Context:
Task 1. Represent in the form of an irreducible fraction

$$
7 \frac{19}{2015} \times 6 \frac{19}{2016}-13 \frac{1996}{2015} \times 2 \frac{1997}{2016}-9 \times \frac{19}{2015}
$$
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Context:
There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats such that neither little boy sits next to either little girl. Determine the number of different ways these six people can be seated.
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Context:
Given $a$, $b$, and $c$ are the sides of $\triangle ABC$, and $a^2-a-2b-2c=0$, $a+2b-2c+3=0$, find the largest angle of this triangle.
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Context:
Ed had 5 more marbles than Doug. Doug lost 3 of his marbles at the playground. If Ed had 27 marbles. How many marbles did Doug have initially?
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Context:
A man sells a car to his friend at a 14% loss. The friend sells it for Rs. 54,000 and gains 20%. What was the original cost price of the car?
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Context:
Out of the 300 senior students, five percent received a full merit college scholarship, ten percent got a half merit scholarship, three percent received a sports scholarship, and seven percent were granted need-based scholarships. Some students received more than one type of scholarship. Among these students, 1% were given both full merit and sports scholarships, 2% received both half merit and need-based scholarships, and 0.5% obtained a sports scholarship and a need-based scholarship. No students were granted all three types of scholarships. Calculate the number of senior students who did not receive any scholarships.
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Context:
In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that:
1. Contains more than two cities,
2. Does not repeat any city,
3. Ends where it begins,

It must include all types of airports. What is the maximum number of flights in this country?
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Context:
A certain project requires 24 days for Team A to complete on their own, and 18 days for Team B to complete on their own. Now, if 6 people from Team B are transferred to Team A, then Team A alone would need 18 days to complete the project, and Team B alone would need 24 days. If Team B starts working alone for 6 days, and then Teams A and B work together for 4 days, after which Team B has another task and takes their 6 people back, leaving Team A to finish the work alone, how many more days does Team A need?
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Context:
求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为自然数。
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Context:
Given real numbers \(a_{1}, a_{2}, \cdots, a_{5}\) such that
\[
\sum_{i=1}^{5} a_{i}^{2} = 1,
\]
find the maximum value of
\[
T = \sum_{i=1}^{5} \left|a_{i} - a_{i+1}\right| \quad \text{where} \quad a_{6} = a_{1}.
\]
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Context:
An integer is called ''parity-monotonic'' if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is [[odd]], and $a_{i}>a_{i+1}$ if $a_{i}$ is [[even]]. How many four-digit parity-monotonic integers are there?
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Context:
The lateral sides of a trapezoid lie on perpendicular lines. Find the area of the quadrilateral with vertices at the midpoints of the diagonals and at the midpoints of the bases of the trapezoid, given that its lateral sides are equal to \(a\) and \(b\).
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Context:
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the ball inscribed in the center of the pyramid, touching six identical balls, if the radius of the sphere in which the pyramid of balls is inscribed is $\sqrt{6}+1$.
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Context:
Find the number of three-digit numbers such that each of the first two digits is divisible by the third digit.
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Context:
When $x \in (0, +\infty)$, the power function $y = (m^2 - m - 1)x^m$ is a decreasing function. Find the value of the real number $m$ that satisfies this condition. Express your answer as a single real number.
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Context:
In a rectangular parallelepiped with dimensions AB = 4, BC = 2, and CG = 5, point M is the midpoint of EF, calculate the volume of the rectangular pyramid with base BDFE and apex M.
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Context:
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once?
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Context:
Let \( X \) be the set of consisting of twenty positive integers \( n,n + 2,\ldots ,n + {38} \) . The smallest value of \( n \) for which any three numbers \( a,b,c \in  X \) , not necessarily distinct, form the sides of an acute-angled triangle is:
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Context:
What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order?
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Context:
椭圆 $\frac{x^{2}}{6}+\frac{y^{2}}{2}=1$, 其右焦点为 $F(2,0)$. 直线 $l$ 过 $F$ 交椭圆于 $A, B$ 两点, $P$ 在 $x=3$ 上, 若 $\triangle A B P$ 为正三角形，则 $S_{\triangle P A B}$ 的最大值的原始答案形式为 \frac{k \sqrt{n}}{m}，请给出k + n + m的值。
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Context:
In how many ways can you divide the set of eight numbers  $\{2,3,\cdots,9\}$  into  $4$  pairs such that no pair of numbers has  $\text{gcd}$  equal to  $2$ ?
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Context:
Consider a nilpotent $4 \times 4$ real matrix $T$. If the dimensions of the nullspaces of $T^2$ and $T^3$ are 3 and 4, respectively, determine the possible dimension of the nullspace of $T$.
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Context:
How many permutations in $S_n$ have the first $k$ items $1,2,...,k$ all in the same cycle?
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Context:
Given that the greatest common divisor (gcd) of two numbers a and b is 1, determine the number of ways the number \(3^3\cdot 5^7 \cdot 11^{13}\) can be expressed as a product of two factors, \(m\) and \(n\), such that \(\text{lcm}(m,n)=3^3\cdot 5^7\cdot 11^{13}\) and \(\gcd(m,n)=1\).
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Context:
In a city, 2019 metro stations have been constructed. Some pairs of stations are connected by tunnels, and it is possible to travel from any station to any other station via these tunnels. The mayor has ordered the organization of several metro lines: each line must include multiple distinct stations connected sequentially by tunnels (one tunnel can be part of multiple lines). Additionally, each station must be part of at least one line. To save costs, the number of lines should be no more than $k$.

It turned out that the mayor's order is impracticable. What is the largest $k$ for which this could happen?
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Context:
How many subsets of the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \) have the property that no two of its elements differ by more than 5? For example, count the sets \( \{3\} \), \( \{2, 5, 7\} \), and \( \{5, 6, 7, 8, 9\} \) but not the set \( \{1, 3, 5, 7\} \).
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Context:
There is a prism whose sum of the number of edges, faces, and vertices is 50. How many edges of the pyramid have the same base shape as the prism?
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Context:
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number?
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Context:
Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$
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Context:
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
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Context:
In the convex quadrilateral \(ABCD\), it is known that \(\angle BCD = 80^\circ\), \(\angle ACB = 50^\circ\), and \(\angle ABD = 30^\circ\). Find the angle \(\angle ADB\).
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Context:
The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$?
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Context:
A is located 60 kilometers west of B. Individuals A and B depart from location A while individuals C and D depart from location B at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After n hours, B and C meet, and n hours later, A catches up to D at location C. Calculate the distance between locations B and C in kilometers.
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Context:
## Task B-3.3.

Given vectors $\vec{u}=\vec{m}+3 \vec{n}, \vec{v}=7 \vec{m}-5 \vec{n}, \vec{w}=\vec{m}-4 \vec{n} \text{ and } \vec{z}=7 \vec{m}-2 \vec{n}$, where $\vec{m}, \vec{n} \neq \overrightarrow{0}$. If vectors $\vec{u} \text{ and } \vec{v}$, and $\vec{w} \text{ and } \vec{z}$ are perpendicular, determine the angle between vectors $\vec{m} \text{ and } \vec{n}$.
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Context:
For how many rational numbers  $p$  is the area of the triangle formed by the intercepts and vertex of  $f(x) = -x^2+4px-p+1$  an integer?
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Context:
There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
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Context:
Let point $O$ be the origin of a two-dimensional coordinate system, and let points $A$ and $B$ be located on positive $x$ and $y$ axes, respectively. If $OA = \sqrt[3]{54}$ and $\angle AOB = 45^\circ,$ compute the length of the line segment $AB.$
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Context:
Find the number of real solutions to the equation [2x] + [3x] = 8x - 6, where [x] denotes the greatest integer less than or equal to the real number x.
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Context:
At a conference, the 2016 participants were registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration form. How many hands did the 2016th participant shake?
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Context:
Vasya and Petya started running simultaneously from the starting point of a circular track and ran in opposite directions. While running, they met at some point on the track. Vasya ran a full lap and, continuing to run in the same direction, reached the point where they had previously met at the moment when Petya completed a full lap. How many times faster was Vasya running compared to Petya?
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Context:
已知直线l：(a+1)x+(2a+1)y=5a+3恒过一定点P，$F _ { 1 } , F _ { 2 }$为双曲线$\frac { x ^ { 2 } } { 2 } - \frac { y ^ { 2 } } { b } = 1$的焦点，若$\Delta F _ { 1 } P F _ { 2 }$总是钝角三角形，则b的取值范围为 ·
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Context:
Given the vertices of a regular 100-sided polygon \( A_{1}, A_{2}, A_{3}, \ldots, A_{100} \). In how many ways can three vertices be chosen that form an obtuse triangle?
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Context:
Find the value of $(\frac{\gamma-\delta}{\gamma+\delta})^{(\frac{\gamma+\delta}{\gamma-\delta})}$ where $\gamma$ and $\delta$ are defined as:

$$\gamma=\sqrt{3}+\frac{1}{\sqrt{3}+\frac{1}{\sqrt{3}+\frac{1}{\sqrt{3}+\frac{1}{...}}}}$$

$$\delta=\sqrt{3}-\frac{1}{\sqrt{3}-\frac{1}{\sqrt{3}-\frac{1}{\sqrt{3}-\frac{1}{...}}}}$$
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Context:
Let a positive integer $n$ be \textit{nice} if there exists a positive integer $m$ such that:
\[
n^3 < 5mn < n^3 + 100.
\]
Find the number of \textit{nice} positive integers.
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Context:
Four mayors decide to build a circular highway that passes within the limits of their cities. Since the four cities are not on the same circle, the mayors hire a company to design a project for constructing a circular highway that is equidistant from all four cities. What is the greatest number of geographically distinct projects that the company can create?
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Context:
A shopkeeper purchases 20 pens, each with a different discount ranging from 5% to 25% applied on their initial cost price. After applying the discounts, the total cost price for 20 pens is equal to the total selling price for 12 pens. In addition, a sales tax of 8% is applied to the selling price of the pens. Find the gain % or loss % for the shopkeeper considering the varying discounts and the sales tax.
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Context:
The distance from a fixed point $P$ on a plane to two vertices $A$ and $B$ of an equilateral triangle $ABC$ are $AP=2$ and $BP=3$. Determine the maximum possible length of the line segment $PC$.
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Context:
12. Six positive integers $a, b, c, d, e, f$ are arranged in alphabetical order to form a sequence, where $a=1$. If a certain positive integer is greater than 1, then the number that is one less than this positive integer must appear to its left. For example, if $d>1$, then one of $a, b, c$ must be $d-1$. Examples: $1,1,2,1,3,2$ satisfies the condition; $1,2,3,1,4,1$ satisfies the condition; $1,2,2,4,3,2$ does not satisfy the condition. The number of different permutations that satisfy the condition is $\qquad$.
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Context:
Find the distance from the point $M_{0}$ to the plane passing through the three points $M_{1}, M_{2}, M_{3}$.

$M_{1}(-3 ; 4 ;-7)$

$M_{2}(1 ; 5 ;-4)$

$M_{3}(-5 ;-2 ; 0)$

$M_{0}(-12 ; 7 ;-1)$
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Context:
The arithmetic-geometric sequence {a_n} satisfies a_1 = 3 and a_1 + a_3 + a_5 = 21. Find the value of a_2 * a_4. Express your answer as a single integer.
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Context:
Given that point P (3, -1) and Q (-1, 2) are on the same side of the line $ax+2y=1$, calculate the range of the real number $a$ as an interval.
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Context:
How many unique ways are there to color the balls in a triangular array with 10 balls, arranged such that the first row has 1 ball, the second row has 2 balls, the third row has 3 balls, and the fourth row has 4 balls, using only 2 colors, considering rotations as equivalent?
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Context:
Example 26 Let $n(\geqslant 3)$ be a given positive integer, 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 the above $m$ under the condition that $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=$ 1.
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Context:
All triangles have the same value, and all circles have the same value. Determine the sum of two circles and a triangle. The equations are as follows:
\begin{align*}
\Delta + 3 \bigcirc &= 18 \\
2 \Delta + \bigcirc &= 14 \\
\Delta + 2 \bigcirc &= \ ?
\end{align*}
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Context:
In the arithmetic sequence {a_n}, it is known that a_5 > 0 and a_4 + a_7 < 0. Find the term number n that corresponds to the maximum value of the sum of the first n terms, S_n. Express your answer in the form S_n.
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Context:
If two distinct members of the set $\{ 3, 7, 21, 27, 35, 42, 51 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 63? Express your answer as a common fraction.
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Context:
Given the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $({{a}\_{n+1}}-{{a}\_{n}}={{(-1)}^{n+1}}\dfrac{1}{n(n+2)})$, find the sum of the first 40 terms of the sequence $\{(-1)^{n}a_{n}\}$.
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Context:
Consider a $2 \times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?
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Context:
Within the interval $1 \leqslant n \leqslant 10^{6}$, find the number of integers $n$ such that the equation $n = x^y$ has non-negative integer solutions $x$ and $y$ with $x \neq n$.
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Context:
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? If the area can be expressed as $x\pi$, find the value of $x$.
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Context:
求满足$\overline { a b c } ( a + b + c ) ^ { 3 }$的所有三位数abc.
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Context:
Starting with a sequence of $n 1's$, you can insert plus signs to get various sums. For example, when $n = 10$,  you can get the sum $1 + 1 + 1 + 11 + 11 + 111 = 136$, and the sum $1 + 1 + 11 + 111 + 111 = 235$.  Find the number of values of $n$ so that the sum of $1111$ is possible.
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Context:
已知函数 $f(x)=|2 x+1|-|3 x-a|(a>0)$
若 $f(x)$ 的图像与 $x$ 轴围成的三角形的面积大于 $\frac{5}{3}$, 求 $a$ 的取值范围.
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Context:
A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly.
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Context:
Find the maximum of the expression:
\[
| \ldots | |x_1 - x_2| - x_3| - \ldots | - x_{2023}|
\]
where \(x_1, x_2, \ldots, x_{2023}\) are distinct natural numbers between \(1\) and \(2023\).
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Context:
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right corner. What is the length \( x \) in centimeters?
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Context:
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?
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Context:
An 8 by 8 grid of numbers obeys the following pattern: 1) The first row and first column consist of all 1s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i-1)$ by $(j-1)$ sub-grid with row less than $i$ and column less than $j$. What is the number in the 8th row and 8th column?
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Context:
在正整数 1,2,\cdots,20210418 中，有多少个数的数码和是 8？
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Context:
For what values of $x$ in the following series, does the series converge?
\begin{align}
\sum^{\infty}_{n=1}\dfrac{(-1)^n x^n}{n[\log (n+1)]^2},\;\;-3<x<17
\end{align}
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Context:
Given a regular quadrangular prism \(ABCD-A_1B_1C_1D_1\) with the base \(ABCD\) being a unit square, if the dihedral angle \(A_1-BD-C_1\) is \(\frac{\pi}{3}\), find \(AA_1\).
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Context:
A trapezoidal plot has its parallel sides measuring 60 meters and 80 meters, and the remaining two sides measuring 50 meters each. The plot is to be enclosed by wire fencing with poles placed at varying intervals: 5 meters apart along the parallel sides and 7 meters apart along the other two sides. How many poles will be needed for this fencing?
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Context:
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly  $ 5\over 4$  times greater than  Vasya's result. What minimum number of portraits can be on the wall?

