange each. Considering that bananas were distributed first, followed by apples and then oranges, find the actual number of students in the school.
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Context:
A steamboat travels from point $A$ to point $B$ in $a$ hours and back in $b$ hours. How many hours will a raft take to drift from $A$ to $B$?
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Context:
求最大正整数 $m$, 使得可以在 $m$ 行 8列的方格表的每个方格中填人 C、G、M、O 这四个字母之一，并且具有如下性质：对于方格表的任意不同两行，至多存在一列，使得这两行在此列处的字母相同。
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Context:
A survey of $120$ teachers determined the following:

$70$ had high blood pressure

$40$ had heart trouble

$20$ had both high blood pressure and heart trouble

What percent of the teachers surveyed had neither high blood pressure nor heart trouble?
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Context:
Parker sleeps for 7 hours everyday. If today he woke up at 9 in the morning, by what time would he would go to bed?
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Context:
What are the last two digits of the decimal representation of \(9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}\)?
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Context:
Let $a$, $b$, and $c$ be integers simultaneously satisfying the equations:
\[
4abc + a + b + c = 2018
\]
and
\[
ab + bc + ca = -507
\]
Find \(|a| + |b| + |c|\).
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Context:
已知[x]表示不超过x的最大整数，则方程$3 ^ { 2 x } - [ 1 0 \cdot 3 ^ { x + 1 } ] +$ $\sqrt { 3 ^ { 2 x } - [ 1 0 \cdot 3 ^ { x + 1 } ] + 8 2 } = - 8 0$的整数解的个数为
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Context:
Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$?
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Context:
There are \(2n\) complex numbers satisfying \(z^{28} - z^8 - 1 = 0\). They have the form \(z_{m} = \cos \theta_{m} + i \sin \theta_{m}\), \(m = 1, 2, \cdots, 2n\), where \(0 \leqslant \theta_{1} < \theta_{2} < \cdots < \theta_{2n} < 2\pi\). Find the value of \(\theta_{1} + \theta_{2} + \cdots + \theta_{2n}\).
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Context:
Consider a counting problem involving an \(n \times m\) grid, where each square contains the number of rectangles that can be formed inside the grid with that square as part of its boundary. For instance, in a \(2 \times 3\) grid, a green square is contained in 8 rectangles. Let \(f(n, m)\) be the sum of these numbers for all squares in the grid. Find a general formula for \(f(n, m)\).
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Context:
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
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Context:
已知点A(1,1),$B ( \frac { 1 } { 2 } ,$ $, 0 ) , C ( \frac { 3 } { 2 } , 0 )$经过点A、B的直线和经过点A、C的直线与直线y=a(0<a<1)所围成的平面区域为G.已知平面矩形区域{(x,y)|0<x<2,0<y<1}中的任意一点进入区域G的可能性为$\frac { 1 } { 1 6 }$，则a=
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Context:
6 阶置换群 S_6 中的 6 阶元素的个数为__________.
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Context:
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.
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Context:
Let $Q(x)$ be a polynomial satisfying the functional equation $Q(x)=Q\left(\frac{1}{1-x}\right)$ for all $x\neq0,1$. Suppose that $P(x)$ is a polynomial such that $P(x)=x(1-x)Q(x)$ for all real $x$. If $P(x)$ is a cubic polynomial, what is the degree of $Q(x)$?
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Context:
Given two positive numbers $a$, $b$ such that $a<b$. Let $A.M.$ be their arithmetic mean and let $G.M.$ be their positive geometric mean. Then $A.M.$ minus $G.M.$ is always less than:
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Context:
Triangle $ABC$ has an area 1.  Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.
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Context:
A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 5 greater than the other.
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Context:
OpenAI runs a robotics competition that limits the weight of each robot. Each robot can be no more than twice the minimum weight and no less than 5 pounds heavier than the standard robot. The standard robot weighs 100 pounds. Additionally, each robot must use a specific battery that weighs 20 pounds, and they can only use one battery. The robots must also carry a payload of 10 up to 25 pounds as part of the competition. The minimum weight of a robot is affected by the payload requirements. Taking into account the battery, payload, and aforementioned constraints, what is the maximum weight of a robot in the competition, including the battery and payload?
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Context:
For each integer $k \ge 2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. For example, $X_2 = 10241048576$. If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n \ge 2$ is an integer.
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Context:
There are 13 3-year-olds, 20 4-year-olds, 15 5-year-olds, 22 6-year-olds, 18 7-year-olds, 25 8-year-olds, 30 9-year-olds, and 16 10-year-olds at a particular Sunday school. Classes are organized as follows:

