Context:
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light?
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Context:
An equilateral triangle is constructed externally on the side \(AB\) of triangle \(ABC\). Find the distance between its center and vertex \(C\), given \(AB=c\) and \(\angle C = 120^\circ\).
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Context:
Find the minimum value of the function \((\sqrt{2} \sin x - 3 \tan y)^{2} + (\sqrt{2} \cos x - 3 \cot y)^{2}\), where \(x, y \in \left(0, \frac{\pi}{2}\right)\).
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Context:
Given a $2 \times 2$ tile and seven dominoes ($2 \times 1$ tile), find the number of ways to tile a $2 \times 7$ rectangle using some of these tiles without leaving gaps or overlapping any two tiles.
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Context:
Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).
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Context:
Given the hyperbola $C$: $\frac{x^{2}}{4}-y^{2}=1$ with left and right foci $F\_1$ and $F\_2$ respectively, find the area of the quadrilateral $P\_1P\_2P\_3P\_4$ for a point $P$ on the hyperbola that satisfies $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=0$.
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Context:
1. Each of the 4 colors has 6 pencils. Each of the six children received 4 pencils. For what smallest $k$ can we always choose $k$ children and decorate their pencils so that we have at least one in each color?
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Context:
Let \(p\) be a prime number and \(\mathbb{F}_{p}\) be the field of residues modulo \(p\). Let \(W\) be the smallest set of polynomials with coefficients in \(\mathbb{F}_{p}\) such that the polynomials \(x+1\) and \(x^{p-2}+x^{p-3}+\cdots+x^{2}+2x+1\) are in \(W\), and for any polynomials \(h_{1}(x)\) and \(h_{2}(x)\) in \(W\) the polynomial \(r(x)\), which is the remainder of \(h_{1}\left(h_{2}(x)\right)\) modulo \(x^{p}-x\), is also in \(W\). How many polynomials are there in \(W\) ?
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Context:
Let $x_{1}, x_{2}, \cdots, x_{1993}$ satisfy $\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993$ and let $y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}$ for $k=1,2, \cdots, 1993$. What is the maximum possible value of $\left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|+\cdots+\left|y_{1992}-y_{1993}\right|$?
(1993 Macau Math Olympiad problem)
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Context:
If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls.
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Context:
If the solution set of the inequality $-1 < ax^2 + bx + c < 1$ is $(-1, 3)$, then the range of the real number $a$ is __________.
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Context:
A class has 60 students. After an exam, the math scores $\xi$ are normally distributed with mean 110 and variance 102. If $P(100 \leq \xi \leq 110) = 0.35$, then estimate the number of students scoring above 120 in math. Provide your answer as a whole number, noting that an approximate answer is acceptable.
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Context:
Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\]
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Context:
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$?
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Context:
Let \( n > m \geq 1 \) be natural numbers such that the groups of the last three digits in the decimal representation of \( 1978^m \) and \( 1978^n \) coincide. Find the ordered pair \( (m, n) \) of such \( m \) and \( n \) for which \( m+n \) is minimal.
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Context:
A large collection of congruent right triangles is given, each with side lengths $3$, $4$, and $5$. Find the maximal number of such triangles you can place inside a $20 \times 20$ square, with no two triangles intersecting in their interiors.
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Context:
What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles?
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Context:
From a solid cylinder, whose height is $$ 8\mathrm{cm} $$ and radius is $$ 6\mathrm{cm}, $$ a conical cavity of height $$ 8\mathrm{cm} $$ and of base radius 6cm is hollowed out. Find the total surface area of the remaining solid. [Use $$ \pi=3.14 $$]
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Context:
What is the greatest number of positive integers less than or equal to 2016 that can be chosen such that no two of them differ by 1, 2, or 6?
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Context:
What is $x-y$ if a town has 2017 houses, 1820 have a dog, 1651 have a cat, 1182 have a turtle, $x$ is the largest possible number of houses that have a dog, a cat, and a turtle, and $y$ is the smallest possible number of houses that have a dog, a cat, and a turtle?
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Context:
在一个圆周上给定 $8$ 个点 $A_{1}, A_{2}, \cdots$, $A_{8}$. 求最小的正整数 $n$, 使得以这 $8$ 个点为顶点的任意 $n$ 个三角形中，必存在两个有公共边的三角形。
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Context:
In a plane, there are 5 points, no three of which are collinear. Connecting these points with line segments, there are at least $\qquad$ intersection points other than the 5 points themselves.
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Context:
Samara and six of her friends heard alligators spotted on the local river and decided to join a search organized by the wildlife service to capture the animals. They split into three groups: Samara, two friends in Group 2, and three friends in Group 3. After searching the whole day, Samara had seen 30 alligators, while her friends in Group 2 had seen a total of 54 alligators with one friend spotting 8 more alligators than the other. The friends in Group 3 had spotted an average of 12 alligators each when in the afternoon, but due to changing conditions, they increased their spotting rate by 25% for the rest of the day, resulting in an additional number of sightings for each friend. Given that each friend in Group 3 ended up spotting the same total number of alligators, calculate the total number of alligators Samara and her friends saw.
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Context:
IMO 2000 Problem B1 100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?
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Context:
## Task A-3.5.