*Author: V. Frank*
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Context:
For how many integers $n$ between $1$ and $50$, inclusive, is
$$\frac{(n^2-1)!}{(n!)^n}$$
an integer? (Recall that $0! = 1$.)
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Context:
Two spheres are inscribed in a dihedral angle so that they touch each other. The radius of one sphere is twice the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. In your answer, provide the cosine of this angle, rounded to two decimal places if necessary.
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Context:
How many ways can you divide a set of $n^2$ elements into $n$ subsets?
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Context:
A farmer has twice as many pigs as cows, 4 more cows than goats, and 3 times as many chickens as pigs. Moreover, there are half as many ducks as the sum of cows and goats. If the farmer has 172 animals total, how many goats does he have?
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Context:
Find the radius of a circle in which a segment corresponding to a chord of length 6 cm contains a square with a side length of 2 cm.
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Context:
Given that $| \overrightarrow{a}|=6$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$.
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Context:
Given the set \( S = \{1, 2, \cdots, 1000\} \), a subset \( A \) of \( S \) with 201 elements is called a "good subset" if the sum of the elements in \( A \) is divisible by 5. Find the number of all good subsets of \( S \).
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Context:
In a court-mandated traffic class, there are a certain number of drunk drivers, 3 less than 7 times that many speeders, and twice the number of drunk drivers who are seatbelt violators. If there are 105 students total, how many drunk drivers are there?
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Context:
If $\overrightarrow{a}=(2,-1),\overrightarrow{b}=(t,3)$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtuse, then the range of real number $t$ is ____.
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Context:
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements.  Let $P(n)=n^2+n+1$.  What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
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Context:
Given the general term formula of the sequence ${a_n} = -n + t$ and the general term formula of the sequence ${b_n} = {2^n}$, the sequence ${c_n}$ satisfies ${c_n} = \frac{{{a_n} + {b_n}}}{2} + \frac{{\left| {{a_n} - {b_n}} \right|}}{2}$. It is also known that ${c_n} \geqslant {c_3}(n \in {N^*})$, find the range of values of the real number $t$.
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Context:
9. Given that $A B C D$ is a tetrahedron, and each edge is colored red or blue with equal probability, the probability that point $A$ can reach point $B$ through the red edges of the tetrahedron $A B C D$ is $\qquad$ .
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Context:
How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by 1 and then multiplied, the result is $N+1$?
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Context:
已知 $n$ 为正整数, 使得存在正整数 $x_{1}$, $x_{2}, \cdots, x_{n}$ 满足
$x_{1} x_{2} \cdots x_{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right)=100 n$.
求 $n$ 的最大可能值.
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Context:
If two functions $y=f(x)$ and $y=g(x)$ always satisfy $f(x)g(x) \geqslant 0$ over the same domain, then these two functions are called "harmonious functions". Given $f(x)=ax-20$ and $g(x)=\lg \left(\frac{x}{a}\right)$ ($a\in \mathbf{R}$) are harmonious functions over $x\in N^{*}$, the range of values for $a$ is ______.
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Context:
Determine all real numbers  $a$  for which the system of equations
\begin{align*}
3x^2+2y^2+2z^2&=a
4x^2+4y^2+5z^2&=1-a
\end{align*}
has at least one solution  $(x,y,z)$  in the real numbers.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[ \frac{2}{7}, \frac{3}{7} \right]}


Context:
4. Sharik and Matroskin are skiing on a circular track, half of which is an uphill climb and the other half is a downhill descent. Their speeds on the climb are the same and four times slower than their speeds on the descent. The minimum distance by which Sharik lags behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track.
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Context:
Given the sequence $\{a_n\}$, where $a_1=1$ and $a_{n+1}=3a_n+1$ for $n\in\mathbb{N}_+$, and let $b_n= \frac{n}{(3^n-1)\cdot2^{n-2}}\cdot a_n$, $T_n$ is the sum of the first $n$ terms of $\{b_n\}$. If the inequality $2^n\lambda < 2^{n-1}T_n+n$ holds for all $n\in\mathbb{N}_+$, then the range of the real number $\lambda$ is ______.
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Context:
Calculate the limit:

$$\lim_{n\to\infty}n\int_0^1 nx^{n-1}\left(\frac{1}{1+x}-\frac{1}{2}\right)\,dx$$
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Context:
Given a rectangle with a length of 8 and a width of 4, it is folded along a diagonal and pressed flat. Find the area of the overlapping part (grey triangle).
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Context:
Five different products, A, B, C, D, and E, are to be arranged in a row on a shelf. Products A and B must be placed together, while products C and D cannot be placed together. How many different arrangements are possible?
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Context:
Find the set of values of the parameter \(a\) for which the discriminant of the equation \(a x^{2} + 2x + 1 = 0\) is 9 times the square of the difference of its two distinct roots.
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Context:
Let $G = \mathbb{Z}/30\mathbb{Z} \oplus \mathbb{Z}$ and $H = \mathbb{Z}/15\mathbb{Z} \oplus \mathbb{Z}/7\mathbb{Z}$ be abelian groups. Determine the number of group homomorphisms from $G$ to $H$, i.e., find the number of elements in $\text{Hom}_{\mathbb{Z}}(G, H)$. \(\text{Hom}_{\mathbb{Z}}(G, H)\)
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Context:
Points \( M \), \( N \), and \( K \) are located on the lateral edges \( A A_{1} \), \( B B_{1} \), and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{3}{7} \), \( \frac{B N}{B B_{1}} = \frac{2}{5} \), and \( \frac{C K}{C C_{1}} = \frac{4}{9} \). Point \( P \) lies within the prism. If the volume of the prism is 40, find the maximum possible volume of the pyramid \( M N K P \).
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Context:
In a psychology experiment, an image of a cat or an image of a dog is flashed briefly onto a screen and then Anna is asked to guess whether the image showed a cat or a dog. This process is repeated a large number of times with an equal number of images of cats and images of dogs shown. If Anna is correct 95% of the time when she guesses "dog" and 90% of the time when she guesses "cat", determine the ratio of the number of times she guessed "dog" to the number of times she guessed "cat".
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Context:
A positive integer $k$ is said to be \textit{good} if there exists a partition of $\{1, 2, 3, \ldots, 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many \textit{good} numbers are there?
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Context:
24. In a $100 \times 100$ square table, some cells are shaded. Each shaded cell is the only shaded cell either in its column or in its row. What is the maximum number of cells that can be shaded?

## 10th grade
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Context:
A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7?
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Context:
If the maximum value of the function $f(x) = |a\sin x + b\cos x - 1| + |b\sin x - a\cos x|$ is $11$, then $a^{2} + b^{2} =$ ____.
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Context:
Evaluate $\lfloor 3.998 \rfloor + \lceil 7.002 \rceil$.
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Context:
Positive integers $a$, $b$, and $c$ satisfy the equation:
\[
\frac{ab}{a - b} = c
\]
What is the largest possible value of $a + b + c$ that does not exceed 99?
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Context:
Given $A \subseteq \{1, 2, \cdots, 2000\}$, and the absolute value of the difference between any two numbers in $A$ is not equal to 4 or 7, find the maximum value of $|A|$.
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Context:
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, and $\overrightarrow{b} \perp (2\overrightarrow{a}+ \overrightarrow{b})$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$.
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Context:
A board with dimensions $2018 \times 2018$ is divided into unit cells of size $1 \times 1$. In some cells of the board, black chips are placed, and in others, white chips are placed (each cell contains at most one chip).

First, we remove all black chips from columns that contain white chips. Then, we remove all white chips from rows that contain black chips.

Let $W$ be the number of remaining white chips, and $B$ be the number of remaining black chips on the board. Define $A = \min\{W, B\}$. Determine the maximum value of $A$.
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Context:
A positive integer $n$ is infallible if it is possible to select $n$ vertices of a regular 100-gon so that they form a convex, non-self-intersecting $n$-gon having all equal angles. Find the sum of all infallible integers $n$ between 3 and 100, inclusive.
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Context:
How many coefficients in the binomial expansion of $(x+1)^{65}$ are not divisible by $65$? Express your answer as a single integer.
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Context:
求最小的实数 C，使得对任意正实数 a_1, a_2, a_3, a_4, a_5（允许相同），总可以选择不同的下标 i,j,k,l，满足 \left|\df{a_i}{a_j}-\df{a_k}{a_l}\right|\leq C.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given a convex n-gon, circles are drawn with each of its sides as diameters, ensuring that the n-gon can be completely covered by these n circles. Determine the maximum value of n for which this is possible. Express your answer as a single integer.
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Context:
After a storm deposits 115 billion gallons of water into the city reservoir, the reservoir is 80% full. The original contents of the reservoir totaled some gallons. The reservoir was approximately 54.44444444444444% full before the storm. What is the total capacity of the reservoir?
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Context:
Mr. Rocky went to the market to sell his handmade crafts on a particular week. He was selling jewelry at $30 each and paintings at $100 each. However, he realized the income from the sales wasn't worth the labor and materials he had used, so he decided to increase the price of jewelry by $10 each and the cost of each painting by 20%. A 6% sales tax was applied to each jewelry item and an 8% sales tax was applied to each painting. Mr. Rocky also decided to offer a 10% discount on the total price if a buyer purchases more than one piece of jewelry, at least three paintings and spends a minimum of $800. Calculate the total price a buyer who takes two pieces of jewelry and five paintings would pay after applying the sales tax and the discount.
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Context:
How many squares of side at least $8$ have their four vertices in the set $H$, where $H$ is defined by the points $(x,y)$ with integer coordinates, $-8 \le x \le 8$ and $-8 \le y \le 8$?
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Context:
Given the sequence $\{n(n+4)\left(\frac{2}{3}\right)^{n}\}$, find the value of $k$ such that the $k$-th term is the largest term in the sequence. Express your answer as a single integer.
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Context:
In parallelogram  $ABCD$ ,  $AC=10$  and  $BD=28$ . The points  $K$  and  $L$  in the plane of  $ABCD$  move in such a way that  $AK=BD$  and  $BL=AC$ . Let  $M$  and  $N$  be the midpoints of  $CK$  and  $DL$ , respectively. What is the maximum walue of  $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$  ?
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Context:
A screenwriter is collaborating with James Nesbitt on a new movie project. They need to write 120 pages of the script. The screenwriter wants to write 3 pages every day, while James suggests writing 5 pages on weekends. If the screenwriter follows this plan and writes every day, how many weekends will it take them to complete the script?
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Context:
A 16-step path is to go from $( -4, -4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. Find the number of such paths that stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.
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Context:
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$ and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
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Context:
Given that $\alpha, \beta \in \{1, 2, 3, 4, 5\}$, find the total number of pairs $(\alpha, \beta)$ such that $\sin(\alpha) \cdot \cos(\beta) < 0$. Express your answer as a single integer.
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Context:
There are ten cards, each with one of the digits from 0 to 9. They are arranged in a sequence to display the number 9072543681. If swapping the positions of two adjacent cards is considered one operation, what is the minimum number of operations needed to rearrange the original number into a number that is divisible by 99?
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Context:
John and Jane went out for a dinner and they ordered the same dish. John used a 10% discount coupon while Jane used a 15% discount coupon. Both were subjected to a varying tax rate, with John having to pay an 8% tax and Jane having to pay a 6% tax over the original price of the dish. John paid a 15% tip over the original price of the dish, while Jane paid the tip over the discounted price for the coupon. If John paid $1.00 more than Jane, what was the original price of the dish?
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Context:
As shown in the diagram, \(ABCD\) is a square with side length \(6\). \(ADGH\) is a trapezoid, and points \(E\) and \(F\) are the midpoints of \(AD\) and \(GH\), respectively. It is given that \(HF = 6\), \(EF = 4\), and \(EF \perp GH\). Line segment \(HE\) is extended to intersect \(CD\) at point \(I\). A line segment \(IJ\) is drawn perpendicular to \(HA\). Find the length of \(IJ\).
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Context:
Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
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Context:
What is the largest area of the projection of a rectangular parallelepiped with edges \(a\), \(b\), and \(c\) onto a plane?
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Context:
求下述反三角式子的和：

