Context:
How many five-character license plates consist of a consonant, followed by a vowel, followed by a consonant, a digit, and then a special character from the set {$, #, @}? (For this problem, consider Y as both a consonant and a vowel.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围。标准答案格式为$\left(-\infty, \frac{a}{b}\right)$，请给出区间端点a+b的和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A \cdot B$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When the mean, median, and mode of the list
\[10,2,5,2,4,2,x\]
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n > 1$ be an integer that is divisible by exactly two distinct primes, $p$ and $q$. If $G$ is a group with exactly $n$ elements of order $n$, find the maximum possible number of cyclic subgroups of order $n$ in $G$.
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Context:
若直线 \(y = x + m\) 与半圆 \(y = \sqrt{1 - x^2}\) 有2个交点，则 \(m\) 的取值为
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(A B C D\), the side \(A B\) is equal to the diagonal \(A C\). On the smaller arc \(A D\) of the circumcircle of triangle \(A B D\), a point \(E\) is selected such that \(A B = A E\). Find the angle \(\angle C E D\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知抛物线$E:{{y}^{2}}=2px(p>0)$的焦点为$F$，点$A$在$E$上，以$A$为圆心的圆与$y$轴相切，且交$AF$于点$B$，若$\left| AB \right|=2\left| BF \right|$，则圆$A$截线段$AF$的垂直平分线所得弦长为$\sqrt{7}$，则$p=$______．
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Context:
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is the length of $AC$? Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $$\sum_{n=1}^{210} nf(n)$$ is divided by $211$.

[i]Proposed by ApraTrip[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cylinder has a height of 8 centimeters. If its height increases by 2 centimeters, the surface area increases by 25.12 square centimeters. What is the original surface area of the cylinder in square centimeters?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{125.6 \text{ square centimeters}}


Context:
Given that the function $f(x) = \log\_2 \frac{1+nx}{1-2x}$ defined on the interval $[-m, m]$ is an odd function, and $f(-\frac{1}{4}) \neq f(\frac{1}{4})$, determine the range of $n^m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of $$ \frac{1}{K} \int_0^K K^{\frac{1}{2x+1}} \textrm{d} x $$ as $K \to +\infty$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given triangle $ABC$, the altitude on side $AB$ is equal to the length of side $AB$. Find the maximum value of $\frac{AC}{BC} + \frac{BC}{AC} + \frac{AB^2}{BC \cdot AC}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola $C_{1}$: $y^{2}=16x$ and the circle $C_{2}$: $\left(x-4\right)^{2}+y^{2}=1$, where point $M$ has coordinates $\left(8,0\right)$, and points $P$ and $Q$ are moving points on $C_{1}$ and $C_{2}$ respectively, satisfying $|PM|=|PQ|$. Find the range of the x-coordinate of point $P$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of 63.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\cos \left( 2x+\dfrac{\pi}{3} \right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\cos 2x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the symbols , m ” n = (n ^ 2 − m) / x + y for all values of m, n, x, and y. If m = 4, n = 3, x = 2, and y = 5, what is the value of 4 ” 3?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Brian starts at the point $(1,0)$ in the plane. Every second, he performs one of two moves: he can move from $(a,b)$ to $(a-b,a+b)$ or from $(a,b)$ to $(2a-b,a+2b)$. How many different paths can he take to end up at $(28,-96)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer n not smaller than 2, let n lines divide a plane into at most M_n parts, and let n+2 rays divide a plane into at most N_{n+2} parts. Determine the relationship between M_n and N_{n+2} for all n ≥ 2. Express your answer as a mathematical relationship (e.g., <, >, =).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $ω\in\mathbb{N}^{*}$ and $ω\leqslant 15$, find the number of values of $ω$ such that the function $y=\sin(ωx)$ is not monotonic in the interval $\left[\frac{π}{4}, \frac{π}{3}\right]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If for the real numbers $x, y, z, k$ the following conditions are valid: $x \ne y \ne z \ne x$ and \[ x^3 + y^3 + k(x^2 + y^2) = y^3 + z^3 + k(y^2 + z^2) = z^3 + x^3 + k(z^2 + x^2) = 2008, \] find the product $xyz$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an orchard, there are 200 apples, 150 oranges, and 100 pears. The apples are 50% red and 50% green, the oranges are 40% orange and 60% yellow, and the pears are 30% green and 70% brown. Among these fruits, 40 percent of apples, 25 percent of oranges, and 35 percent of pears are rotten. Further, 70 percent of the rotten apples, 50 percent of the rotten oranges, and 80 percent of the rotten pears have a strong smell. How many green rotten fruits in the orchard did not smell?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the exact value of the alternating series $\displaystyle \sum_{n=0}^\infty \frac{(-1)^{n+1}n}{(2n+1)!} \cdot \pi^{2n}$, which is known to converge based on the ratio test.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A printing house determines the cost of printing a book by adding the cost of the cover to the cost of each page and rounding the result up to the nearest whole ruble (for example, if the result is 202 rubles and 1 kopek, it is rounded up to 203 rubles). It is known that the cost of a book with 104 pages is 134 rubles, and a book with 192 pages costs 181 rubles. What is the cost of printing the cover, given that it is a whole number of rubles and the cost of a single page is a whole number of kopeks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When $\frac{1}{909}$ is expressed as a decimal, what is the sum of the first 30 digits after the decimal point?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dmitry's parents decided to buy him a laptop for his birthday. They calculated that they could save the required amount in two ways. In the first case, they need to save one-tenth of their salary each month for six months. In the second case, they would save half of one month's salary and then deposit it in a bank for ten months at 3% interest per month (simple interest). In the first case, the saved money will be just enough for the laptop. In the second case, after buying the laptop, there will be leftover money sufficient to buy a computer table for 2875 rubles. What is the mother's salary, if the father's salary is 30% higher? (Provide the answer as an integer, without spaces or units of measurement.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence of the following power series: $$1 + 3z + 2z^2 + 9z^3 +...+ 2^nz^{2n} + 3^{n+1}z^{2n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers $a$ and $N$, let $r(a, N) \in\{0,1, \ldots, N-1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \leq 1000000$ for which $r(n, 1000)>r(n, 1001)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that $$ N(T) \leq a T^{2}+b T+c $$ for all $T>0$ and all initial points $P_{0}$. The original answer is in \(\pi\) format, please give the integer part of the answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many nonempty subsets of {1, 2, ..., 10} have the property that the sum of their largest element and smallest element is 11?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players are playing a game. One of them thinks of a sequence of integers ($x_{1}, x_{2}, \ldots, x_{n}$), which are single-digit numbers, both positive and negative. The other player is allowed to ask for the value of the sum $a_{1} x_{1}+\ldots+a_{n} x_{n}$, where $(a_{1}, \ldots, a_{n})$ is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The lengths of the altitudes of $\triangle ABC$ are the roots of the polynomial $x^3 - 34x^2 + 360x - 1200.$ Find the area of $\triangle ABC.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( n \) new students such that among any 3 students, there are at least 2 students who know each other, and among any 4 students, there are at least 2 students who do not know each other. Determine the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the expression $\sqrt{3}-a$ is meaningful within the real number range, then the real number $a$ can be _______. (Fill in one number)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\frac{x-b}{x+2}$ defined on the interval $(a, b+4)$ where $b<-2$, and its range is $(2, +\infty)$, find the value of $a^b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 4 people in the family. If Masha's scholarship is doubled, the total family income will increase by 5%. If the mother's salary is doubled instead, the total family income will increase by 15%. If the father's salary is doubled, the total family income will increase by 25%. By how many percent will the family's income increase if the grandfather's pension is doubled?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系 $x O y$ 中， $F_{1} 、 F_{2}$ 分别是椭圆 $\frac{x^{2}}{2}+y^{2}=1$ 的左、右焦点. 设不经过焦点 $F_{1}$ 的直线 $l$ 与椭圆交于两个不同的点 $A 、 B$, 焦点 $F_{2}$ 到直线 $l$ 的距离为 $d$. 如果直线 $A F_{1} 、 l 、 B F_{1}$ 的斜率依次成等差数列，已知 $d$ 的取值范围形如$(\sqrt{m},n)$，求$m+n$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest positive integer $n < 30$ such that $\frac{1}{2}(n^8 + 3n^4 - 4)$ is not divisible by the square of any prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A baseball team has averaged 15 hits per game over their first 5 games. There are 11 players on the team. Their best player has some total hits. If the other players continue their average over the next 6 games, each player will average 6 hits across the 6 games in total. How many total hits does the best player have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find the value of \(m + n + p\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mark does a gig every other day for 2 weeks. For each gig, he plays 3 songs. 2 of the songs are 5 minutes long and the last song has a certain relationship to the length of the first two songs. He played for 280 minutes. What is the ratio of the length of the last song to the length of the first two songs?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Ebbinghaus forgetting curve describes the law of human brain forgetting new things. Based on this, a research team found that after learning course $A$, 20% of the memorized content is forgotten every week. In order to ensure that the memorized content does not fall below $\frac{1}{12}$, the content needs to be reviewed after $n$ ($n\in N$) weeks. Find the value of $n$. ($\lg 3\approx 0.477$, $\lg 2\approx 0.3$)
Please reason step by step, and put your final answer within \boxed{}.