Class 1: 3 and 4-year-olds
Class 2: 4 and 5-year-olds
Class 3: 5, 6, and 7-year-olds
Class 4: 8-year-olds
Class 5: 9 and 10-year-olds

Given this distribution of students, what is the average class size?
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Context:
If x and y are positive integers, and y = x^2 + 8, what is the greatest common divisor that x and y cannot have?
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Context:
Rohan spends 30% of his salary on food, 20% on house rent, and the rest on other expenses. His entertainment expenses are half of his food expenses, while his conveyance expenses are 25% more than his entertainment expenses. He spends 5% of his salary on education and 10% of his salary on utilities. If his savings at the end of a month are Rs. 2500, then what is his monthly salary?
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Context:
In the village, three types of cultivated land are classified: Category X, Category Y, and Category Z. Each attracts farm tax rates of 45%, 55%, and 65%, respectively. The tax department succeeded in collecting a total of $6500 through the farm tax. Mr. Willam, a farmer in the village, owns lands of all three categories. He paid $1200 as farm tax for his Category X land, $1600 for his Category Y land, and $1900 for his Category Z land.

Given the total farm tax from the village and Mr. Willam's farm tax payment for each category of land, find the percentage of total land Mr. Willam owns over the total taxable land in the village. Assume that the total taxable land in the village is 1000 acres.
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Context:
Find the number of roots of the equation

$$
Q_{7}(z) \equiv z^{7}-2z-5=0
$$

in the right half-plane.
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Context:
A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth?
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Context:
March 9, 2014 is a Sunday. Based on this information, determine which day of the week occurs most frequently in the year 2014.
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Context:
## Task Condition

Find the derivative.

$$
y=\frac{1}{4 \sqrt{5}} \ln \frac{2+\sqrt{5} \tanh x}{2-\sqrt{5} \cdot \tanh x}
$$
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Context:
作出正四面体每个面（三角形）的中位线，共得12条线段，在这些线段中，相互成异面直线的“线段对”有几个.
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Context:
From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit?
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Context:
Consider a rectangular grid of  $ 10 \times 10$  unit squares. We call a *ship* a figure made up of unit squares connected by common edges. We call a *fleet* a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.
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Context:
Given that \( x, y, z \in \mathbb{R}_{+} \), and \( s = \sqrt{x+2} + \sqrt{y+5} + \sqrt{z+10} \), \( t = \sqrt{x+1} + \sqrt{y+1} + \sqrt{z+1} \), find the minimum value of \( s^{2} - t^{2} \).
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Context:
Andrey found the product of all the numbers from 1 to 11 inclusive and wrote the result on the board. During a break, someone accidentally erased three digits, leaving the number $399 * 68 * *$. Help restore the missing digits without recalculating the product.
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Context:
Rhonda invested $4725 for one year, part at a certain annual interest rate, and the balance at 8% annual interest. She earned twice as much interest from the higher interest rate investment as from the 8% investment. She invested $1925 at the higher interest rate and the rest at 8%. What was the higher annual interest rate?
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Context:
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $3:1$. Calculate the percent of the rectangle's area that is contained within the square.
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Context:
A piece of rope is randomly cut into two parts. Find the probability that the longer segment is at least x times the shorter segment. Express your answer as a fraction or ratio.
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Context:
Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
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Context:
On the radius \( AO \) of a circle with center \( O \), point \( M \) is chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \( BC \), given that the radius of the circle is 15 and \(\sin \alpha = \frac{\sqrt{21}}{5}\)?
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Context:
Calculate the following limit:
$$\lim_{n \to \infty} \left[\prod_{k=1}^{n}\left(1+\frac{(k+1)^{n}}{n^{n+1}}\right)\right]$$
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Context:
Given $(1+x) + (1+x)^2 + \ldots + (1+x)^5 = a_0 + a_1(1-x) + a_2(1-x)^2 + \ldots + a_5(1-x)^5$, calculate the value of $a_1 + a_2 + a_3 + a_4 + a_5$.
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Context:
A math test consists of 12 multiple-choice questions, each worth 5 points. It is known that a student is confident in correctly answering 6 of these questions. For another three questions, the student can eliminate one incorrect option. For two questions, the student can eliminate two incorrect options. For the last question, due to a lack of understanding, the student has to guess randomly. Estimate the score of this student in this test.
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Context:
In a crown, five diamonds are embedded at five positions, equally spaced around a circle. If there are three different colors of diamonds to choose from, how many different ways are there to embed the diamonds? Assume that the five positions are indistinguishable from each other.
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Context:
Given a complex number $z=\frac{4}{5}-\sin \theta +(\cos \theta -\frac{3}{5})i$ is purely imaginary (where $i$ is the imaginary unit), find $\tan \theta$.
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Context:
For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array},  \] for all $n\geq 1$.

Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.
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Context:
Initially, the cookie jar had a random but fixed amount of money, M dollars. Doris spent 15% of the initial amount from the cookie jar, Martha spent 10% of the remaining amount after Doris's spending, and John spent 20% of the amount left after Martha's spending. If there were $15 left in the cookie jar after John's spending, how much money, in dollars, was there in the cookie jar at first?
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Context:
There are 101 nonzero integers written in a circle such that each number is greater than the product of the two numbers that follow it in a clockwise direction. What is the maximum number of negative numbers that could be among these 101 written numbers?
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Context:
Given Gilda starts with a bag of marbles, she gives $30\%$ of them to Pedro, $10\%$ of the remaining to Ebony, $15\%$ of the remaining to Jimmy, and $5\%$ of the remaining to Maria. Calculate the percentage of her original bag of marbles that Gilda has left.
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Context:
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled  $A$ . The three vertices adjacent to vertex  $A$  are at heights 10, 11, and 12 above the plane. The distance from vertex  $A$  to the plane can be expressed as  $\tfrac{r-\sqrt{s}}{t}$ , where  $r$ ,  $s$ , and  $t$  are positive integers, and  $r+s+t<1000$ . Find  $r+s+t$ .
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Context:
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\).
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Context:
It will cost $60 to rent a sailboat and $80 per hour to rent a ski boat on weekdays. During weekends or holidays, the sailboat rental costs $90 and the ski boat rental costs $120 per hour. Ken rented a sailboat while Aldrich rented a ski boat. Additionally, there are fuel costs of $10 per hour for Ken's sailboat rental and $20 per hour for Aldrich's ski boat rental. The company is offering a discount of 10% on the second day of rental. Ken rented the sailboat on a weekday, while Aldrich rented the ski boat during the weekend. How much more did it cost Aldrich to rent the ski boat than Ken to rent a sailboat for three hours a day in two days, accounting for different hourly rates and any applicable discounts?
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Context:
If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\).
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Context:
A solid wooden rectangular prism measures $3 \times 5 \times 12$. The prism is cut in half by a vertical cut through four vertices, creating two congruent triangular-based prisms. What is the surface area of one of these triangular-based prisms?
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Context:
In the Cartesian coordinate plane \( xOy \), given a moving line \( l \) that is tangent to the parabola \( \Gamma: y^{2}=4x \) and intersects the hyperbola \( \Omega: x^{2}-y^{2}=1 \) at one point on each of its left and right branches at points \( A \) and \( B \), find the minimum area of \( \triangle OAB \).
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Context:
在四面体 $A B C D$ 中, $\triangle A B D$ 为等边三角形， $\angle B C D=90^{\circ}, B C=C D=1, A C=\sqrt{3}, E$ 、 $F$ 分别为 $B D 、 A C$ 的中点. 则直线 $A E$ 与 $B F$ 所成角的余弦值为原始答案是\frac{\sqrt{m}}{n}的形式，请给出m + n的值。
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Context:
In the quadrilateral \(ABCD\), the sides \(AB\) and \(CD\) are parallel, with \(AB\) being twice as long as \(CD\). Point \(S\) is the intersection point of the diagonals of quadrilateral \(ABCD\), and triangles \(ABS\) and \(CDS\) are both equilateral. Point \(E\) lies on the segment \(BS\) such that the angle \(ACE\) measures \(30^{\circ}\).