In a scalene triangle $ABC$, the lengths of two altitudes are equal to the lengths of two medians. What is the ratio of the lengths of the remaining altitude and the remaining median?
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Context:
设f:R→R满足：对任意实数x、y，有f(2x)+f(2y)=f(x+y)f(x-y)，又f(π)=0，且f(x)不恒等于零，则f(x)的周期是 ·
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Context:
Patty is picking peppermints off a tree. They come in two colors: red and white. She picks fewer than $100$ total peppermints but at least one of each color. The white flavor is stronger, so she prefers red to white. Thus, she always picks fewer white peppermints than ten times the number of reds. How many different combinations of peppermints can she go home with?
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Context:
Find the sum of all fractions of the form $\frac{1}{xy}$ where $\operatorname{gcd}(x, y) = 1$, $x \leq n$, $y \leq n$, and $x + y > n$.
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Context:
Shapovalov A.V.

Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them receives nine handfuls, after which the other takes all the remaining coins (the division may also end if the coins are divided before anyone gets nine handfuls). Hapok can grab any number of coins in a handful. What is the maximum number of coins he can guarantee himself regardless of Glazok's actions?
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Context:
Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.
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Context:
Usain runs one lap around the school stadium at a constant speed, and photographers Arina and Marina are positioned near the track. After the start, for 4 seconds, Usain was closer to Arina, then for 21 seconds he was closer to Marina, and then until the finish, he was again closer to Arina. How long does it take for Usain to run a whole lap?
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Context:
Given the equation $ax^{2}+x+3a+1=0$ with respect to $x$, if there is a root within the interval $(0,3]$, find the range of values for the real number $a$. Express your answer as an interval in standard notation.
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Context:
Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\pi$.
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Context:
A car traveled a certain percentage of the way from town X to town Y at an average speed of 35 mph. The car traveled at an average speed of 65 mph for the remaining part of the trip. The average speed for the entire trip was 50 mph. What percentage of the trip was traveled at 35 mph?
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Context:
I2.2 The surface area of a cube is $b \mathrm{~cm}^{2}$. If the length of each side is increased by $3 \mathrm{~cm}$, its volume is increased by $(2 b-a) \mathrm{cm}^{3}$, find the value of $b$.
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Context:
10. Teacher Yangyang's city has 4 subway lines, each of which intersects with every other line at exactly one point, serving as the transfer station for those two lines. One day, Teacher Yangyang had a sudden idea to start from the subway station near her home (which is not a transfer station), make at least one transfer at each transfer station, and finally return home. How many transfers does Teacher Yangyang need to make to achieve her goal? $\qquad$ times.
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Context:
How many positive integers $n < 10^6$ are there such that $n$ is equal to twice the square of an integer and is equal to three times the cube of an integer?
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Context:
What is the highest power of 3 that divides the number whose decimal representation consists of $3^n$ ones?
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Context:
Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes 8 equal rows with the pebbles remaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes he can make 9 equal rows with the remaining pebbles. After Kramer eats another, he finds he can make 10 equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the box in the beginning.
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Context:
Given that the equation $\frac{x-3}{x-2}=\frac{m}{2-x}$ has no solution, calculate the exact value of $m$ that makes the equation undefined. Express your answer as a single integer.
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Context:
Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m) = 20$ and $s(33m) = 120$, what is the value of $s(3m)$?
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Context:
The lateral face of a regular truncated triangular pyramid forms an acute angle $\alpha$ with the plane of the base. Find the angle between the height and the lateral edge of the pyramid.
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Context:
The hypotenuse of a right triangle measures $9$ inches, and one angle is $30^{\circ}$. What is the number of square inches in the area of the triangle?
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Context:
To pull a block up an inclined plane, a force three times greater than that needed to drag the block downwards is required, and the sum of these forces is equal to the weight of the block. Determine the coefficient of friction between the block and the plane.
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Context:
Two trucks each 250 m long are running in opposite directions on parallel paths. One truck has a certain speed, and the other truck has a speed of 30 km/hr. The time taken by the slower truck to pass the driver of the faster one is 35.997120230381576 seconds. What is the speed of the slower truck?
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Context:
Given a sorted list $L$ with $n$ numbers, where indices from $\frac{n}{4}$ to $\frac{3n}{4}$ contain "good" numbers, find the probability that the median of three randomly selected numbers from $L$ (with repetitions allowed) is a "good" number. The probability of selecting a "good" number is $\frac{1}{2}$, a "low bad" number is $\frac{1}{4}$, and a "high bad" number is also $\frac{1}{4}$. Calculate this probability considering all possible cases.
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Context:
Let \( p(x) = c_1 + c_2 \cdot 2^x + c_3 \cdot 3^x + c_4 \cdot 5^x + c_5 \cdot 8^x \). Given that \( p(k) = k \) for \( k = 1, 2, 3, 4, 5 \), compute \( p(6) \).
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Context:
In $\triangle ABC$, point $D$ is on side $AB$, and $\overrightarrow{BD}=\dfrac{1}{2}\overrightarrow{DA}$. Let $\overrightarrow{CB}=a$ and $\overrightarrow{CA}=b$. Calculate $\overrightarrow{CD}$ in terms of $a$ and $b$, expressing your answer as a linear combination of these vectors.
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Context:
Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$. For each $i$, where $1 \leq i \leq 200$, calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. Determine the sum of the distinct values of $g_i$.
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Context:
A positive integer \( n \) is said to be good if \( 3n \) is a re-ordering of the digits of \( n \) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11.
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Context:
Given the function $f(x) = 2\sin(\omega x - \frac{\pi}{3})$ ($\omega > 0$) and $g(x) = \cos(2x + \varphi)$ ($0 < \varphi < \pi$) have the same axis of symmetry, then the value of $g(\frac{\pi}{3})$ is \_\_\_\_\_\_.
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Context:
Determine the value of
$$
\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot \ln \left(1+\frac{1}{2 n}\right) \cdot \ln \left(1+\frac{1}{2 n+1}\right)
$$
The original answer is in the form of \frac{m}{n} \ln^3(2). Please provide the value of m + n.
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Context:
8,9