Context:
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1$, 前 $n$ 项和为 $S_{n}, S_{n+1}=4 a_{n}+2$, 求 $a_{2013}$ 的值。原始答案为$x \times y^{z}$的形式，其中$y$不能是2的倍数，请给出$x + y + z$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dan is holding one end of a 26 inch long piece of light string that has a heavy bead on it with each hand. If he starts with his hands together at the start and leaves his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the surface of a cube, there is a closed broken line with eight segments, whose vertices coincide with the vertices of the cube.

What is the minimum number of segments of this broken line that can coincide with the edges of the cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady's $2010$th step, considering the procedure implies hexadecimal (base 16) operations rather than quinary (base 5)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square with side length \(a\), on each side of the square, a trapezoid is constructed outside such that the top bases and the lateral sides of these trapezoids form a regular dodecagon. Compute its area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(AA_1B_1B\) be the axial section of a cylinder, and let \(C\) be a point on the circumference of the base. Given \(AA_1 = 1\), \(AB = 4\), and \(\angle BAC = 60^\circ\), find the dihedral angle between plane \(A_1CB_1\) and the base of the cylinder \(AB\).
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Context:
A $20 \times 20 \times 20$ block is cut up into 8000 non-overlapping unit cubes, and a number is assigned to each cube. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Find the sum of all numbers of the cubes outside these slices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\left\{a_{n}\right\}$ with $k$ terms $(k\in N^{*})$, where $a_{1}=2$, define vectors $\overrightarrow {c_{n}}=(a_{n}, a_{n+1})$ and $\overrightarrow {d_{n}}=(n, n+1)$ for $n=1,2,3,\ldots, k-1$. If $|\overrightarrow {c_{n}}|=|\overrightarrow {d_{n}}|$, calculate the number of sequences $\left\{a_{n}\right\}$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\triangle X Y Z$ be a right triangle with $\angle X Y Z=90^{\circ}$. Suppose there exists an infinite sequence of equilateral triangles $X_{0} Y_{0} T_{0}, X_{1} Y_{1} T_{1}, \ldots$ such that $X_{0}=X, Y_{0}=Y, X_{i}$ lies on the segment $X Z$ for all $i \geq 0, Y_{i}$ lies on the segment $Y Z$ for all $i \geq 0, X_{i} Y_{i}$ is perpendicular to $Y Z$ for all $i \geq 0, T_{i}$ and $Y$ are separated by line $X Z$ for all $i \geq 0$, and $X_{i}$ lies on segment $Y_{i-1} T_{i-1}$ for $i \geq 1$. Let $\mathcal{P}$ denote the union of the equilateral triangles. If the area of $\mathcal{P}$ is equal to the area of $X Y Z$, find $\frac{X Y}{Y Z}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
a and b run around a circular track. a beats b by one round or some minutes. In this race, they had completed 4 rounds. If the race was only of one round, a's time over the course would be 7.5 minutes. How many minutes does a beat b by?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知正方体 $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ 的棱长为 1, 点 $M, N$ 分别为线段 $A B^{\prime}, A C$ 上的动点, 点 $T$在屏幕 $B C C^{\prime} B^{\prime}$ 内，求 $|M T|+|N T|$ 的最小值。原始答案为 $\frac{a\sqrt{b}}{c}$ 格式，请给出 $a + b+c$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知正整数数列 $a, b, c, d$ 严格递增, $d \leq 101$, 且 $101 \mid a+b+c+d$, 则这样的数组 $(a, b, c, d)$ 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $E$ is on $A_{1} D_{1}$, point $F$ is on $C D$, and $A_{1} E = 2 E D_{1}$, $D F = 2 F C$. Find the volume of the triangular prism $B-F E C_{1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2016 spheres in space, none of which are identical. Some of the spheres are red, while the others are green. Each point of contact between a red sphere and a green sphere is painted blue. Find the maximum possible number of blue points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
61. (5) In a line, numbers $1,2, \ldots, n$ are written in some order. A pair of numbers is called a pit if these numbers are adjacent, or there are only numbers between them that are smaller than each of them. What is the maximum number of pits? (A number can be part of several pits.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f\left(x\right)=ax^{2}+bx+c$, $x\in \left[-2a-5,1\right]$ is an even function, then $a+2b=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the midline of an isosceles trapezoid with height $h$, given that the lateral side is seen from the center of the circumscribed circle at an angle of $120^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an examination, 30% of total students failed in Hindi, 45% failed in English, 25% failed in Math, and 40% failed in Science. Additionally, 12% in both Hindi and English, 15% in both Hindi and Math, 18% in both Hindi and Science, 20% in both English and Math, 22% in both English and Science, and 24% in both Math and Science. Finally, 10% failed in all four subjects. Calculate the percentage of students who passed in all of the subjects.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ and $b$ be natural numbers, and $a = 1999b$. The sum of the greatest common divisor and the least common multiple of $a$ and $b$ is ____.
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Context:
The center of the city is a rectangle of \(5 \times 8\) km, consisting of 40 blocks of size \(1 \times 1\) km, with the boundaries being streets that form 54 intersections. What is the minimum number of police officers needed at intersections such that each intersection can be reached by a police officer driving no more than 2 km on the city streets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the following expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
the h.c.f. of two numbers is 120 and the other three factors of their l.c.m. are 13, 17 and 23. The larger of the two numbers is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
For given natural numbers \( k_{0}<k_{1}<k_{2} \), determine the minimum number of roots in the interval \([0, 2\pi)\) that the equation

$$
\sin \left(k_{0} x\right) + A_{1} \cdot \sin \left(k_{1} x\right) + A_{2} \cdot \sin \left(k_{2} x\right) = 0
$$

can have, where \( A_{1}, A_{2} \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You have two blackboards, $A$ and $B$. You need to write some integers on them, where each integer is greater than or equal to $2$ and less than or equal to $20$. The condition is that each number on blackboard $A$ must be co-prime with each number on blackboard $B$. Determine the maximum possible value of the product of the number of integers written on $A$ and the number of integers written on $B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain number with a sum of digits equal to 2021 was divided by 7 and resulted in a number composed exclusively of the digit 7. How many digits 7 can this number contain? If there are multiple answers, provide their sum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A teacher grades students' tests by using a certain method. Student B answers each of the 100 questions on her test and receives a score of 73. She answered 91 questions correctly. What is the ratio of the number of incorrect responses subtracted to the number of correct responses in the grading method?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$. If $PQ$ is a focal chord and the equation of the line $PQ$ is $y=mx+c$, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each half of this figure is composed of $3$ red triangles, $5$ blue triangles, and $8$ white triangles. When the upper half is folded down over the centerline, $2$ pairs of red triangles coincide, as do $3$ pairs of blue triangles. There are $2$ red-white pairs. How many white pairs coincide? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \(i^2 = -1\), then find the sum
\[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \]
\[ + \cdots + i^{40}\cos{3645^\circ} \]
The original answer is in the form \(\frac{k}{m}(\sqrt{2})(a - bi)\), please calculate the value of a + b + k + m.
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Context:
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
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Context:
7. On the side $B C$ of triangle $A B C$, a point $K$ is marked.