Determine the ratio of the areas of quadrilateral \(ABCD\) and triangle \(EBC\).
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Context:
Rachelle picks a positive integer \(a\) and writes it next to itself to obtain a new positive integer \(b\). For instance, if \(a=17\), then \(b=1717\). To her surprise, she finds that \(b\) is a multiple of \(a^{2}\). Find the product of all the possible values of \(\frac{b}{a^{2}}\).
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Context:
Find the sum of the digits of the following product (as a function of \( n \)):
$$
9 \cdot 99 \cdot 9999 \cdot \cdots \cdot \left(10^{2^{n}} - 1\right).
$$

In this product, the number of digits of each factor is twice the number of digits of the previous factor.
(21st United States of America Mathematical Olympiad, 1992)
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Context:
For the fractional equation involving x, $\frac{x}{x-2}-2=\frac{m}{x-2}$, find the value of m when the equation has a root. Express your answer as a single integer.
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Context:
Find the Laplace transform of the function $$f(x) = (1 + 2ax)x^{-\frac{1}{2}}e^{ax}$$
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Context:
The marked price of a book was 30% less than the suggested retail price.  Alice purchased the book for half the marked price at a Fiftieth Anniversary sale.  What percent of the suggested retail price did Alice pay?
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Context:
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of this ellipse is ___.
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Context:
Given a positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil \frac{n}{2} \rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $x$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a + b$.
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Context:
Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?
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Context:
A rectangle is inscribed within a semi-circle with a perimeter of 126 cm. One pair of opposite sides of the rectangle are tangent to the semi-circle. The length of the rectangle is three times its width. What is the radius of the semi-circle?
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Context:
Mona has 12 match sticks of length 1, and she has to use them to make regular polygons, with each match being a side or a fraction of a side of a polygon, and no two matches overlapping or crossing each other. What is the smallest total area of the polygons Mona can make?
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Context:
Given the function $f(x) = x^3 + ax + \frac{1}{4}$, if the x-axis is the tangent to the curve $y = f(x)$, then the value of $a$ is __________.
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Context:
Find the smallest positive real number $\lambda$ such that for any three complex numbers $\mathrm{z}_{1}$, $\mathrm{z}_{2}$, $\mathrm{z}_{3} \in \{\mathrm{z} \in \mathbb{C} \mid |\mathrm{z}| < 1\}$, if $\mathrm{z}_{1} + \mathrm{z}_{2} + \mathrm{z}_{3} = 0$, then $\left| \mathrm{z}_{1} \mathrm{z}_{2} + \mathrm{z}_{2} \mathrm{z}_{3} + \mathrm{z}_{3} \mathrm{z}_{1} \right|^{2} + \left| \mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}_{3} \right|^{2} < \lambda$.
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Context:
The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of triangle $ABC$?