$\arcsin \frac { \sqrt { 3 } } { 2 } + \arcsin \frac { \sqrt { 8 } - \sqrt { 3 } } { 6 } + \arcsin \frac { \sqrt { 1 5 } - \sqrt { 8 } } { 1 2 } +$

$\cdots + \arcsin \frac { \sqrt { ( n + 1 ) ^ { 2 } - 1 } - \sqrt { n ^ { 2 } - 1 } } { n ( n + 1 ) } ,$

这里n为给定的正整数.
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\boxed{$ \arccos \frac { 1 } { n + 1 } .$}


Context:
In trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 30 units, respectively, and the altitude is 18 units. Points $T$ and $U$ are the midpoints of sides $PR$ and $QS$, respectively. What is the area of quadrilateral $TURS$?
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Context:
Find the closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ and express it in the form $1 - \frac{a^{n+b}}{3^{2^{n+c}}-1}$, where $a$, $b$, and $c$ are integers. Find $a+b+c$.
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Context:
Points $A$, $B$, and $C$ form an isosceles triangle with $AB = AC$. Points $M$, $N$, and $O$ are the midpoints of sides $AB$, $BC$, and $CA$ respectively. Find the number of noncongruent triangles that can be drawn using any three of these six points as vertices.
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Context:
If j is the least positive integer that is divisible by every integer from 10 to 15 inclusive, what is the result when j is divided by 26?
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Context:
In a certain cleaning solution, the ratio by volume of bleach to detergent to water is 2 : 25 : 100. The solution will be altered so that the ratio of bleach to detergent is tripled, and the ratio of detergent to water is halved. If the altered solution will contain 80 gallons of water, how many quarts of detergent will it contain? Note that 1 gallon equals 4 quarts.
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Context:
ABCD is a cyclic quadrilateral with circumcenter $O$ and circumradius 7. The line segments $AB$ and $CD$ intersect at point $E$, while $DA$ and $CB$ intersect at point $F$. Given that $OE = 13$ and $OF = 14$, let $\cos\angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime integers. Find the sum $p + q$.
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Context:
Let's call integer square-free if it's not divisible by  $p^2$  for any prime  $p$ . You are given a square-free integer  $n>1$ , which has exactly  $d$  positive divisors. Find the largest number of its divisors that you can choose, such that  $a^2 + ab - n$  isn't a square of an integer for any  $a, b$  among chosen divisors.

*(Proposed by Oleksii Masalitin)*
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Context:
For the sequence $\{a_n\}$, we have $a_1 = 5$ and $a_n = 2a_{n-1} + 2^{n} - 1$ for $n \ge 2$ and $n \in \mathbb{N}^*$. If there exists a real number $\lambda$ such that the sequence $\left\{ \frac{a_n + \lambda}{2^n}\right\}$ is an arithmetic progression, then find the value of $\lambda$.
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Context:
the average runs scored by a batsman in 15 matches is 30 . in the next 20 matches the batsman scored an average of 15 runs . find his average in all the 30 matches ?
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Context:
Given a quadratic function $f(x)=ax^{2}+2x+c$ $(x \in \mathbb{R})$ whose range is $[0, +\infty)$, find the minimum value of $\frac{a+1}{c} + \frac{c+1}{a}$.
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Context:
Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of $N$.
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Context:
Let $X,$ $Y,$ and $Z$ be points on the line such that $\frac{XZ}{ZY} = 3$. If $Y = (2, 6)$ and $Z = (-4, 8)$, determine the sum of the coordinates of point $X$.
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Context:
Right triangle $XY Z$ has right angle at $Y$ and $XY = 228$, $Y Z = 2004$. Angle $Y$ is trisected, and the angle trisectors intersect $XZ$ at $P$ and $Q$ so that $X$, $P$, $Q$,$Z$ lie on $XZ$ in that order. Find the value of $(PY + Y Z)(QY + XY )$.
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Context:
In a hockey tournament, there are $n$ teams participating. Each team plays with every other team exactly once, where the winning team is awarded 2 points, a draw gives each team 1 point, and the losing team gets 0 points. What is the maximum point difference that can occur between teams that are ranked in consecutive positions?
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Context:
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}, BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
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Context:
Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked?
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Context:
Given that $ \frac{\pi}{2} < \beta < \alpha < \frac{3\pi}{4} $, and $ \cos (\alpha - \beta) = \frac{12}{13} $, $ \sin (\alpha + \beta) = -\frac{3}{5} $. Find the value of $ \sin \alpha + \cos \alpha $.
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Context:
In the "Turn Back Difference" game, two polynomials $-m$ and $-n$ are operated on sequentially. 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. Express your answer as a polynomial.
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Context:
Let natural numbers \( k \) and \( n \) be coprime, where \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line formed by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting the same endpoints. For example, a \((5 ; 2)\)-star has 5 points of self-intersection, which are the bold points in the drawing. How many self-intersections does the \((2018 ; 25)\)-star have?
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Context:
## Task 5 - 230935

In a triangle $A B C$, a line parallel to $A B$, whose position is otherwise not specified, intersects the side $A C$ at a point $A_{1}$ between $A$ and $C$, and it intersects the side $B C$ at $B_{1}$. Furthermore, let $P$ be a point on $A B$ between $A$ and $B$, whose position is also not otherwise specified.

The area of triangle $A B C$ is $F_{0}$, and the area of triangle $A_{1} B_{1} C$ is $F_{1}$.