It is known that $\angle B+\angle C=\angle A K B, \quad A K=5, \quad B K=6, \quad K C=2$.

Find the area of the circle inscribed in triangle $A B C$.
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Context:
In triangle \( ABC \), the midpoints of sides \( AC \) and \( BC \) are marked as points \( M \) and \( N \) respectively. The angle \( MAN \) is \( 15^{\circ} \), and the angle \( BAN \) is \( 45^{\circ} \).

Find the angle \( ABM \).
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Context:
Given $\triangle ABC$, with $\sin A : \sin B : \sin C = 1 : 1 : \sqrt{2}$, the degree measure of the largest interior angle of this triangle is \_\_\_\_\_.
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Context:
已知四面体 PABC，点 A_1 在 \triangle PBC 内，满足 \triangle A_1BP, \triangle A_1CP, \triangle A_1BC 的面积之比为 3:2:1. G 在线段 AA_1 上，直线 PG 交平面 ABC 于点 M，且 \df{AG}{GA_1}=\df{PG}{GM}，则四面体 PABC 与 A_1AMB 的体积之比为__________.
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Context:
In trapezoid \(ABCD\), the bases \(AD = 12\) and \(BC = 8\) are given. A point \(M\) is chosen on the extension of side \(BC\) such that \(CM = 2.4\).

In what ratio does the line \(AM\) divide the area of trapezoid \(ABCD\)?
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Context:
Calculate $$\int^{1}_{-1}\frac{e^\frac{-1}{x}dx}{x^2(1+e^\frac{-2}{x})}$$
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Context:
For any integer \( n (n \geqslant 4) \), consider the set \(\{1, 2, \cdots, n\}\) with \( m \) subsets \( A_{1}, A_{2}, \cdots, A_{m} \), where \( A_{1} \) has one element, \( A_{2} \) has two elements, ..., \( A_{m} \) has \( m \) elements, and no subset is contained within another subset. Find the maximum value of \( m \).
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Context:
Triangle  $ABC$  is a triangle with side lengths  $13$ ,  $14$ , and  $15$ . A point  $Q$  is chosen uniformly at random in the interior of  $\triangle{ABC}$ . Choose a random ray (of uniformly random direction) with endpoint  $Q$  and let it intersect the perimeter of  $\triangle{ABC}$  at  $P$ . What is the expected value of  $QP^2$ ?
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Context:
设 $A=\{1,2,3, \cdots, 4 n+2\}, M=\{2 n+1,4 n+3,6 n+5\}$. 对 $A$ 的任意非空子集 $B$, 当 $B$ 的任意两个数之和不属于 $M$ 时, 称 $B$ 为 $M$-自由集。如果 $A=A_1 \cup A_2, A_1 \cap A_2=$ $\varnothing$, 且 $A_1, A_2$ 均为 $M$-自由集, 那么称有序对 $\left(A_1, A_2\right)$ 为 $A$ 的一个 $M$ - 划分. 试求 $A$ 的所有 $M-$ 划分的个数.
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Context:
For which values of the parameter \( a \), the equation

\[
3^{x^{2}+6 a x+9 a^{2}} = a x^{2}+6 a^{2} x+9 a^{3}+a^{2}-4 a+4
\]

has exactly one solution?
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Context:
What is the generating function for the sequence 1,1,1,1,1,1?
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Context:
Round down the largest four-digit number you can make by playing four 7, 9, 3, and 6 cards only once, to the hundredth place.
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Context:
Red spends his Sunday at home with the following schedule: 30 minutes on the computer, 20 minutes listening to music, 35 minutes cooking (including 3 minutes for rice washing), 15 minutes tidying up the room, and 25 minutes practicing the piano. Among these activities, how many minutes are needed for the most reasonable arrangement?
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Context:
已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是以$\frac{m}{n}$的形式表示，请给出$m+n$的值。
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Context:
Calculate the following expression:
$$2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))))))$$
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Context:
设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点,已知 $B F=4, B D=2, B E=5, \angle B A D=\angle A C D, \angle B A F=\angle C A E$, 求 $B C$ 的长。
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Context:
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$.
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Context:
Consider all functions $f: \mathbb{N} \to \mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $\mathbb{N}$. Determine the least possible value of $f(1998)$.
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Context:
A semicircular sponge with a diameter of $20 \text{ cm}$ is used to wipe a corner of a room's floor such that the ends of the diameter continuously touch the two walls forming a right angle. What area does the sponge wipe?
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Context:
If $x > 0$, $y > 0$, and $\frac{1}{2x+y} + \frac{4}{x+y} = 2$, find the minimum value of $7x + 5y$.
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Context:
Let $\{a_n\}$ be a geometric sequence, $\{b_n\}$ be an arithmetic sequence, and $b_1=0$, $c_n=a_n+b_n$. If $\{c_n\}$ is $1$, $1$, $2$, $\ldots$, find the sum of the first $10$ terms of the sequence $\{c_n\}$.
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Context:
Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle?
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Context:
marts income is 60 percent more than tims income and tims income is 50 percent less than juans income. additionally, katies income is 20 percent more than marts income, and jasons income is 30 percent less than katies income. what percentage of jasons income is juans income
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Context:
一个 10\times10 的数表初始每个位置的数字均为 1，现在做有限次操作，每次把某行或某列的所有数换成相反数. 称非负整数 N 是好的，如果最终数表中 -1 的个数恰为 N. 则好的 N 的个数为__________.
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Context:
Determine the number of isolated subgroups of a value group with rank $n$, where $n$ is a positive integer.
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Context:
In order to investigate the number of students from various classes participating in the extracurricular calligraphy group at a school, 5 classes were randomly selected, and the number of students from each class participating in the group was taken as sample data. It is known that the sample mean is 7, the sample variance is 4, and the sample data are all different from each other. Find the maximum value among the sample data.
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Context:
Calculate the dimension of the irreducible highest weight module for the simple Lie algebra of type $G_2$ using the Weyl dimension formula.
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Context:
Let $ABC$ be a triangle with $AB = 2$ and $AC = 3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2 = \frac{a + b\sqrt{c}}{d}$ for positive integers $a, b, c$, and $d$, where $\gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Compute $1000a + 100b + 10c + d$.
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Context:
Among the four-digit numbers, the number of four-digit numbers that have exactly 2 digits repeated is.
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Context:
Evaluate the following limit:

$$\lim_{n \to \infty} \frac{1}{n}\left((m+1)(m+2) \cdots (m+n)\right)^{\frac{1}{n}}$$