Enter your answer in the form $\frac{a \sqrt{b} - c}{d},$ and simplified as usual.
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Context:
If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam?
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Context:
In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the trees: two configurations are considered identical if one can be converted into the other by swapping the trees (along with the corresponding groups) or by rotating each group around its tree. In how many ways can the children be divided into groups?
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Context:
Given the total number of stations is 6 and 3 of them are selected for getting off, calculate the probability that person A and person B get off at different stations.
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Context:
The starting and ending points of a tram route are Station A and Station B, respectively. A tram departs from Station A for Station B every 5 minutes, and the entire journey takes 15 minutes. A person starts cycling from Station B towards Station A along the tram route. When he starts, a tram just arrives at Station B. On the way, he encounters 10 trams coming towards him before reaching Station A. Upon reaching Station A, there is a tram just departing from Station A. How many minutes did it take for him to cycle from Station B to Station A?
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Context:
12. Given $S$ as a binary string of $10^{4}$ bits containing only $0$ and $1$, a positive integer $k \leqslant 10^{4}$, a $k$-block of $S$ is a substring of $S$ consisting of $k$ consecutive bits. Two $k$-blocks $a_{1} a_{2} \cdots a_{k}=b_{1} b_{2} \cdots b_{k}$ are equal if and only if $a_{i}=b_{i}(i=1$, $2, \cdots, k)$. Consider all $10^{4}$-bit binary strings that contain at most seven different 3-blocks. Find the maximum number of different 10-blocks such a string can contain.
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Context:
Plot points $A, B, C$ at coordinates $(0,0),(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $A B$ and $B C$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise 45 degrees about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise 45 degrees about point $A$. Find $\frac{X_{1}+X_{2}}{2}$.
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Context:
How many pairs of integers \((x, y)\) are there such that \(x \geq y\) and \(\frac{1}{x}+\frac{1}{y}=\frac{1}{211}\) ?
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Context:
Determine the numeral system in which the following multiplication is performed: \(352 \cdot 31 = 20152\).
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Context:
Let $C$ be the unit circle $x^{2}+y^{2}=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$-and $y$-axes with diagonal $p q$. What is the probability that no point of $R$ lies outside of $C$? The original answer is in $\frac{n}{\pi^m}$format, please give the value of $m + n$.
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Context:
What is the limit of the following sum as $n$ approaches infinity?
$$
\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?
$$
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Context:
The graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight line and a curve. Find the sum of the coordinates of the intersection points of a straight line and a curve, rounded down.
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Context:
Petya, Sasha, and Misha are playing knockout tennis. A knockout game means that in each round, two players play while the third waits. The loser of the round gives place to the waiting player and becomes the one waiting in the next round. Petya played a total of 12 rounds, Sasha played 7 rounds, and Misha played 11 rounds. How many times did Petya win against Sasha?
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Context:
If x, y, and z are positive integers, and 6z = some value = 2x, then the least possible value of x + y + z is 26. What is the value that 6z is equal to?
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Context:
Solve the equation $$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\ldots+\sqrt{4^{2008}x+3}}}}}-\sqrt{x}=1$$
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Context:
The sale price of an article including the sales tax is some amount. The rate of sales tax is 10%. If the shopkeeper has made a profit of 16%, and the cost price of the article is Rs. 531.03, what is the sale price of the article including the sales tax?
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Context:
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.
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Context:
A school arranges four classes in the morning, each lasting 40 minutes. The first class starts from 8:00 to 8:40, with a 10-minute break between classes. If a student arrives late, and he arrives at the classroom between 9:10 and 10:00, what is the probability that he listens to the second class for no less than 10 minutes? Express your answer as a simplified fraction.
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Context:
Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a *single* and that between a boy and a girl was called a *mixed single*. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?
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Context:
Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and $2S_{n}$ is the arithmetic mean of $6$ and $a_{n}$. If for any positive integer $n$, $3S_n-\frac{1}{S_n} \in [k,t]$, then the minimum value of $t-k$ is ____.
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Context:
Jacob's water tank can hold up to 350 liters of water. Jacob collects water from the river and rain to fill his water tank. The daily water collection varies based on weather conditions and river flow. On average, he can collect 300 to 600 milliliters of water from the rain and 900 to 1500 milliliters of water from the river every day. How many days does Jacob need to fill up his water tank, considering the variable rate of daily water collection?
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Context:
设 $a_{1}, a_{2}, \cdots, a_{20} \in\{1,2, \cdots, 5\}, b_{1}, b_{2}, \cdots, b_{20} \in\{1,2, \cdots, 10\}$, 集合 $X=\{(i, j) \mid$ $\left.1 \leqslant i<j \leqslant 20,\left(a_{i}-a_{j}\right)\left(b_{i}-b_{j}\right)<0\right\}$, 求 $X$ 的元素个数的最大值.
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Context:
What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)