In a triangular pyramid $P A B C$, the lateral edge $P B$ is perpendicular to the plane of the base $A B C$, $P B=6, A B=B C$ $=\sqrt{15}, A C=2 \sqrt{3}$. A sphere with center $O$ lying on the face $A B P$ touches the planes of the other faces of the pyramid. Find the distance from the center $O$ of the sphere to the edge $A C$.
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Context:
2. Given $A, B, C$ are any three points on a plane, and $BC=a, CA=b, AB=c, y=$ $\frac{c}{a+b}+\frac{b}{c}$, the minimum value of $y$ is . $\qquad$
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Context:
In a $N \times N$ square table, all integers are arranged according to the following rule: 1 is placed in any cell, 2 is placed in the row whose number is equal to the column number containing 1, 3 is placed in the row whose number is equal to the column number containing 2, and so on. By how much does the sum of the numbers in the column containing $N^{2}$ differ from the sum of the numbers in the row containing 1?
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Context:
If in the quadratic equation \( x^{2} - 2(a - 3)x - b^{2} + 9 = 0 \), \( a \) and \( b \) are the numbers obtained from rolling a die, what is the probability \( P \) that this quadratic equation has two positive roots?
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Context:
A 250 m long train is running at a speed of 55 km/hr. It crossed a platform of a certain length in 50.395968322534195 seconds. What is the length of the platform?
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Context:
12.224. In a regular triangular pyramid, the dihedral angle at the base is equal to $\alpha$, the lateral surface area is $S$. Find the distance from the center of the base to the lateral face.
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Context:
在矩形 $A B C D$ 中, $A B=2, A D=1$, 边 $D C$ 上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$ 延长线上 (包含点 $B$ ) 的动点 $Q$ 满足 $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, 则向量 $\overrightarrow{P A}$ 与向量 $\overrightarrow{P Q}$ 的数量积 $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ 的最小值为 $\qquad$.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:
For how many values of $n$ with $3 \leq n \leq 12$ can a Fano table be created?
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Context:
If \( x \) is positive, find the minimum value of \(\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\).
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Context:
A few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its neighboring integers. If the sum of all the integers is positive, what is the minimal value of this sum?
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Context:
Let