Determine the area $F$ of the quadrilateral $A_{1} P B_{1} C$ in terms of $F_{0}$ and $F_{1}$!
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Context:
In Class 1 of Grade 12, there are a total of 56 students, numbered consecutively from 1 to 56. A systematic sampling method is used to draw a sample of 4 students. It is known that students with the numbers 6, 34, and 48 are included in the sample. Determine the number of the fourth student in the sample.
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Context:
Given that the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy the conditions $|\overrightarrow{a}| = 2|\overrightarrow{b}| = 2$ and $|\overrightarrow{a} - 4\overrightarrow{b}| = 2\sqrt{7}$, find the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.
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Context:
A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.
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Context:
At a tanning salon, customers are charged a certain amount for their first visit in a calendar month and $8 for each visit after that in the same calendar month. In the last calendar month, 100 customers visited the salon, of which 30 made a second visit, and 10 made a third visit. All other customers made only one visit. The revenue for the last calendar month at the salon was $1240. What is the charge for the first visit?
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Context:
Find the smallest real number \(\lambda\) such that the inequality
$$
5(ab + ac + ad + bc + bd + cd) \leq \lambda abcd + 12
$$
holds for all positive real numbers \(a, b, c, d\) that satisfy \(a + b + c + d = 4\).
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Context:
A natural number A has three digits added to its end, forming a new number. This new number is equal to the sum of all natural numbers from 1 to A. Find the value of A.
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Context:
Given that the area of △ABC is S, and $| \overrightarrow {BC}|^{2}$ = $\overrightarrow {CA}$ • $\overrightarrow {CB}$ + 2S, determine the value of ∠B.
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Context:
Given a $202 \times 202$ matrix $A$ with $\|A\|_2 = 100$ and $\|A\|_f = 101$, determine the sharpest lower bound on the 2-norm condition number $k(A)$. Assume $k(A) = \|A\|_2 \cdot \|A^{-1}\|_2$. Provide your answer as a single numerical value.
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Context:
Consider a geometric progression with a non-zero natural number as the common ratio. The logarithm of the nth term, with the base equal to the common ratio, is 6. The logarithm of the product of the first n terms, with the same base, is 20. The product of the first and the nth term of the sequence is 243. Determine the sum of the first n terms of this sequence.
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Context:
Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations.
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Context:
Let real numbers x and y satisfy the following equations:
x^3 = 3y^2x + 5 - sqrt(7)
y^3 = 3x^2y + 5 + sqrt(7)

Find the value of x^2 + y^2. Express your answer as a single number.
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Context:
1. In each cell of a $75 \times 75$ table, a natural number is written. In each row, there are at least 15 different numbers, and in any three consecutive rows, there are no more than 25 different numbers. What is the maximum number of different numbers that can be in the table?
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Context:
A sequence of $28$ letters consists of $14$ of each of the letters $A$ and $B$ arranged in random order. The expected number of times that $ABBA$ appears as four consecutive letters in that sequence is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
2. Solve the equation $\frac{x}{3}+\frac{x}{\sqrt{x^{2}-9}}=\frac{35}{12}$. Write the sum of all obtained solutions in the answer.

(5 points)
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Context:
Robert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the $8$ coins.
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Context:
Let \( f:(0,1) \rightarrow (0,1) \) be a differentiable function with a continuous derivative such that for every positive integer \( n \) and odd positive integer \( a < 2^n \), there exists an odd positive integer \( b < 2^n \) such that \( f\left( \frac{a}{2^n} \right) = \frac{b}{2^n} \). Determine the set of possible values of \( f^{\prime} \left( \frac{1}{2} \right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players, A and B, are competing in a table tennis match. The winner of each game scores 1 point, and the loser scores 0 points. The match ends when one player has 2 points more than the other or after 6 games, whichever comes first. The probability of A winning a game is $\frac{2}{3}$, and the probability of B winning a game is $\frac{1}{3}$. The outcomes of the games are independent. Find the expected number of games $\xi$ played when the match ends, $\mathrm{E} \xi$.
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Context:
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$?
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Context:
In a campus singing competition, 15 students participated and each student had a unique score. The top 7 students based on their scores will advance to the final round. What statistical measure should Xiao Ming know, along with his own score, to determine whether he can advance to the final round?
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Context:
Simplify the following expression:

$$
\frac{bx(a^{2}x^{2} + 2a^{2}y^{2} + b^{2}y^{2})}{bx + ay} + \frac{ay(a^{2}x^{2} + 2b^{2}x^{2} + b^{2}y^{2})}{bx + ay}
$$

Express your answer as a simplified algebraic expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The equation
$$
1 - 3 \sin \theta + 5 \cos 3\theta = 0 \quad (0 < \theta < 2\pi)
$$
has how many distinct solutions? Express your answer as a single integer.
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Context:
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three unit vectors such that $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = 0,$ and the angle between $\mathbf{b}$ and $\mathbf{c}$ is $60^\circ.$  Compute $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|.$
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Context:
In the equilateral triangle \(PQR\), \(S\) is the midpoint of \(PR\), and \(T\) is on \(PQ\) such that \(PT=1\) and \(TQ=3\). Many circles can be drawn inside the quadrilateral \(QRST\) such that no part extends outside of \(QRST\). The radius of the largest such circle is closest to:
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Context:
In $\triangle ABC$, $\angle A = 42^{\circ}$. If the trisectors of $\angle B$ and $\angle C$ meet at points $D$ and $E$ respectively, then what is the measure of $\angle BDC$?
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Context:
At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make?
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Context:
Chang's Garden has six kinds of apples: sweet, sour, crunchy, soft, juicy, and tangy apples. The sale prices for each apple variety depend on their weight. The prices and proportions for each kind of apple are as follows:

Sweet apples:
- Light (under 100 grams): $.4
- Heavy (100 grams or more): $.6
- Proportion of total apples: 40%

Sour apples:
- Light: $.1
- Heavy: $.15
- Proportion of total apples: 20%

Crunchy apples:
- Light: $.25
- Heavy: $.35
- Proportion of total apples: 15%

Soft apples:
- Light: $.15
- Heavy: $.25
- Proportion of total apples: 10%

Juicy apples:
- Light: $.2
- Heavy: $.3
- Proportion of total apples: 10%

Tangy apples:
- Light: $.05
- Heavy: $.1
- Proportion of total apples: 5%

For each type of apple, 60% of them are light, and 40% are heavy. If Chang earns $120, how many apples did his trees give him?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $n$-plant to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it perfect if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.

Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a six-digit number $\overline{xy243z}$ that is divisible by 396.
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Context:
There are several chairs inside the church. If 9 students sit per chair, one student cannot sit, and if 10 students sit per chair, 1 chair becomes vacant. How many students went to church?
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Context:
A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2 - 4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
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Context:
Given the series $\sum_{n=1}^{\infty} \left(\sqrt[4]{n^4+2n^2}-\sqrt[3]{p(n)}\right)$, find a polynomial $p(n)$ such that this series converges.
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Context:
Let $x, y, z$ be positive real numbers satisfying $x^3 + y^3 + z^3 = 3$. Find the minimum value of the expression \[ P = \frac{1}{2-x} + \frac{1}{2-y} + \frac{1}{2-z}. \]
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Context:
A large circular table has 60 chairs around it. What is the largest number of people who can sit around the table so that each person is only sitting next to exactly one other person?
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Context:
A cashier, upon checking the account before leaving work, finds that the cash is 153 yuan less than the account book. She knows the actual amount collected cannot be wrong, so it must be due to a decimal point error during bookkeeping. What is the actual amount of the cash that was recorded incorrectly?
Please reason step by step, and put your final answer within \boxed{}.


Context:
a group of people participate in some curriculum , 25 of them practice yoga , 18 study cooking , 10 study weaving , 4 of them study cooking only , 5 of them study both the cooking and yoga , 4 of them participate all curriculums . how many people study both cooking and weaving ?
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Context:
The altitudes $A D$ and $B E$ of an acute-angled triangle $A B C$ intersect at point $H$. The circumcircle of triangle $A B H$ intersects sides $A C$ and $B C$ at points $F$ and $G$ respectively. Find $F G$, if $D E=5$ cm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The tax on a commodity is diminished by 20%, and its consumption increased by some percentage. The effect on revenue is 8%. What is the percentage by which the consumption increased?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let's determine the area enclosed by the curves $y=\sin x$ and $y=(4 / \pi)^{2} \sin (\pi / 4) x^{2}$ (the latter being a quadratic function)!
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Context:
Solve the linear recurrence relation with binomial coefficients:

\[ a_n = \binom{9}{1}a_{n-1}-\binom{9}{2}a_{n-2}+\binom{9}{3}a_{n-3}-\dots+\binom{9}{9}a_{n-9} \]