Alexandru Mihalcu
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Context:
In a cube with edge length 1, two cross-sections in the form of regular hexagons are made. Find the length of the segment at which these cross-sections intersect.
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Context:
7 distinct primes can be written as a, b, c, a + b + c, a + b - c, a - b + c, -a + b + c, where two of a, b, c sum to 800. Let d be the difference between the largest and the smallest of the 7 primes. What is the largest possible value of d?
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Context:
A cricket bat is sold for $850, making a profit of $205. What is the profit percentage?
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Context:
Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots.
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Context:
The domain of the inverse function of $f(x) = 2 - x$ $(0 < x \leq 3)$ is
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Context:
Let $S$ be a set. We say $S$ is $D^\ast$-finite if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a witness of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have?
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Context:
Ravi has some coins. He has 2 more quarters than nickels and 4 more dimes than quarters. If he has 6 nickels, how much money does he have?
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Context:
John gets his pool cleaned every 3 days throughout the year. In summer (3 months), it costs $150 per cleaning, and in the remaining 9 colder months, it costs only $120 per cleaning. He gives a 10% tip to the cleaner every time he comes. In addition to cleaning, he needs to use $250 worth of chemicals twice a month during the summer and $150 worth of chemicals twice a month during the rest of the year. Furthermore, he spends $100 a month on equipment rental for pool maintenance tools, and his pool pump consumes an average of $75 worth of electricity per month. John also subscribes to a quarterly $200 repair and inspection service.

Taking all these factors into account, calculate John's total pool maintenance and operation expenses for an entire year.
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Context:
The opposite sides of a quadrilateral inscribed in a circle intersect at points \( P \) and \( Q \). Find the length of the segment \( |PQ| \), given that the tangents to the circle drawn from \( P \) and \( Q \) are \( a \) and \( b \) respectively.
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Context:
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
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Context:
Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its 4 digits when expressed in mmdd format (e.g. the digit sum of 13 May is $0+5+1+3=9$). Find the number of dates in the year 2021 with digit sum equal to the positive integer $n$.
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Context:
A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd?
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Context:
Consider the set of all bijective functions from the set of non-negative integers to the set of rational numbers. For each function, define its maximum step size as the maximum absolute difference between the values of the function at consecutive integers. What is the greatest lower bound of the maximum step sizes of all such functions?
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Context:
A beadshop has a varied profit distribution throughout the week: On Monday, it earns 20% of its profit, followed by 18% on Tuesday, 10% on Wednesday, 25% on Thursday, and 15% on Friday. However, on Saturday, there are no sales, but the shop offers a 20% discount on their items, resulting in a 5% loss in overall profit. The shop makes a total profit of $3,500 during the six days (from Monday to Saturday; Sunday is not included). Calculate the actual profit made on Friday, in dollars, after factoring in the discount applied on Saturday that affects the overall profit obtained.
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Context:
Audrey has 26 peaches and decides to buy 3 times as many. Paul has 48 peaches and decides to buy twice as many. Meanwhile, their friend Maya has 57 peaches and buys an additional 20. What's the difference between the total number of peaches Audrey, Paul, and Maya have now?
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Context:
Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin.
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Context:
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \]
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Context:
Is there a closed form for the general recurrence relation \( f(n) = an + b\sum_{k=0}^{n-1}f(k) \), where \( a \) and \( b \) are constants?
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Context:
The wholesale department operates a certain product, with a wholesale price (selling price) of 500 yuan per unit, and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank, with a monthly interest rate of 4.2‰. The storage and operating cost of the product is 0.30 yuan per unit per day. Find the maximum average storage period for the product without incurring a loss. Express your answer in days as a whole number.
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Context:
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$.
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Context:
Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
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Context:
Find the range of the independent variable x in the function y = 1 / (x - 1). Express your answer as an inequality or a statement about x.
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Context:
We wrote the reciprocals of natural numbers from 2 to 2011 on a board. In one step, we erase two numbers, \( x \) and \( y \), and replace them with the number