where $m$ is a positive constant.
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Context:
In a rectangular solid with edge lengths of 3, 8, and 9, if a cylindrical hole is drilled through the top surface without changing its surface area, then the radius of the hole is ______.
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Context:
Let \( f(x) = \cos (\cos (\cos (\cos (\cos (\cos (\cos (\cos x))))))) \), and suppose that the number \( a \) satisfies the equation \( a = \cos a \). Express \( f'(a) \) as a polynomial in \( a \).
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Context:
For which smallest $n$ does there exist a convex $n$-gon such that the sines of all its angles are equal, but the lengths of all its sides are different?
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Context:
Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline {AB}$, $\overline{CD}$, and $\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$?
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Context:
What is the base of the numeral system in which the decimal number 1728 is represented as 3362?
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Context:
Find the smallest positive integer $n$ such that for every integer $m$, with $0<m<1993$, there exists an integer $k$ for which
$$
\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of a rectangle divided by a diagonal is 576 square centimeters (cm2). If the width of the rectangle is 32 centimeters (cm), what is the height in centimeters (cm)?
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Context:
An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.
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Context:
Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers $n$, $k$, and $l$, where $n! = 1 \cdot 2 \cdots n$. In your answer, indicate 0 if there are no solutions, specify the value of $n$ if there is only one solution, or provide the sum of all $n$ values if there are multiple solutions. Recall that a solution is a triple $(n, k, l)$; if they differ in at least one component, they are considered different solutions.
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Context:
Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.
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Context:
Let $A$ be the set of all sequences of 0's and 1's with length 4. What is the minimal number of sequences that can be chosen, such that any arbitrary sequence from $A$ differs in at most one position from one of the chosen sequences?
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Context:
Let $\left(x_{1}, x_{2}, \cdots, x_{20}\right)$ be a permutation of $(1,2, \cdots, 20)$ that satisfies $\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right)=620$. How many such permutations are there?
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Context:
In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900?
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Context:
If the sum of the coefficients of the expansion of $(x+y)^{3}(2x-y+a)^{5}$ is $256$, find the coefficient of the term which contains the letter $x$ and in which $x$ is raised to the power of $1$.
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Context:
Given $\sin (α- \frac{π}{3})= \frac{3}{5} $, where $a∈(\frac{π}{4}, \frac{π}{2})$, find $\tan α=\_\_\_\_\_\_\_\_\_$.
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Context:
Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?
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Context:
Paulo wants to get a burger meal that costs $6. Aside from that, he also wants to get a soda which costs 1/3 as much as the burger meal. While on the counter, Jeremy asked Paulo to get him 2 of each item Paulo is going to get. Stephanie also wants to order, and she is going to get 3 burger meals and 1 soda. There is a 10% discount on all burger meal purchases during the Happy Hour promo when buying more than 2 items. In addition, a 5% sales tax is applied to the total order cost before any discounts. How much will they be charged for their orders combined?
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Context:
John had a stock of 900 books in his bookshop. He sold 75 books on Monday, 50 books on Tuesday, 64 books on Wednesday, 78 books on Thursday, and 135 books on Friday. However, he also had daily customer returns and discounts. On Monday, 5 customers returned books, and he provided a 10% discount to 20 customers. On Tuesday, 10 customers returned books, and he provided a 15% discount to 15 customers. On Wednesday, 3 customers returned books, and he provided a 5% discount to 10 customers. On Thursday, 7 customers returned books, and he provided a 20% discount to 5 customers. On Friday, 15 customers returned books, and he provided a 25% discount to 25 customers. What percentage of the books were not sold by the end of the week?
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Context:
Find the smallest volume bounded by the coordinate planes and  by a tangent plane to the ellipsoid  $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1. $$
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Context:
1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?
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Context:
Several identical boxes weigh a total of 10 tons, with each box weighing no more than 1 ton. What is the minimum number of 3-ton trucks needed to transport all this cargo in one trip?
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Context:
Find the smallest positive integer $k$ such that there exist positive integers $M, O > 1$ satisfying:
\[
 (O \cdot M \cdot O)^k = (O \cdot M) \cdot \underbrace{(N \cdot O \cdot M) \cdot (N \cdot O \cdot M) \cdot \ldots \cdot (N \cdot O \cdot M)}_{2016 \text{\ (N \cdot O \cdot M)s}},
\]
where $N = O^M$.
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Context:
对于一个由至少两个正整数构成的集合，若每个元素均与其他至少一个元素不互质，则称该集合为"芳香集".设 P(n)=n^2+n+1. 求最小的正整数 b，使得存在非负整数 a，满足集合 \{P(a+1),P(a+2),\cdots,P(a+b)\} 是芳香集.
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Context:
A tailor trims 6 feet from opposite edges of a square piece of cloth, and some feet from the other two edges. If 120 square feet of cloth remain, and the length of a side of the original piece of cloth was 22 feet, how many feet did the tailor trim from the other two edges?
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Context:
Find $$\lim_{x\to\infty}\frac{\sqrt{1-\cos^3(1/x)}\cdot(3^{1/x}-5^{-1/x})} {\log_2( 1+x^{-2} + x^{-3})}$$
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Context:
A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their non-negative difference. When just one number remains, it is compared to \(x\). If it has the same parity A wins, otherwise B wins. Find the probability that A wins.
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Context:
Find the Taylor series expansion of $\sqrt{\frac{1-x}{1+2x}}$ around $x=0$ up to the term in $x^2$.
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Context:
Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ .
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Context:
Calculate
\[\prod_{n = 1}^{13} \frac{n(n + 2)}{(n + 4)^2}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A host standing at the golden ratio point of the stage appears more natural and appropriate. If the length of stage AB is 20 meters and the host currently stands at point A, then they should walk at least ____ meters to reach the ideal position. (Round the result to 0.01 meters)
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Context:
Next year's 'leap day', 29th February 2024, may be written in 'ddmmyy' format as the number 290224. What is the sum of all the two-digit factors of 290224?
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Context:
There are  $63$  houses at the distance of  $1, 2, 3, . . . , 63 \text{ km}$  from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants,  $63$  elfs named  $E_1, E_2, . . . , E_{63}$  , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf  $E_n$   takes  $n$  minutes to travel  $1 \text{ km}$  for each  $n = 1,2,...,63$  , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?
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Context:
Suppose point $P$ is on the parabola $y^2=x$, and point $Q$ is on the circle $(x-3)^2+y^2=1$, then the minimum value of $|PQ|$ is $\_\_\_\_\_\_\_\_\}.$
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Context:
In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\angle LMN$ is $60^{\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12.
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Context:
4. Given an equilateral triangle $ABC$ with side $12 \, \text{cm}$. Two circles with the same radius equal to $\sqrt{3} \, \text{cm}$ touch the side $BC$ from the outside of the triangle, touch each other, and one of the circles touches the line $AB$. Find the area of the triangle whose vertices are the centers of the circles and the midpoint of side $AB$.
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Context:
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. He would like a set of shirts that includes either 3 of the same color or 3 of different colors. What is the minimum number of shirts that Zebadiah has to pull out to guarantee that he has such a set? Express your answer as a whole number.
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Context:
Positive real numbers \(a\) and \(b\) have the property that
\[
\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100
\]
and all four terms on the left are positive integers, where \(\log\) denotes the base \(10\) logarithm. The original answer is in the form \(10^k\). Please find the value of \(k\).
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Context:
In a class with $23$ students, each pair of students has watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
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Context:
双曲线 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$ 的左、右焦点分别为 $A, B, P$ 是双曲线上一点, 若 $\triangle P A B$ 的内切圆圆心为 $(3,1)$, 则 $\triangle P A B$ 外接圆的半径为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
已知函数$f\left( x \right)=\frac{1}{3}{{x}^{3}}+{{x2}}-2ax+1$，若函数$f\left( x \right)$在$\left( -2,2 \right)$上有极值，则实数$\textit{a}$的取值范围为______.
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Context:
Given a cylinder with height $OO_1 = 12$ and a base radius $r = 5$. There are points $A$ and $B$ on the circumferences of the top and bottom bases respectively, with $AB = 13$. Find the distance between the axis $OO_1$ and line segment $AB$.
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Context:
Let a positive integer \(n\) be called a cubic square if there exist positive integers \(a, b\) with \(n=\operatorname{gcd}\left(a^{2}, b^{3}\right)\). Count the number of cubic squares between 1 and 100 inclusive.
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Context:
If the square roots of a positive number $a$ are $3x-2$ and $5x+6$, then $a=\_\_\_\_\_\_.$
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Context:
Evaluate the limit: $$\lim_{x\rightarrow 0}\frac{\sin{(\tan{\sin x})}-\sin{(\sin x)}}{\tan{(\tan{\sin x})}-\tan{(\sin x)}}$$
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Context:
Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
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Context:
A stacking of circles in the plane consists of a base, or some number of unit circles centered on the $x$-axis in a row without overlap or gaps, and circles above the $x$-axis that must be tangent to two circles below them (so that if the ends of the base were secured and gravity were applied from below, then nothing would move). How many stackings of circles in the plane have 4 circles in the base?
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Context:
Given the function $f\left(x+ \frac {1}{2}\right)= \frac {2x^{4}+x^{2}\sin x+4}{x^{4}+2}$, calculate the value of $f\left( \frac {1}{2017}\right)+f\left( \frac {2}{2017}\right)+\ldots+f\left( \frac {2016}{2017}\right)$.
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Context:
Given that the fixed point $A(a,3)$ is outside the circle $x^{2}+y^{2}-2ax-3y+a^{2}+a=0$, determine the range of values for $a$.
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Context:
For which $x\in \mathbb{R}$ does the power series $$\sum_{n=0}^{\infty}\binom{2n}{n}(x-2)^n$$ converge?
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Context:
The sexagenary cycle, originating from China, consists of ten heavenly stems and twelve earthly branches. The ten heavenly stems are: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui. The twelve earthly branches are: Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai. In the sexagenary cycle, a heavenly stem is paired with an earthly branch in sequential order, with the heavenly stem preceding the earthly branch. The heavenly stems start with "Jia" and the earthly branches start with "Zi". For example, the first year is "Jia Zi", the second year is "Yi Chou", the third year is "Bing Yin", and so on. After "Gui You", the heavenly stems return to "Jia" and the earthly branches return to "Zi" to start over. Given that the year 2023 is a year of "Gui Mao", then the year 3023 is the year of _______.
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Context:
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
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Context:
When $x \gt 0$, $y \gt 0$, and satisfy $\frac{1}{x}+\frac{2}{y}=1$, then $2x+y\geqslant k$ always holds. Find the range of values for $k$.
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Context:
In tetrahedron \( ABCD \), the dihedral angle between face \( ABC \) and face \( BCD \) is \( 60^\circ \). Vertex \( A \)'s projection onto face \( BCD \) is point \( H \), which is the orthocenter of \( \triangle BCD \). Point \( G \) is the centroid of \( \triangle ABC \). Given that \( AH = 4 \) and \( AB = AC \), find \( GH \).
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Context:
Given that |m| = 3, |n| = 2, and mn < 0, find the value of m + n. Express your answer as ±a, where a is an integer.
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Context:
Perpendiculars \( B E \) and \( D F \), dropped from the vertices \( B \) and \( D \) of parallelogram \( A B C D \) onto sides \( A D \) and \( B C \) respectively, divide the parallelogram into three parts of equal area. On the extension of diagonal \( B D \) past vertex \( D \), a segment \( D G \) is laid off equal to segment \( B D \). Line \( B E \) intersects segment \( A G \) at point \( H \). Find the ratio \( A H: H G \).
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Context:
Positive real numbers \( x, y, z \) satisfy: \( x^{4} + y^{4} + z^{4} = 1 \). Find the minimum value of the algebraic expression \( \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} \).
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Context:
15.2. In trapezoid $K L M N$, diagonal $K M$ is equal to 1 and is also its height. Perpendiculars $K P$ and $M Q$ are drawn from points $K$ and $M$ to sides $M N$ and $K L$ respectively. Find $L M$, if $K N = M Q$ and $L M = M P$.
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Context:
Let $b_0 = 0$, $b_1 = 1$, $b_2=0$ and for $n>2$, $b_n = \dfrac{b_{n-1}+b_{n-2}+b_{n-3}}{3}$. Find the value of $\lim\limits_{n \to +\infty} b_n$.
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Context:
There are 2160 cards in a box. If 7/12 of the cards are red, exactly 11/19 of the remainder are black, and the rest are green, calculate the number of green cards in the box.
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Context:
There is a beach soccer tournament with 17 teams, where each team plays against every other team exactly once. A team earns 3 points for a win in regular time, 2 points for a win in extra time, and 1 point for a win in a penalty shootout. The losing team earns no points. What is the maximum number of teams that can each earn exactly 5 points?
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Context:
Given a right triangular rectangular pyramid (P-ABC) with a side length of $a$, all of its sides are right triangles, and all four vertices are on the surface of a sphere. Find the surface area of this sphere.
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Context:
To obtain the graph of the function $$y=2\sin(x+ \frac {\pi}{6})\cos(x+ \frac {\pi}{6})$$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$.
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Context:
Kanga labelled the vertices of a square-based pyramid using \(1, 2, 3, 4,\) and \(5\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \(7, 8, 9,\) and \(10\). What is the sum for the fifth face?
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Context:
In a middle school mathematics competition, three problems $A, B, C$ were posed and 25 students participated. Each student solved at least one problem. Among students who did not solve problem $A$, the number of students who solved problem $B$ is twice the number who solved problem $C$. Among students who solved problem $A$, the number of students who solved only problem $A$ is one more than the number of those who, in addition to problem $A$, solved other problems. Additionally, among students who solved only one problem, half of them did not solve problem $A$. How many students solved only problem $B$?
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Context:
Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly $10$ operations.
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Context:
In an isosceles triangle \(ABC\) (\(AC = BC\)), an incircle with radius 3 is inscribed. A line \(l\) is tangent to this incircle and is parallel to the side \(AC\). The distance from point \(B\) to the line \(l\) is 3. Find the distance between the points where the incircle touches the sides \(AC\) and \(BC\).
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Context:
In a regular tetrahedron O-ABC with each edge length equal to 1, if point P satisfies $$\overrightarrow {OP}=x \overrightarrow {OA}+y \overrightarrow {OB}+z \overrightarrow {OC}(x+y+z=1)$$, find the minimum value of $$| \overrightarrow {OP}|$$.
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Context:
On the radius $AO$ of a circle with center $O$, a point $M$ is chosen. On one side of $AO$ on the circle, points $B$ and $C$ are chosen such that $\angle AMB = \angle OMC = \alpha$. Find the length of $BC$ if the radius of the circle is 9 and $\cos \alpha = \frac{1}{3}$.
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Context:
Let the universal set be U = {1,2,3,4,5,6,7,8,9,10}. Given the conditions: ① A ⊆ U; ② if x ∈ A, then 2x ∉ A; ③ if x ∉ A, then 2x ∉ complement of A. Calculate the exact number of sets A that satisfy all three conditions simultaneously.
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Context:
If 8% more is gained by selling an article for Rs. 420 than by selling it for Rs. 380, what is the cost of the article?
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Context:
Calculate: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$
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Context:
There are $14$ players participating in a chess tournament, each playing one game with every other player. After the tournament, players are ranked in descending order based on their points. The sum of the points of the first three players is equal to the sum of the points of the last nine players. What is the highest possible number of draws in the tournament? (For a victory, a player gets $1$ point, for a loss $0$ points, and in a draw, both players get $0.5$ points.)
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Context:
What is the formula for $\sum_{n_{k-1}=1}^{n_k} \sum_{n_{k-2}=1}^{n_{k-1}} ... \sum_{n_0=1}^{n_1} n_0(n_0+1)(n_0+2)$?
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Context:
Teams A and B each have 7 players who compete in a go tournament following a predetermined sequence. The first player from each team competes first, with the loser being eliminated. The winner then competes against the next player from the losing team, and this process continues until all players on one team are eliminated. The other team wins, thus forming a potential sequence of the match. What is the total number of possible match sequences?
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Context:
Given $f(\sin x) = -2x + 1$, where $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, find the value of $f(\cos 10)$.
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Context:
40. Find the best constant $k$ such that the following inequality holds for all positive real numbers $a, b, c$.
$$\frac{(a+b)(b+c)(c+a)}{a b c}+\frac{k(a b+b c+c a)}{a^{2}+b^{2}+c^{2}} \geq 8+k \quad(\text{Pham Kim Hung})$$
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Context:
What is the smallest number which, when diminished by 12, is divisible by 8, 12, some number, and 24, and the result is 276?
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Context:
The negation of the proposition "$\forall x\in \left[-2,+\infty \right)$, $x+3\geqslant 1$" is ______.
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\boxed{\exists x\in \left[-2,+\infty \right), x+3 < 1}