Given the initial conditions:
\[ a_0=a_1=a_2=\dots=a_7=0, \quad a_8=1, \quad \text{for} \quad n\ge9 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find (in terms of $n \geq 1$) the number of terms with odd coefficients after expanding the product: $\prod_{1 \leq i<j \leq n}\left(x_{i}+x_{j}\right)$
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Context:
When a wire was divided by intervals of 120 centimeters (cm), it was split into two parts, with 2.4 centimeters (cm) being left. Find how many centimeters (cm) of wire you'll get if you divide this wire into three parts.
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Context:
The function $f(x) = 2^{x-1} - \ln x - a$ has exactly two distinct zeros. Find the range of $a$.
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Context:
Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\angle G_{3} G_{1} G_{4}=30^{\circ}$. Let $G_{3} G_{4}$ and $G_{1} G_{2}$ meet at $G_{5}$. Determine the length of segment $G_{2} G_{5}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=a_{2}=1$, 且 $a_{n+2}=\frac{1}{a_{n+1}}+a_{n}, n=1,2, \cdots$. 求 $a_{2004}$. 原始答案是 \(\frac{m!!}{n!!}\) 格式，请给出 m + n 的值。
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Context:
给定整数 $m 、 n(2 \leqslant m<n,(m, n)=1)$ 。求最小的整数 $k$ ，满足对集合 $\{1,2, \cdots, n\}$ 的任意 $m$ 元子集 $I$, 若 $\sum_{i \in I} i>k$, 则存在 $n$ 个实数 $a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, 使得
$$  \frac{1}{m} \sum_{i \in I} a_{i}>\frac{1}{n} \sum_{i=1}^{n} a_{i}  $$  ，设m=2，n=3，求出k的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots?
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Context:
Two trains of length 100 m and 160 m are running towards each other on parallel lines at 42 kmph and another speed respectively. They will be clear of each other in 12.998960083193344 seconds from the moment they meet. What is the speed of the second train in kmph?
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Context:
Evaluate \( \frac{1}{\int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\, dx} \).
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Context:
设 $x_{1}=3, x_{n+1}=\sqrt{x_{n}+14}-\sqrt{x_{n}+2}\left(n \in N^{*}\right)$, 求当$n=100$时$\left[x_{1}+x_{2}+\cdots+x_{n}\right]$ 的值。
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Context:
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
记$F = \underset { 1 \leq x \leq 3 }\max\vert x ^ { 3 } - a x ^ { 2 } - b x - c \vert$.当$a$、$b$、$c$取遍所有实数时，求$F$的最小值．
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Context:
Points $M$ and $N$ are taken on the sides $AB$ and $AD$ of parallelogram $ABCD$ such that the lines $MC$ and $NC$ divide the parallelogram into three equal areas. Find $MN$, if $BD = d$.
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Context:
A trapezoid with a top base of 5 and a side of 6 can have an inscribed circle and a circumscribed circle. Calculate the area of the pentagon formed by the radii of the inscribed circle, perpendicular to the sides of the trapezoid, its bottom base, and the corresponding segments of the side lengths.
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Context:
Find the smallest integer $N \in \mathbb{N}$ such that the Taylor polynomial $T_Nf(x,0)$ of $f(x) = \cos(x)e^x$ centered at $x = 0$ satisfies the inequality $$|f(x) - T_Nf(x,0)| \leq 10^{-6}$$ for all $x \in [-\frac{1}{10}, \frac{1}{10}]$ without using the Lagrange remainder theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( x, y, z \in \mathbf{Z}_{+} \) and \([x, y] = 72\), \([x, z] = 600\), \([y, z] = 900\), where \([m, n]\) represents the least common multiple of integers \( m \) and \( n \), find the number of ordered triples \((x, y, z)\) satisfying these conditions. Express your answer as a single integer.
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Context:
Consider a hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Let $F_1$ be its left focus, and let $MN$ be a chord perpendicular to the real axis with $A$ as the right vertex. If $\overrightarrow{AM} \cdot \overrightarrow{AN} = 0$, find the eccentricity of the hyperbola.
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Context:
Tracy initially had a bag of candies. She first ate $\frac{1}{4}$ of them, then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. After that, Tracy and her mom each ate 12 candies from what was left. Lastly, her brother took between two to six candies, leaving Tracy with five candies. How many candies did Tracy have at the beginning?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train 110 m long running at the speed of 90 km/hr takes 9.679225661947045 seconds to cross a bridge of a certain length. What is the length of the bridge in meters?
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Context:
What is the largest number of digits that can be erased from the 1000-digit number 201820182018....2018 so that the sum of the remaining digits is 2018?
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Context:
A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where the bug is located, uses no edge multiple times, and uses at most two of the edges adjacent to any particular face. Find the number of healthy paths.
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Context:
Determine the domain of the function $y= \sqrt {\sin x}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
When \(0 < x < \frac{\pi}{2}\), the value of the function \(y = \tan 3x \cdot \cot^3 x\) cannot take numbers within the open interval \((a, b)\). Find the value of \(a + b\).
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Context:
Given that $x > 0$, $y > 0$, if ${2}^{x}\cdot {8}^{y}=16$, then ${2}^{-1+{\mathrm{log}}_{}\left(2x\right)}+{\mathrm{log}}_{}^{27y}=\_\_\_\_\_\mathrm{.}$
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Context:
Find the mass of the solid $E$ where $E$ is outside the sphere $x^2 + y^2 + z^2 = 2z$, but inside the hemisphere $x^2 + y^2 + z^2 = 4$, $z\geq 0$ and its density is $\rho(x,y,z)=z\sqrt{x^2 +y^2 +z^2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum possible value of the probability $P[Y=0]$, given that $Y$ is a non-negative integer-valued random variable with $E[Y]=1$, $E[Y^2]=2$, and $E[Y^3]=5$?
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Context:
5. Given the number $500 \ldots 005$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 165. In how many ways can this be done?
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Context:
Find the value of \(x\) to three decimal places for the equation \(4^x = \frac{5}{x}\).
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Context:
Find the unique 3-digit number $N=\underline{A}\ \underline{B}\ \underline{C}$, whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}\ \underline{B}\ \underline{C} \times \underline{A}\ \underline{B} \times \underline{A}$ is divisible by $1000$.
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Context:
Given that point P (a, b) is outside the circle C: x^2 + y^2 = 1, describe the positional relationship between the line ax + by + 1 = 0 and circle C. Express your answer as a single geometric term (e.g. Tangent, Separate, Intersect).
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Context:
When the child was born, neither of his parents was yet 40 years old, but they were already adults. When the child turned 2 years old, the age of exactly one of the parents was divisible by 2; when the child turned 3 years old, the age of exactly one of the parents was divisible by 3; and so on. How long could this pattern have continued?
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Context:
Given that \( x - \frac{1}{x} = 5 \), find the value of \( x^4 - \frac{1}{x^4} \).
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Context:
Simplify the following propositional logic expression to the form with the least atomic propositions:

$$\lnot ((P \Rightarrow (Q \Rightarrow R)) \Rightarrow (Q \Rightarrow (P \Rightarrow R)))$$
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Context:
Mark Watney tests the durability of a new dome designed for the Mars expedition. The dome is in the shape of a hemisphere with a radius of 20 meters. Mark turns to the north and shoots at an angle of $45^{\circ}$ to the ground, then turns to the south and also shoots at an angle of $45^{\circ}$ (see figure). What values can $\left|P_{1} P_{2}\right|$ - the distance between the points of impact, take?
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Context:
Given a grid formed by $n$ horizontal lines and $n$ vertical lines, how many distinct closed paths with $2n$ segments can be drawn such that each path passes through each horizontal line and each vertical line exactly once?
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Context:
Given that the original value of the house is $12000, Mr. A sells it at a 15% loss, and then Mr. B sells it back to Mr. A at a 20% gain after holding it for one year and accruing a 5% annual interest, calculate the total amount Mr. A spends to buy back the house including the interest accrued.
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Context:
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB|  = 4$. Let $E $ be the intersection  point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.
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Context:
Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$?
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Context:
Given the function f(x)=sin(ωx+φ) (ω>0, |φ|≤π/2), x=-π/4 is a zero point of f(x), x=π/4 is the symmetric axis of the graph of y=f(x), and f(x) is monotonic in (π/18, 5π/36), find the maximum value of ω.
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Context:
Given that $S$, $A$, $B$, $C$ are distinct points on the surface of a sphere $O$, $SA \perp$ plane $ABC$, $AB \perp BC$, $AB=1$, $BC= \sqrt{2}$, and the surface area of sphere $O$ is $4\pi$, calculate the value of $SA$. Express your answer as a single number.
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Context:
For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i=\sqrt{-1}$?
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Context:
The line $x+ay=3$ is tangent to the circle $(x-1)^{2}+y^{2}=2$. Find the value of $a$.
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Context:
In a certain kingdom, there were 32 knights. Some of them were vassals of others (a vassal could have only one suzerain, and a suzerain was always richer than his vassal). A knight who had at least four vassals bore the title of baron. What is the maximum number of barons that could be under these conditions?

(The kingdom had a law: "a vassal of my vassal is not my vassal".)