$$
\frac{xy}{xy + (1 - x)(1 - y)}
$$

By repeating this process 2009 times, only one number remains. What could this number be?
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Context:
Trapezoid $EFGH$ has sides $EF=100$, $FG=55$, $GH=23$, and $HE=75$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the length of $EQ$ if it is expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers.
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Context:
Indecisive Andy starts out at the midpoint of the 1-unit-long segment \(\overline{H T}\). He flips 2010 coins. On each flip, if the coin is heads, he moves halfway towards endpoint \(H\), and if the coin is tails, he moves halfway towards endpoint \(T\). After his 2010 moves, what is the expected distance between Andy and the midpoint of \(\overline{H T}\)?
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Context:
A 90-degree angle is covered by some angles with the same vertex in two or three layers, and the sum of the angles is 290 degrees. Find the measure of the angle covered in three layers.
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Context:
已知集合S={1,2,⋯,2005}.A⊆S,A中任两数之和不被117整除，试求｜A｜的最大值.
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Context:
A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. Calculate the ratio of the perimeter of the rectangle to the perimeter of the rhombus. Express your answer as a ratio (e.g., 1:1, 2:1, etc.).
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Context:
$已知f(x)=|x-a|x+|x-2|(x-a).$
$若x\in (-\infty ,1)时， f(x)<0，求a的取值范围.$
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Context:
Two circles \(C_{1}\) and \(C_{2}\) touch each other externally and the line \(l\) is a common tangent. The line \(m\) is parallel to \(l\) and touches the two circles \(C_{1}\) and \(C_{3}\). The three circles are mutually tangent. If the radius of \(C_{2}\) is 9 and the radius of \(C_{3}\) is 4, what is the radius of \(C_{1}\)?
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Context:
Find the volume of a regular quadrilateral prism if its diagonal makes an angle of \(30^{\circ}\) with the plane of the lateral face, and the side of the base is equal to \(a\).
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Context:
Among the 12 lines where the edges of a hexagonal pyramid are located, how many pairs of skew lines exist? Express your answer as a whole number.
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Context:
记如下关于实数变量 $x, y$ 的函数
$$
f(x, y)=\sqrt{4 y^{2}-12 y+10}+\sqrt{18 x^{2}-18 x+5}+\sqrt{18 x^{2}+4 y^{2}-12 x y+6 x-4 y+1}
$$
的最小值为 $a$. 求取整 $\left[8 a^{2}\right]$ 的值.
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Context:
There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done?
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Context:
Stringing 6 red balls, 1 green ball, and 8 yellow balls into a necklace, how many possible different types can appear (balls of the same color are not distinguished)?
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Context:
At a certain pizzeria, there are five different toppings available, and a pizza can be ordered with any (possibly empty) subset of them on it. Determine the number of ways to order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither.
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Context:
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
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Context:
According to the directions on a packet of smoothie mix, 1 3-ounce packet of smoothie mix is to be combined with 15 ounces of water to make a smoothie. To prepare 150 smoothies of a certain size, 180 3-ounce packets of smoothie mix are required. What is the size of each smoothie in ounces?
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Context:
For transporting cargo from one place to another, a certain number of trucks with the same capacity were required. Due to road conditions, each truck had to carry 0.5 tons less than originally planned, thus 4 additional trucks of the same capacity were required. The mass of the transported cargo was at least 55 tons but did not exceed 64 tons. How many tons of cargo were transported by each truck?
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Context:
Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet.
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Context:
What fraction of the area of triangle $ABC$ is occupied by those points $K$ which lie inside the triangle and for which the point-similarity with center $K$ and ratio $-1/2$ maps $A$ and $B$ to internal points of the triangle, but maps $C$ to an external point?
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Context:
A regular  $17$ -gon with vertices  $V_1, V_2, . . . , V_{17}$  and sides of length  $3$  has a point  $ P$  on  $V_1V_2$  such that  $V_1P = 1$ . A chord that stretches from  $V_1$  to  $V_2$  containing  $ P$  is rotated within the interior of the heptadecagon around  $V_2$  such that the chord now stretches from  $V_2$  to  $V_3$ . The chord then hinges around  $V_3$ , then  $V_4$ , and so on, continuing until  $ P$  is back at its original position. Find the total length traced by  $ P$ .
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Context:
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.