$$p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.$$

Suppose that

$$p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0.$$

There is a point $(\frac {a}{c},\frac {b}{c})$ for which $p\left(\frac {a}{c},\frac {b}{c}\right) = 0$ for all such polynomials, where $a, b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$. Find $a + b + c$.
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Context:
A certain type of plant is suitable for growth on hillsides with temperatures between 16℃ and 20℃. It is known that for every 100m increase in altitude, the temperature drops by 0.5℃. If the average temperature at the foot of the mountain is 22℃, calculate the suitable altitude range for planting this type of plant. Express your answer as an inequality in meters (m).
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Context:
56. Probability of divisibility. Find the probability that if the digits $0,1,2, \ldots, 9$ are placed in random order in the empty spaces in the sequence of digits

$$
5-383-8-2-936-5-8-203-9-3-76
$$

then the resulting number will be divisible by 396.
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Context:
The watermelon seller weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) one by one while waiting for customers, balancing a watermelon on one pan of the scale with one or two weights on the other pan (possibly identical). At the same time, the seller noted down which weights he used. What is the minimum number of different numbers that could have appeared in his records if the weight of each weight is an integer number of kilograms?
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Context:
Tommy's mother lets him ride his bike 2 blocks north of his block, 3 blocks east, 2 blocks west, and 2 blocks south. Each block is rectangular in shape, with a length of 200 meters and a width of 150 meters. Due to a special rule enforced for Tommy, he needs to alternate between going north and east or west for every block he rides.

His friend's father lets his friend ride his bike in an area of 80 square blocks, with square-shaped blocks that have a side length of 180 meters. Tommy's friend has no restrictions on his riding pattern. How many times greater is the area his friend can ride in compared to Tommy?
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Context:
If $x$ and $y$ are two negative numbers, and $x < y$, then $|x|$ \_\_\_ $|y|$.
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Context:
Let $f(x,y)$ be a polynomial of degree $3$. If $f(0,0)=f(\pm 1,0)=f(0, \pm 1)=0$ and $f(1,-1)=1$, then what is the value of $f(-1,1)$?
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Context:
We say that a number of 20 digits is [i]special[/i] if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.
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Context:
Jan's three-eyed lizard has 3 times more wrinkles than eyes, and seven times more spots than wrinkles. The lizard's cousin has 2 times more wrinkles than eyes, and 5 times more spots than wrinkles. How many fewer eyes do both lizards have combined, compared to the combined total of spots and wrinkles on both lizards?
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Context:
Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.If the length of the interval in which \(t\) lies is \(\frac{a}{b}\), find \(a + b\).
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Context:
Someone forgot the last digit of a phone number and dialed randomly. The digits can be $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. Calculate the probability of connecting the call in no more than two attempts. Express your answer as a simplified fraction.
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Context:
What is the maximum number of principals that can serve at Lincoln High School during an 8-year period, given that each principal serves exactly one 3-year term? Provide your answer as a whole number.
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Context:
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$. At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E \rfloor$, the greatest integer less than or equal to $E$?
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Context:
Baker made 61 pastries and 167 cakes. He sold 108 cakes and some pastries. Baker still has 59 cakes. How many pastries did Baker sell?
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Context:
From the digits 1 to 9, three single-digit and three double-digit numbers are formed, with no repeated digits. Find the smallest possible average of the resulting set of numbers.
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Context:
The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$.
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Context:
A finite increasing sequence $a_{1}, a_{2}, \ldots, a_{n}$ ($n \geq 3$) of natural numbers is given, such that for all $k \leq n-2$, the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-1$ holds. The sequence must contain the term $a_{k}=2021$. Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain.
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Context:
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of $Q \left( - \frac{1}{2} \right)$.
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Context:
Given the polynomial f(x) = x^5 + 2x^3 + 3x^2 + x + 1, apply Horner's method to calculate the value at x = 3. What is the value of v_3? Express your answer as a single numerical value.
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Context:
If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?
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Context:
Two circles touch each other internally at point K. Chord \( AB \) of the larger circle touches the smaller circle at point L, with \( A L = 10 \). Find \( B L \) given that \( A K : B K = 2 : 5 \).
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Context:
A loaf of sausage has thin transverse rings drawn on it. If you cut along the red rings, you get 5 pieces, if along the yellow rings, you get 7 pieces, and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors?
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Context:
In a drawer Sandy has $5$ pairs of socks, each pair a different color.  On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer.  On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random.  The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers,  Find $m+n$.
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Context:
A class has a group of 7 students, and now select 3 of them to swap seats with each other, while the remaining 4 students keep their seats unchanged. Calculate the number of different ways to adjust their seats.
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Context:
ABC is an acute-angled triangle. B' is a point on the perpendicular bisector of AC on the opposite side of AC to B such that angle AB'C = 2∠A. A' and C' are defined similarly (with ∠CA'B = 2∠C, ∠BC'A = 2∠B). The lines AA' and B'C' meet at A". The points B" and C" are defined similarly. Find AA'/A"A' + BB'/B"B' + CC'/C"C'.
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Context:
Find the maximum real number \( M \) such that for all real numbers \( x \) and \( y \) satisfying \( x + y \geqslant 0 \), the following inequality holds:
$$
\left(x^{2}+y^{2}\right)^{3} \geqslant M\left(x^{3}+y^{3}\right)(xy - x - y).
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Context:
Find the value of the integral $$\int_{0}^{\infty} \frac{\ln^2 z}{1+z^2} \, dz$$ without using contour integration. Utilize standard special functions, such as the zeta, gamma, or beta functions.
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Context:
Consider the sequence: $x_1 = 19$, $x_2 = 95$, $x_{n+2} = \text{lcm}(x_{n+1}, x_n) + x_n$, for $n > 1$, where $\text{lcm}(a, b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
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Context:
已知向量$\overrightarrow{a}=(0,2,1)$，$\overrightarrow{b}=(-1,1,2)$，则$\overrightarrow{a}$与$\overrightarrow{b}$的夹角为________
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Context:
A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now these external spheres have radii larger than those in the original setup. Assume new radii are 50% larger than the radius of the inscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. Compute the probability that $P$ lies inside one of these external spheres.
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Context:
Given the function $f(x)=\ln(e^{x}+e^{-x})+x^{2}$, find the range of $x$ for which $f(x) > f(2x-1)$ holds.
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Context:
The square  $BCDE$  is inscribed in circle  $\omega$  with center  $O$ .  Point  $A$  is the reflection of  $O$  over  $B$ . A "hook" is drawn consisting of segment  $AB$  and the major arc  $\widehat{BE}$  of  $\omega$  (passing through  $C$  and  $D$ ). Assume  $BCDE$  has area  $200$ .  To the nearest integer, what is the length of the hook?