Context:
正三棱雉 $S-A B C$ 的侧棱与底面边长相等, 如果 $E, F$ 分别为 $S C, A B$ 的中点, 那么异面直线 $E F$ 与 $S A$ 所成的角等于多少?
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Context:
Given a sequence $\{a_{n}\}$ where $a_{1}=1$, and ${a}_{n}+(-1)^{n}{a}_{n+1}=1-\frac{n}{2022}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\{a_{n}\}$. Find $S_{2023}$.
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Context:
Call a positive integer $n \geq 2$ "junk" if there exist two distinct $n$-digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that:

- $a_1 + a_2 = b_1 + b_2$,
- $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \leq i \leq n-1$,
- $a_{n-1} + a_n = b_{n-1} + b_n$.

Find the number of junk positive integers less than or equal to $2016$.
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Context:
在侧棱长和底面边长都是 4 的正四棱锥 $P-A B C D$ 的表面上与顶点 $P$ 的距离为 3的动点所形成的所有曲线段的长度之和为$m \pi$，求$m$的值。
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Context:
Given \( z \in \mathbf{C} \), if the equation \( 4x^2 - 8zx + 4i + 3 = 0 \) (where \( i \) is the imaginary unit) has real roots, what is the minimum value of the modulus \( |z| \) of the complex number \( z \)?
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Context:
There are 8 different positive integers. Among them, at least 6 are multiples of 2, at least 5 are multiples of 3, at least 3 are multiples of 5, and at least 1 is a multiple of 7. To minimize the largest number among these 8 integers, what is this largest number?
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Context:
12. Simplify: $\sum_{k=1}^{n} \frac{\cos 3^{k} x+3 \cos 3^{k-1} x}{3^{k-1} \sin 3^{k} x}$.
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Context:
The numbers from 1 to 600 are divided into several groups. It is known that if a group contains more than one number, then the sum of any two numbers in this group is divisible by 6. What is the minimum number of groups?
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Context:
Consider each permutation of the nine digits \(1, 2, \ldots, 9\) as a nine-digit number. How many of these nine-digit numbers are divisible by 11?
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Context:
Let $T$ be a trapezoid with two right angles and side lengths $4$, $4$, $5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$.
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Context:
Two trains of length 140 m and another length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. In 20.99832013438925 seconds, they will be clear of each other from the moment they meet. What is the length of the second train?
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Context:
In trapezoid \(ABCD\), the angles \(A\) and \(D\) at the base \(AD\) are \(60^{\circ}\) and \(30^{\circ}\) respectively. Point \(N\) lies on the base \(BC\) such that \(BN : NC = 2\). Point \(M\) lies on the base \(AD\), the line \(MN\) is perpendicular to the bases of the trapezoid and divides its area in half. Find the ratio \(AM : MD\).
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Context:
Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions?
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Context:
The first 14 terms of the sequence $\left\{a_{n}\right\}$ are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, \ldots$. Following this pattern, calculate the value of $a_{18}$. Express your answer as a single integer.
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Context:
You have 5000 distinct finite sets. Their intersection is empty. However, the intersection of any two sets is nonempty. What is the smallest possible number of elements contained in their union?
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Context:
Find the smallest positive period of the function f(x) = 4 sin^3 x - sin x + 2 (sin x/2 - cos x/2)^2. Express your answer in terms of pi.
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Context:
Points \( M \), \( N \), and \( K \) are located on the lateral edges \( AA_1 \), \( BB_1 \), and \( CC_1 \) of the triangular prism \( ABC A_1 B_1 C_1 \) such that \( \frac{AM}{AA_1} = \frac{2}{3} \), \( \frac{BN}{BB_1} = \frac{3}{5} \), \( \frac{CK}{CC_1} = \frac{4}{7} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( MNKP \), if the volume of the prism is 27.
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Context:
Five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers, with each child choosing any one of the other four fathers except their own. What is the total number of different combinations of choices for the five children? Express your answer as a single integer.
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Context:
XLVI OM - III - Problem 2

The diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal areas?
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Context:
Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project.
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Context:
Will stands at a point \(P\) on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of \(n^{\circ}\) and \((n+1)^{\circ}\) with the tangent at \(P\), where \(n\) is a positive integer less than 90. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach \(P\) again. (\(P\) is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?
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Context:
Noa scored 30 points to win a bowl, Phillip scored twice the number of points Noa did plus 20% of Noa's score, and Lucy scored the square root of the sum of Noa and Phillip's points multiplied by 10. What is the total number of points Noa, Phillip, and Lucy scored to win the bowl?
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Context:
In $\triangle ABC$, it is known that $a^2+b^2+ab=c^2$. Find the measure of $\angle C$.
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Context:
The cells of a $100 \times 100$ table are painted white. In one move, you are allowed to choose any 99 cells from one row or one column and repaint each of them in the opposite color – from white to black or from black to white. What is the minimum number of moves needed to obtain a table with a checkerboard pattern of cells?
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Context:
There are 2 prime numbers between a certain number and 84 / 6. What is the lower limit of the range?
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Context:
From vertex $C$ of rhombus $A B C D$, with side length $a$, two segments $C E$ and $C F$ are drawn, dividing the rhombus into three equal areas. Given that $\cos C = \frac{1}{4}$, find the sum of $C E + C F$.
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Context:
A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$.
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Context:
A man is 45 years older than his son. In 10 years, his age will be four times the age of his son. Additionally, the son's age will double in 15 years. What is the present age of the son?
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Context:
24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \( a_k \) based on their ranking \( k \) in that round, where \( a_{k} \in \mathbb{N}_+, k = 1, 2, \ldots, n, a_1 > a_2 > \cdots > a_n \). After all the rounds are completed, the overall ranking is determined based on the total score each person has accumulated over the 12 rounds. Find the smallest positive integer \( n \) such that no matter the ranking in the penultimate round, at least 2 participants have the potential to win the championship.
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Context:
Given that there are 20 cards numbered from 1 to 20 on a table, and Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card, find the maximum number of cards Xiao Ming can pick.
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Context:
A company pays a certain percentage as dividend to its investors. An investor buys Rs. 60 shares and gets 25% on investment. The investor bought the shares at Rs. 30. What is the dividend percentage paid by the company?
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Context:
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.
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Context:
A pedestrian and a cyclist set off from point $A$ to point $B$ simultaneously. At point $B$, the cyclist turns back and meets the pedestrian 20 minutes after starting. Without stopping, the cyclist continues to point $A$, turns around, and catches up with the pedestrian 10 minutes after their first meeting. How long will it take for the pedestrian to travel from $A$ to $B$?
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Context:
If "Labor Day" is on Monday, then "National Day" is on what day of the week?
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Context:
In the TV show "Dad, Where Are We Going?", five dads each bring one child (son or daughter) to experience rural life. One day, the village chief arranges for one dad to take three kids to complete a certain task, and at least one girl (among the 5 kids, there are 3 boys and 2 girls) must be chosen. Kimi (a boy) says, "If my dad goes, I'll go; if my dad doesn't go, I won't go." Stone (a boy) is angry with his dad and says, "If my dad goes, I won't go; if my dad doesn't go, I'll go." The others have no objections. How many possible selections are there?
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Context:
A natural number greater than 1 is defined as nice if it is equal to the product of its distinct proper divisors. A number \( n \) is nice if:
1. \( n = pq \), where \( p \) and \( q \) are distinct prime numbers.
2. \( n = p^3 \), where \( p \) is a prime number.
3. \( n = p^2q \), where \( p \) and \( q \) are distinct prime numbers.
Determine the sum of the first ten nice numbers under these conditions.
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Context:
Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$?
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Context:
Given that the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is $60^{\circ}$, and $\overrightarrow{a}=2\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, find the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{e_{1}}$.
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Context:
A sphere is circumscribed around a right parallelepiped, which has base diagonals equal to \(a\) and \(b\). Determine the entire surface area of the parallelepiped.
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Context:
Azar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has three of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$. How many ways can the board look after the game is over?
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Context:
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$.  Find $\rho^2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A certain factor and 3 ^ 3 are factors of n x ( 2 ^ 5 ) x ( 6 ^ 2 ) x ( 7 ^ 3 ). The smallest possible positive value of n is 75. What is the other factor?
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Context:
In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are placed on top of the grid. One large circle is centered at the center of the grid with a radius equal to 1.5 cm, and five smaller circles each with a radius of 0.5 cm are placed at the center of the outer border of the grid. The area of the visible shaded region can be written in the form $C-D\pi$ square cm. What is the value of $C+D$?
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Context:
Suppose \( a_{i}, b_{i} (i=1,2,\ldots,n) \) are rational numbers, such that for any real number \( x \), the following holds:
\[ x^{2}+x+4=\sum_{i=1}^{n}\left(a_{i} x+b_{i}\right)^{2}. \]