Find the greatest possible value of $S$.
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Context:
已知正方体 \(A B C D-A_{1} B_{1} C_{1} D_{1}\) 的棱长为 1, 棱 \(A A_{1}\) 的中点为 \(E\), \(A C\) 与 \(B D\) 交于点 \(O\)。若平面 \(\alpha\) 过 \(E\) 且与 \(O C_{1}\) 垂直，则 \(\alpha\) 截该正方体所得截面的面积的原始答案是\(\frac{\sqrt{k}}{m}\)的形式，请给出k + m的值。
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Context:
There are four tiles marked X, and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that the two outermost positions in the arrangement are occupied by X and the middle one by O?
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Context:
知曲线 $C$ 上任意一点, 到定点 $A(1,0)$ 与定直线 $x=4$ 的距离之和等于 5 , 对于给定的点 $B(b, 0)$, 在曲线上恰有三对不同的点关于点 $B$ 对称, 求 $b$ 的取值范围.
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Context:
Calculate the volumes of solids formed by the rotation of regions bounded by the graphs of the functions around the y-axis.

$$
y=\arcsin \frac{x}{5}, y=\arcsin x, y=\frac{\pi}{2}
$$
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Context:
Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?
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Context:
Suppose the function \( y= \left| \log_{2} \frac{x}{2} \right| \) has a domain of \([m, n]\) and a range of \([0,2]\). What is the minimum length of the interval \([m, n]\)?
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Context:
Given the universal set U = {0,1,2,3,4}, set M = {0,2,3}, and the complement of N in U, ∁_U N = {1,2,4}, find the intersection of M and N, M ∩ N. Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas.
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Context:
Consider the signal $x(t)=e^{-t}u(t)$ where $u(t)=\mathbb{1}(t\geq0)$, i.e. the Heaviside function. Find the signal $y(t)=x(t)*x(-t)$.
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Context:
Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. Calculate the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $O_1O_2O_3O_4$.
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Context:
Given $x < y < 0$, compare the magnitude of $(x^2+y^2)(x-y)$ with $(x^2-y^2)(x+y)$.
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Context:
The coefficient of $x^{4}y^{2}$ in the expansion of $({x-\frac{1}{x}+y})^8$ is ______ (expressed as a constant).
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Context:
Last year, Manfred received 26 paychecks. Each of his first 6 paychecks was $750. After that, Manfred received a 5% pay raise for his next 10 paychecks. Then, Manfred received another 3% pay raise for his remaining paychecks. Also, Manfred had a one-time bonus of $250 added to his 10th paycheck and a medical deductible of $120 taken from his 20th paycheck. Calculate the average (arithmetic mean) amount of his paychecks for the year, to the nearest dollar, after considering federal tax deductions at a rate of 15% and the other mentioned adjustments.
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Context:
Given $f(x) = x^5 - 2x^3 + 3x^2 - x + 1$, calculate the value of $f(x)$ at $x=3$ using the Horner's method. The value of $v_3$ is \_\_\_\_\_\_.
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Context:
To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$?
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Context:
A circle centered at \( O_1 \) with a radius of 2, inscribed in the triangle \( ABC \), touches side \( BC \) at point \( D \). A second circle centered at \( O_2 \) with a radius of 4 touches the extensions of sides \( AB \) and \( AC \), as well as side \( BC \) at point \( E \). Find the area of the quadrilateral \( O_1 D O_2 E \) if the measure of angle \( ACB \) is \( 120^\circ \). (20 points)
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Context:
11. Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant$ $i<j \leqslant n)$ as $m$. Find the maximum value of $m$ under the condition $\sum_{i=1}^{n} a_{i}^{2}=1$.
Please reason step by step, and put your final answer within \boxed{}.