*Proposed by Evan Chen*
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Context:
Annie was given a pack of crayons. The pack contained some crayons. She already had a box of 36 crayons in her locker. Her friend Bobby gave her half the amount she already had in her locker. She decided to give her sister Mary 25 crayons. How many crayons were in the pack she was given?
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Context:
The equation $x^5-2x^4-1=0$ has five complex roots $r_1,r_2,r_3,r_4,r_5$.  Find the value of \[\dfrac1{r_1^8}+\dfrac1{r_2^8}+\dfrac1{r_3^8}+\dfrac1{r_4^8}+\dfrac1{r_5^8}.\]
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Context:
The surface area of a spherical cap is $n$ times larger than the lateral surface area of the cone that can be inscribed in the corresponding spherical slice. What is the ratio of the height of the slice to the diameter of the sphere?
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Context:
Let $$f(x)= \lim_{n \to \infty}\dfrac{3^n(\overbrace{\sin(\sin(...\sin(x)))}^{\text{n times}}+(\sqrt 2 \cos x+2)^n+2^n\cos x}{3^n+\sin x(\sqrt 2\cos x+2)^n}.$$ If $l = \lim\limits_{x\to \frac{\pi}{4}^+}f(x)$ and $m = \lim\limits_{x\to \frac{\pi}{4}^-}f(x)$, find the value of $l^2+m^2$.
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Context:
The real numbers \( x_1, x_2, \ldots, x_{1997} \) have a sum of \(-318\sqrt{3}\) and satisfy \(-\frac{1}{\sqrt{3}} \le x_i \le \sqrt{3}\). What is the maximum possible value for the sum of their 12th powers?
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Context:
In $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $2$, $3$, and $6$, respectively. The length of the shortest side of $\triangle ABC$ is
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Context:
Find the value of $$\lim_{x\to-\infty} \frac{\int_0^x e^{t^2} dt}{\sqrt{-\pi x}}$$
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Context:
Given points $A(1,m)$ and $B(1,2\sqrt{5}-m)$, if there exists only one point $P$ on the circle $C:x^{2}+y^{2}+2x=0$ such that $PA\perp PB$, then one possible value of the real number $m$ is _____. (Provide one value that satisfies the condition)
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Context:
At an interview, ten people were given a test consisting of several questions. It is known that any group of five people together answered all the questions (i.e., for each question, at least one of the five gave the correct answer), but any group of four did not. What is the minimum number of questions this could have been?
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Context:
Given a convex quadrilateral $ABCD$ inscribed in a circle with diameter $1$. Inside $ABCD$, there exists a point $M$ such that the sum of its squared distances to the vertices is $2$, i.e., $|MA|^2 + |MB|^2 + |MC|^2 + |MD|^2 = 2$. Determine the area of $ABCD$.
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Context:
Reese joined a Pinterest group where members contributed an average of some pins per day. The group owner deleted older pins at the rate of 5 pins per week per person. The group has 20 people and the total number of pins is 1000. After Reese had been a member for a month, there were 6,600 pins. How many pins did members contribute per day on average?
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Context:
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with a side length of 4 cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $60^{\circ}$.
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Context:
Let \( k=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i} \). In the complex plane, the vertices of \(\triangle ABC\) correspond to the complex numbers \( z_{1}, z_{2}, z_{3} \) which satisfy the equation
\[ z_{1}+k z_{2}+k^{2}\left(2 z_{3}-z_{1}\right)=0 \text {. } \]