#
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Context:
A  $3$  by  $3$  determinant has three entries equal to  $2$ , three entries equal to  $5$ , and three entries equal to  $8$ . Find the maximum possible value of the determinant.
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Context:
Determine the number of distinct ways to express 1,000,000 as a product of three natural numbers, where products that differ only in the order of the factors are considered identical.
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Context:
A sphere is circumscribed around a tetrahedron \(ABCD\). Another sphere with radius 1 is tangent to plane \(ABC\), and both spheres are internally tangent at point \(D\). Given \(AD = 3\), \(\cos \angle BAC = \frac{4}{5}\), and \(\cos \angle BAD = \cos \angle CAD = \frac{1}{\sqrt{2}}\), find the volume of the tetrahedron \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x \neq y$, we have $xy \notin A$. Determine the maximum possible size of $A$.
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Context:
Pyramid $EARLY$ has rectangular base $EARL$ and apex $Y$, and all of its edges are of integer length. The four edges from the apex have lengths $1, 4, 7, 8$ (in no particular order), and EY is perpendicular to $YR$. Find the area of rectangle $EARL$.
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Context:
A man can row a boat at 15 kmph in still water. If he rows downstream, where the speed of current is 3 kmph, he takes 17.998560115190784 seconds to cover a certain distance. What is the distance he covers?
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Context:
设三次函数$f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d ( a \neq 0 )$在区间$［0,1］$上，其值的绝对值均不大于1，求$|a|+|b|+|c|+|d|$的最大值.
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Context:
Solve the equation \(\sqrt{4 x^{2}+2 x+1}-\sqrt{4 x^{2}+14 x+5}=6 x+2\) and determine the number of real roots. Express your answer as a single integer.
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Context:
Arrange the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) in some order to form a nine-digit number \(\overline{\text{abcdefghi}}\). If \(A = \overline{\text{abc}} + \overline{\text{bcd}} + \overline{\text{cde}} + \overline{\text{def}} + \overline{\text{efg}} + \overline{\text{fgh}} + \overline{\text{ghi}}\), find the maximum possible value of \(A\).
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Context:
An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon?
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Context:
Consider the piecewise constant interpolation of the function $f(x) = ln(x)$ , $10 ≤ x ≤ 11$ , at points $x_i = 10+ih$, where $h = 0.1$. Thus, our interpolant satisfies $v(x) = ln(10+ih)$, $10 + ih − 0.5h ≤ x < 10 + ih + 0.5h$, for $i = 0, 1, . . . , 11$. Find a bound for the error in this interpolation, i.e. find a (reasonably small) constant M such that $|f(x) − v(x)| < M$ for any $10 ≤ x ≤ 11$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the cost of fencing around a circular field of diameter 20 m at the rate of Rs. 1.50 a meter. What is the total cost of fencing?
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Context:
Cary starts working at Game Stop for $10/hour. In the first year, she works 40 hours a week for 50 weeks, with 2 weeks of vacation. After this year, she gets a 20% raise. Additionally, she earns a performance bonus of $500 at the end of the year.

In the second year, her base pay is cut to 75% of what it used to be due to the company's profits decreasing, but she is now working an average of 50 hours per week, where hours beyond 40 are paid at 1.5 times the normal rate. She takes only 1 week of vacation in the second year. Moreover, she incurs a $300 penalty for underperformance in the second year.

In the third year, she receives another 10% raise of her base pay, almost back to her original starting wage, and her hours go back to 40 hours a week with 2 weeks of vacation. Also, she earns a performance bonus of $1,000 at the end of the third year.

Throughout these three years, the income tax rates change: in the first year, it is 20%; in the second year, 22%; and in the third year, it is 18%. Calculate Cary's total net earnings, after taxes and accounting for bonuses and penalties, over these three years.
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Context:
Simplify the expression \(\frac{\log _{a} \sqrt{a^{2}-1} \cdot \log _{1 / a}^{2} \sqrt{a^{2}-1}}{\log _{a^{2}}\left(a^{2}-1\right) \cdot \log _{\sqrt[3]{a}} \sqrt[6]{a^{2}-1}}\).
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Context:
Given that \(b\) and \(c\) are both integers and \(c < 2000\). If the quadratic equation \(x^{2} - bx + c = 0\) has roots whose real parts are both greater than 1, how many pairs \((b, c)\) satisfy this condition?
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Context:
I live on a very short street with 14 small family houses. The odd-numbered houses from 1 are on one side of the street, and the even-numbered houses from 2 are on the opposite side (e.g., 1 and 2 are opposite each other).

On one side of the street, all families have surnames that are colors, and on the other side, the surnames indicate professions.

Szabó and Fazekas live opposite to Zöld and Fehér, respectively, who are both neighbors of Fekete.

Kovács is the father-in-law of Lakatos.

Lakatos lives in a higher-numbered house than Barna. The sum of the house numbers of Lakatos and Barna is equal to the sum of the house numbers of Fehér and Fazekas. Kádárné's house number is twice the house number of her sister, Kalaposné.

Sárga lives opposite Pék.

If Bordóné's house number is two-digit and she lives opposite her sister, Kádárné, what is the house number of Mr. Szürke?
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Context:
Calculate the volume enclosed by the paraboloid $x^2 + y^2 = z$, the cylinder $y = x^2$, and the planes $z=0$ and $y = 1$. Additionally, consider the revised condition $0 \le z \le 1$.
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Context:
$已知函数f(x)=|x-2|,g(x)=|2x+3|-|2x-1|.$
$若f(x+a)\geq g(x)，求a的取值范围.$
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Context:
Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$.
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Context:
A volunteer organization is recruiting new members. In the fall, they manage to increase their number by 8%. By the spring, however, membership falls by some percentage. The total change in percentage from fall to spring is 12.52%. What was the percentage decrease in the spring?
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Context:
It is known that $\operatorname{tg} \alpha$ and $\operatorname{tg} 3 \alpha$ are integers. Find all possible values of $\operatorname{tg} \alpha$.
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Context:
Let \(ABC\) be a triangle where \(AB = 9\), \(BC = 10\), and \(CA = 17\). Let \(\Omega\) be its circumcircle, and let \(A_1\), \(B_1\), \(C_1\) be the diametrically opposite points from \(A\), \(B\), and \(C\), respectively, on \(\Omega\). Find the area of the convex hexagon with the vertices \(A, B, C, A_1, B_1, C_1\).
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Context:
In a tournament with 5 teams, there are no ties. In how many ways can the $\frac{5 \cdot 4}{2}=10$ games of the tournament occur such that no team wins all their games and no team loses all their games?
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Context:
Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1, z_2$ with $|z_1| = |z_2| = 1$ and $\{z_1, z_2\} \neq \{-i, i\}$ such that:
\[
z_1(z_1^3 + z_1^2 + rz_1 + 1) = z_2(z_2^3 + z_2^2 + rz_2 + 1).
\]
There exist real numbers $a, b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r \le b$. If the value of $|a| + |b|$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $100p + q$.
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Context:
In the geometric sequence {a_n}, a_1 = 9 and a_5 = a_3 * a_4^2. Find the value of a_4. Express your answer as a numerical value, including the ± symbol if necessary.
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Context:
Evaluate the following limit:

$$\lim_{n \to \infty} \sum_{k=n}^{2n-1} \frac{1}{n+\sqrt{k+1}}$$
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Context:
设 f(x)=\ds{n=0}{\infty}2^{-n}\cos nx，记 A=\int_0^{2\pi}|f(x)|^2\dd x，求 [10A].
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Context:
Rhona wrote down a list of nine multiples of ten: $10, 20, 30, 40, 50, 60, 70, 80, 90$. She then deleted some of these multiples so that the product of the remaining multiples was a square number. What is the least number of multiples that she could have deleted? Express your answer as a whole number.
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Context:
2. On a sheet of lined paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, while the second has the opposite. Find the maximum possible area of their common part, if the first rectangle contains 2015 cells, and the second - 2016.
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Context:
10. The base of the right prism $A B C A_{1} B_{1} C_{1}$ is a triangle $A B C$ with angle $B$ equal to $90^{\circ}$ and angle $C$ equal to $30^{\circ}$. Find the area of the section of the prism by a plane passing through the center of the lateral face $A A_{1} C_{1} C$ and vertex $B$ and parallel to the diagonal of the lateral face $A B_{1}$, if the distance from vertex $C$ to the cutting plane is 2, and the hypotenuse of the base of the prism is $\sqrt{14}$?
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Context:
Alice has three daughters, each of whom has two daughters. Each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)
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Context:
Find the fourth order partial derivative $K_{xxxy}$ of the function $K(x,y)$ defined as:

$$ K_x = \lim_{h \to 0} \frac{y[(x+h)(x+1)]^{2019} - y[x(x+h+1)]^{2019}}{h(x+1)^{2019}(x+h+1)^{2019}} $$
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Context:
1. How many five-digit numbers have the first digit equal to the last, the digit in the thousands place equal to the digit in the tens place, and the product of all digits equal to the square of a natural number?
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Context:
The numbers $a, b, c, d$ belong to the interval $[-6.5, 6.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.
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Context:
Find all three-digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! \times b! \times c!$.
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Context:
I have recently come across iterated integrals of the form $$\int_0^tdt_1(t-t_1)^{-1/2}\int_0^{t_1}dt_2 (t_1-t_2)^{-1/2}\cdots\int_{0}^{t_{m-1}}dt_m(t_{m-1}-t_m)^{-1/2}. \tag{1}$$ I am wondering if there is a closed formula for these expressions for general $m$. If $m=2$, then the expression above reduces to $$2\int_0^t dt_1 (t-t_1)^{-1/2} t_1^{1/2} = 2t\int_0^1d\tau(1-\tau)^{-1/2}\tau^{1/2} = \pi t.$$ But it's not clear to me how to generalize this to $m>2$.
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Context:
The radius of a right circular cylinder is decreased by $20\%$ and its height is increased by $25\%$. What is the absolute value of the percent change in the volume of the cylinder?
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Context:
Little Johnny invested $10,000. He put part of it in a bank at 5 percent interest and the remainder in bonds which pay a 9 percent annual return. He invested $6,000 in each method (savings bank and bond). What was his annual income from the two investments?
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Context:
If  $ABCDE$  is a regular pentagon and  $X$  is a point in its interior such that  $CDX$  is equilateral, compute  $\angle{AXE}$  in degrees.

*2020 CCA Math Bonanza Lightning Round #1.3*
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Context:
How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.
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Context:
Peggy has 6 dolls. Her grandmother gives Peggy her own collection of 28 dolls. Over the year, Peggy receives half that amount of dolls for her birthday. She loses 10% of her total dolls during a spring cleaning mishap. For Easter, she receives one-third of the birthday gift amount, and her friend gifts her another doll in exchange for 2 of Peggy's dolls. At Christmas, Peggy gets 20% more dolls than the number of Easter dolls, but her dog chews up and ruins 3 of her dolls. After all these events, how many dolls does Peggy now have?
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Context:
A box contains 5 white balls and 3 black balls. What is the probability that when drawing the balls one at a time, all draws alternate in color starting with a white ball?
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Context:
Let a constant $a$ make the equation $\sin x + \sqrt{3}\cos x = a$ have exactly three different solutions $x_{1}$, $x_{2}$, $x_{3}$ in the closed interval $\left[0,2\pi \right]$. The set of real numbers for $a$ is ____.
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Context:
The smallest two-digit number that can be formed by two of 0, 3, 4, 6, and 8 is subtracted by a certain number and it became 7. Find the number.
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Context:
Square $PQRS$ has side length $2$ units. Points $T$ and $U$ are on sides $PQ$ and $SQ$, respectively, with $PT = SU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and lie on diagonal $RQ$. Find the length of segment $PT$ which can be expressed in the form $\sqrt{k}-m$ units. What is the integer value of $k+m$?
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Context:
In $\triangle ABC$, the sides $a$, $b$, $c$ are opposite to angles $A$, $B$, $C$, respectively. If $a^{2}+b^{2}=2018c^{2}$, then $\dfrac {2\sin A\sin B\cos C}{1-\cos ^{2}C}=$ \_\_\_\_\_\_.
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Context:
Given that the terminal side of angle θ passes through point P(-x, -6) and $$cosθ=- \frac {5}{13}$$, find the value of $$tan(θ+ \frac {π}{4})$$.
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Context:
10. In $\triangle A B C$, let $D, E$, and $F$ be points on sides $B C, C A$, and $A B$, respectively, so that the segments $A D, B E$, and $C F$ are concurrent at point $P$. If $A F: F B=4: 5$ and the ratio of the area of $\triangle A P B$ to that of $\triangle A P C$ is $1: 2$, determine $A E: A C$.
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Context:
Find the smallest positive integer,  $n$ , which can be expressed as the sum of distinct positive integers  $a,b,c$  such that  $a+b,a+c,b+c$  are perfect squares.
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Context:
Let \( S = \{1, 2, 4, 8, 16, 32, 64, 128, 256\} \). A subset \( P \) of \( S \) is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set \( Q \) is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets. (A set \( A \) is said to be a proper subset of a set \( B \) if \( A \) is a subset of \( B \) and \( A \neq B \).)
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Context:
For Gwen's birthday, she received 8 dollars from her mom. Her dad gave her some more money. If she spent 4 dollars and received 3 more dollars from her mom than she did from her dad, how much money did she receive from her dad?
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Context:
Given that 60% of all students in Ms. Hanson's class answered "Yes" to the question "Do you love science" at the beginning of the school year, 40% answered "No", 80% answered "Yes" and 20% answered "No" at the end of the school year, calculate the difference between the maximum and the minimum possible values of y%, the percentage of students that gave a different answer at the beginning and end of the school year.
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Context:
6. Let $\left(x_{1}, x_{2}, \cdots, x_{20}\right)$ be a permutation of $(1,2, \cdots, 20)$, and satisfy
$$
\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right)=620 \text {. }
$$

Then the number of such permutations is. $\qquad$
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Context:
Given a small cube block, each face is painted with a different color. If you want to carve 1, 2, 3, 4, 5, 6 small dots on the faces of the block, and the dots 1 and 6, 2 and 5, 3 and 4 are carved on opposite faces respectively, determine the number of different carving methods.
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Context:
Let \(\left(x_{1}, x_{2}, \cdots, x_{20}\right)\) be a permutation of \((1, 2, \cdots, 20)\), such that it satisfies
\[
\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right) = 620.
\]
How many such permutations exist?
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Context:
George donated 40% of his monthly income to charity. Of the remaining income, he paid 25% in taxes, saved 20% for future use, and spent $50 on groceries and $75 on transportation costs. If he now has $150 left for entertainment and miscellaneous expenses, how much was his monthly income?
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Context:
A part of a book has fallen out. The number of the first fallen page is 387, and the number of the last page consists of the same digits but in a different order. How many sheets fell out of the book?
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Context:
Find the infimum of the set of distances \(\left\{\int_0^1 |A'(t)|\,dt + \int_0^1 |B'(t)|\,dt: (A,B)\in S\right\}\), where \(S\) consists of pairs of smooth paths \(A,B: [0,1]\rightarrow\mathbb{R}^2\) satisfying \(A(0) = (0,0)\), \(B(0) = (1,0)\), and \(B(1) - A(1) = (1,0)\). This infimum represents the shortest total distance that two points in the plane, initially separated by a distance of 1 in the horizontal direction, must travel to become vertically separated by the same distance.
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Context:
Six friends earn $25, $30, $35, $45, $50, and $60. Calculate the amount the friend who earned $60 needs to distribute to the others when the total earnings are equally shared among them.
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Context:
Determine the center of the Lie algebra $\mathfrak{gl}_n(\mathbb{F})$ when $\mathbb{F}$ is a field with positive characteristic $p$ and $n$ is an integral multiple of $p$. Provide your answer in terms of $n$ and $p$.
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Context:
Determine the value of $t$ such that the sum of the probabilities $P(T > -t)$, $P(T > 0)$, $P(T > t)$, and $P(T > 2t)$ for a random variable $T$ from a Student-$t$ distribution with 7 degrees of freedom equals 1.75.
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Context:
In a hockey match, the away team was leading at half-time with a total of six goals scored in the first half. In the second half, the home team scored three goals and won the game. How many goals did the home team score altogether?
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Context:
如图， $A B$ 是单位圆的直径，在 $A B$ 上任取一点 $D$ ，作 $D C \perp A B$, 交圆周于 $C$ 。若点 $D$ 的坐标为 $(x, 0)$,则当 $x \in$ $\qquad$时，线段 $A D 、 B D 、 C D$ 可以构成锐角三角形。请提供区间的两个端点之和。
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Context:
Convert $1729_{10}$ to base 6.
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Context:
A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye is worth 315 points. How many points is hitting the blue (second-to-last) zone worth?
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Context:
What is the greatest number of integers that can be selected from a set  of  $2015$  consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
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Context:
Xiao Ming departs from his home to his grandmother's house on a bicycle at a speed of 12 km/h. After 2.5 hours, his father realizes that Xiao Ming forgot to bring his homework and starts chasing him on a motorcycle at a speed of 36 km/h. Half an hour after Xiao Ming reaches his grandmother's house, his father arrives. What is the distance between Xiao Ming's home and his grandmother's house in kilometers?
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