Find the minimum possible value of \( n \).
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Context:
The function relationship between the total cost $y$ (in ten thousand yuan) of a certain product and the output $x$ (in units) is $y=3000+20x-0.1x^{2}$ ($0<x<240$, $x\in \mathbb{N}^{+}$). If the selling price per unit of the product is 25 thousand yuan, then the minimum output at which the producer does not incur losses (sales revenue is not less than total cost) is ____ units.
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Context:
In a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is tangent to both spheres and intersects the cylindrical surface, forming an ellipse, what is the sum of the lengths of the major and minor axes of this ellipse? ( ).
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Context:
从正九边形中任取三个顶点构成三角形, 则正九边形的中心在三角形内的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\arcsin (3 / \sqrt{10})} \frac{2 \tan x-5}{(4 \cos x-\sin x)^{2}} d x
$$
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Context:
设正数X,y,Z,a,0,c满足cy+bz=a,az+cx=b,bx+ay=c,

求函数$f ( x , y , z ) = \frac { x ^ { 2 } } { 1 + x } + \frac { y ^ { 2 } } { 1 + y } + \frac { z ^ { 2 } } { 1 + z }$的最小值.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{f(x,y,z)的最小值为$\frac { 1 } { 2 }$}


Context:
Cyclic pentagon $ABCDE$ has a right angle $\angle ABC=90^{\circ}$ and side lengths $AB=15$ and $BC=20$. Supposing that $AB=DE=EA$, find $CD$.
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Context:
A, B, and C are partners in a business. A is a working partner, while B and C are sleeping partners. A invests Rs. 5000, B invests Rs. 1000, and C invests Rs. 7000. A receives 12% of the total profit for managing the business, while the remaining profit is divided in the following ratio: A's capital: 35%, B's capital: 25%, C's capital: 40%. Out of a total profit of Rs. 16,500, find the money received by A.
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Context:
On each of the lines \( y=0 \) and \( y=2 \), there are 64 points marked with abscissas \( 1, 2, 3, \ldots, 64 \). In how many ways can three points be chosen from the 128 marked points so that they form the vertices of a right triangle?
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Context:
Adam, Barney, Carrie, and Jerrie are participating in a sit-up competition. Here are their respective performance profiles:

- Adam can perform 40 sit-ups in the first minute, but his stamina decreases after each break, so he loses 8 sit-ups capacity per minute after each one-minute break.
- Barney can perform 45 sit-ups in one minute. Then he takes a 45-second break before starting again.
- Carrie can do twice as many sit-ups per minute as Barney can. However, after two minutes, she needs a 90-second break.
- Jerrie can do 5 more sit-ups per minute than Carrie can do. But Jerrie takes an extra 20-second break after every minute.

During the competition, Adam performs sit-ups for three rounds with two minutes of rest between each round. Barney does sit-ups for five rounds with one 45-second break between them. Carrie does sit-ups for four rounds, with one 90-second break after two rounds. Jerrie performs sit-ups for six rounds, taking a 20-second break after each round.

What would be the combined total number of sit-ups performed by all of them in the competition?
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Context:
The surface of a clock is circular, and on its circumference, there are 12 equally spaced points representing the hours. Calculate the total number of rectangles that can have these points as vertices.
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Context:
Given a sequence $\{a_n\}$, where $a_{n+1} + (-1)^n a_n = 2n - 1$, find the sum of the first 12 terms of the sequence $\{a_n\}$.
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Context:
Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. Calculate the total number of songs these trios sang.
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Context:
How can we evaluate the following integral:
$$ \int_0^1 \frac{\tan^{-1}(x)}{x+1}dx $$
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Context:
A man buys 1 liter of milk for 12 rs, mixes it with 20% water, and sells it for a certain price. The percentage of gain is 50%. What is the selling price of the milk?
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Context:
The security gate at a storage facility requires a 6-digit lock code. If the lock code must consist only of digits from 1 through 9, inclusive, with no repeated digits, and the first, third, and fifth digits of the code must be even, how many lock codes are possible?
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Context:
Find the expected degree of the root of binary trees using probability generating functions.
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Context:
Calculate the limit as $n$ approaches infinity for the following integral:

$$\lim_{n \to \infty} \int_0^3 \sin\left(\frac{\pi}{3} \sin\left(\frac{\pi}{3} \sin \left(\frac{\pi}{3} \dots \sin\left(\frac{\pi}{3} x\right) \dots \right)\right)\right)dx$$

where the sine function is applied $n$ times.
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Context:
How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? Find the number of such sequences.
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Context:
All dwarves are either liars or knights. Liars always lie, while knights always tell the truth. Each cell of a $4 \times 4$ board contains one dwarf. It is known that among them there are both liars and knights. Each dwarf stated: "Among my neighbors (by edge), there are an equal number of liars and knights." How many liars are there in total?
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Context:
已知复数 $z_{1} 、 z_{2}$ 满足 $\left|z_{1}\right|<1$, $\left|z_{2}\right|<1, z_{1}+z_{2}$ 为实数，且 $z_{1}+z_{2}+z_{1} z_{2}=0$ 。求 $z_{1} z_{2}$ 的取值范围。请给出区间端点的倒数的和。
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Context:
Let \( A \) be the \( 2n \times 2n \) matrix whose diagonal elements are all \( x \) and whose off-diagonal elements \( a_{ij} = a \) for \( i + j \) even, and \( b \) for \( i + j \) odd. Find \( \lim_{x \to a} \frac{\det(A)}{(x - a)^{2n - 2}} \).
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Context:
In preparation for a game of Fish, Carl must deal $48$ cards to $6$ players. For each card that he deals, he runs through the entirety of the following process:

1. He gives a card to a random player.
2. A player $Z$ is randomly chosen from the set of players who have at least as many cards as every other player (i.e., $Z$ has the most cards or is tied for having the most cards).
3. A player $D$ is randomly chosen from the set of players other than $Z$ who have at most as many cards as every other player (i.e., $D$ has the fewest cards or is tied for having the fewest cards).
4. $Z$ gives one card to $D$.

He repeats steps $1-4$ for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly $8$ cards?
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Context:
Two trains of length 180 m and another length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. In 22.998 seconds, they will be clear of each other from the moment they meet. What is the length of the second train?
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Context:
Winwin won $50,000 in a lottery. The government imposes a progressive tax system:
- 0% for the amount up to $10,000
- 10% for the amount between $10,001 and $20,000
- 20% for the amount between $20,001 and $50,000

Additionally, there are multiple processing fees:
- $5 for the standard processing
- $200 for mandatory identity verification check
- 2% administrative fee on the final amount she can take home

How much was Winwin able to take home after the taxes and processing fees have been deducted?
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Context:
Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even.

Initially, there are $2021$ integers on the board, each one sampled uniformly at random from the set $\{0, 1, 2, 3, \ldots, 2021\}$. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\gcd(m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.
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Context:
If two distinct members of the set $\{ 2, 4, 12, 14, 21, 28, 98 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 196? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Samantha sleeps for some hours everyday. If today she woke up at 11 in the morning and went to bed at 7, how many hours does she sleep every day?
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Context:
On a line, there are blue and red points, with at least 5 red points. It is known that on any segment with endpoints at red points, containing a red point inside, there are at least 4 blue points. Additionally, on any segment with endpoints at blue points, containing 3 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on a segment with endpoints at red points, that does not contain other red points?
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Context:
In square \( A B C D \), \( P \) and \( Q \) are points on sides \( C D \) and \( B C \), respectively, such that \( \angle A P Q = 90^\circ \). If \( A P = 4 \) and \( P Q = 3 \), find the area of \( A B C D \).
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Context:
In triangle $A B C$ where $A C = B C$ and $\angle A C B = 30^\circ$, point $D$ lies on side $B C$ such that $B D = 3$ and $D C = 6$. Find a point $P$ on side $A C$ such that the angle $D P C$ is 1.5 times the angle $A P B$.
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Context:
In the triangular pyramid \( SABC \) with vertex \( S \), the lateral edges are equal, and the sum of the dihedral angles at the edges \( SA \) and \( SC \) is \( 180^\circ \). Express the length of the lateral edge in terms of the sides \( a \) and \( c \) of triangle \( ABC \).
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Context:
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
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Context:
Find the number of positive integers $n$ that satisfy
\[(n - 1)(n - 3)(n - 5) \dotsm (n - 97) < 0.\]
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Context:
A Tim number is a five-digit positive integer with the following properties:
1. It is a multiple of 15.
2. Its hundreds digit is 3.
3. Its tens digit is equal to the sum of its first three (leftmost) digits.

How many Tim numbers are there?
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Context:
Given that $f(x) = \log_{\frac{1}{2}}(x^2 - ax + 3a)$ is a decreasing function on the interval $[2,+\infty)$, find the range of the real number $a$.
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Context:
Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k.
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Context:
The region $G$ is bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, and the region $g$ is bounded by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly chosen within the region $G$. What is the probability that it belongs to region $g$ (event $A$)?
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Context:
四面体 $A B C D$ 中, $A B \perp B C, C D \perp B C, B C=2$, 且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\circ}$ 。若四面体 $A B C D$ 的外接球半径为 $\sqrt{5}$, 则四面体 $A B C D$ 体积的最大值为 $\qquad$.  原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}}  \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.)

[i]Proposed by Ankit Bisain[/i]
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Context:
Three bodies x, y, and z start moving around a circular track from the same point simultaneously in the same direction at speeds of 12 m/s, 20 m/s, and 36 m/s respectively. They meet for the first time after 120 seconds since they started moving. What is the length of the circular track?
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Context:
Given the function $f(x)=\sin (2x-\frac{\pi }{6})$ whose graph is $C_{1}$, if the graph $C_{1}$ is translated to the left by $\frac{\pi }{4}$ units to obtain the graph $C_{2}$, then the interval where $C_{2}$ is monotonically decreasing on $\left[ 0,\pi \right]$ is.
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Context:
焦点在 $x$ 轴上的椭圆 $\frac{x^2}{m} + \frac{y^2}{4-m} = 1$ 的离心率为 $\frac{6}{3}$，则 $m$ 的值为何？
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Context:
In quadrilateral $ABCD$, let $AB = 7$, $BC = 11$, $CD = 3$, $DA = 9$, $\angle BAD = \angle BCD = 90^o$, and diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. The ratio $\frac{BE}{DE} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Among all pairs of real numbers  $(x, y)$  such that  $\sin \sin x = \sin \sin y$  with  $-10 \pi \le x, y \le 10 \pi$ , Oleg randomly selected a pair  $(X, Y)$ .  Compute the probability that  $X = Y$ .
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Context:
A pentagon is obtained by joining, in order, the points \((0,0)\), \((1,2)\), \((3,3)\), \((4,1)\), \((2,0)\), and back to \((0,0)\). The perimeter of the pentagon can be written in the form \(a + b\sqrt{c} + d\sqrt{e}\), where \(a\), \(b\), \(c\), \(d\), and \(e\) are whole numbers. Find \(a+b+c+d+e\).
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Context:
Example 5 Suppose we have a $4 \times 4$ grid where each cell is colored differently. Each cell is filled with either 0 or 1, such that the product of the numbers in any two adjacent cells is 0. How many different ways are there to fill the grid with numbers?

(2007, Korean Mathematical Olympiad)
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Context:
One day, Raviraj left home and cycled 20 km southwards, turned right and cycled some distance, turned right and cycled 20 km, and turned left and cycled 20 km. He will have to cycle 30 km to reach his home straight. How many kilometers did he cycle after the first turn?
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Context:
Given that the greatest common divisor (GCD) of a pair of positive integers \((a, b)\) is 2015, find the minimum value of \(a+b\) such that \((a+b) \mid \left[(a-b)^{2016} + b^{2016}\right]\).
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Context:
Three of the four endpoints of the axes of an ellipse are, in some order, \[(-2, 4), \; (3, -2), \; (8, 4).\]Find the distance between the foci of the ellipse.
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Context:
A question from an old exam asks to find the number of groups of order 2014. Given that $2014 = 2 · 19 · 53$, how many groups of order 2014 are there?
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Context:
4. In an isosceles triangle $A B C$ the angle opposite the base $\overline{A B}$ is $80^{\circ}$. On the height to the base of this triangle, a point $D$ is chosen such that $|\angle D B A|=30^{\circ}$. On the segment $\overline{B D}$, a point $T$ is chosen such that $|\angle T A D|=20^{\circ}$. Determine the size of the angle $\angle T C A$.
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Context:
设 A_1,A_2,\cdots,A_n 是集合 \{1,2,\cdots,2018\} 的二元子集，使得 A_i+A_j,1\leq i\leq j\leq n 互不相同. 求 n 的最大可能值.
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Context:
How many unordered pairs of prime numbers have a sum of 40?
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Context:
Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent.
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Context:
For any shape, what is the smallest positive integer \( n \) such that the total number of lattice points on and inside a convex \( n \)-gon is at least \( n+1 \)?
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Context:
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Determine the maximum slope of the line $OQ$.
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Context:
Given that the function $f(x)$ is defined on the domain $\mathbb{R}$ and satisfies $f'(x) > f(x)$, determine the solution set of the inequality $e^{f(\ln x)} - xf(1) < 0$ (represent the answer using intervals).
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Context:
In triangle $ABC$, $\tan B= \sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\frac {3 \sqrt {3}}{2}$. Find the length of $AC$.
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Context:
There is a tunnel between locations A and B. A car leaves B for A at 08:16, and a truck leaves A for B at 09:00. It is known that the truck and the car arrive at the ends of the tunnel at the same time, but the truck leaves the tunnel 2 minutes later than the car. If the car arrives at A at 10:56 and the truck arrives at B at 12:20, at what time do they simultaneously arrive at the ends of the tunnel?
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Context:
8. Equilateral triangle $A B C$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $A B$ and $A C$ of $\Omega$ respectively such that $B C=D E$. Given that triangle $A B E$ has area 3 and triangle $A C D$ has area 4 , find the area of triangle $A B C$.
Proposed by: Yuan Yao
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Context:
Five unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending from \((a,0)\) to \((3,3)\), divides the entire region into two regions of equal area. The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.
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Context:
In triangle \(ABC\), it is given that \(|AB| = |BC|\) and \(\angle ABC = 20^\circ\). Point \(M\) is taken on \(AB\) such that \(\angle MCA = 60^\circ\), and point \(N\) is taken on \(CB\) such that \(\angle NAC = 50^\circ\). Find \(\angle NMA\).
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Context:
The probability of each seed germinating is 0.9. Now, 1000 seeds have been sown. For each seed that does not germinate, 2 more seeds need to be sown. Let the number of seeds that need to be sown again be denoted as _X_. Calculate the expected value of _X_. Express your answer as a whole number.
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