Find the radian measure of the smallest interior angle of this triangle.
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Context:
Given the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a \geqslant b > 0)$ passes through the point $(\sqrt{3}, 1)$, find the area of the region formed by all points on these curves that satisfy $y \geqslant 1$.
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Context:
There are two people wearing red clothes, two people wearing yellow clothes, and one person wearing blue clothes. When arranging these five people in a row, people wearing the same color cannot be adjacent. Calculate the total number of different arrangements.
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Context:
求值: \(\lim _{n \rightarrow+\infty}\left(\sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\right) = x\pi\)，其中答案为\(\frac{k}{m}\)的形式，请计算k+m的值。
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Context:
A pair of dice is rolled twice, and the coordinates of point $P(m,n)$ are determined by the sequence of the dots $m$ and $n$ obtained. The probability that point $P$ lies outside the circle $x^2+y^2=17$ is $\_\_\_\_\_\_\_.$
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Context:
Two circles have radii $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$.

A common external tangent is a tangent line to two circles such that the circles are on the same side of the line, while a common internal tangent is a tangent line to two circles such that the circles are on opposite sides of the line.
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Context:
In the coordinate plane, a rectangle is drawn with vertices at $(34,0),(41,0),(34,9),(41,9)$. Find the smallest value of the parameter $a$ for which the line $y=ax$ divides this rectangle into two parts such that the area of one part is twice the area of the other. If the answer is not an integer, write it as a decimal fraction.
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Context:
Charles Dickens' 200th anniversary of birth was celebrated on February 7, 2012, which was a Tuesday. A year is a leap year if it is divisible by 400 or divisible by 4 but not by 100. Using this rule, calculate the day of the week on which Charles Dickens was born. Provide your answer as the full day of the week.
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Context:
In a distant future, humans have built 6 cities on Mars, evenly distributed on a circle. If a straight high-speed track is laid between any two cities, and a station is set up at each intersection of the tracks, with each of the 6 cities also having a station, then a total of $\qquad$ stations will be set up.
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Context:
If the proposition $p$: "There exists $x_0 \in \mathbb{R}$, such that $2^{x_0}-2 \leqslant a^2-3a$" is a false proposition, then the range of values for the real number $a$ is ______.
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Context:
Xiao Wang and Xiao Li have to process the same number of identical parts and start working at the same time. It is known that Xiao Wang processes 15 parts per hour and must rest for 1 hour after every 2 hours of work. Xiao Li works continuously, processing 12 parts per hour. Both of them finish at the exact same time. How many parts did Xiao Wang process?
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Context:
Determine the fixed point through which the graph of the function $f(x)=\log_{a}(2x-1)+1$ (where $a > 0$ and $a \neq 1$) always passes.
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Context:
A trader cheats both his supplier and customer by using faulty weights. When he buys from the supplier, he takes 10% more than the indicated weight. When he sells to his customer, he gives the customer a weight such that some percentage of that is added to the weight, the weight claimed by the trader is obtained. If he charges the cost price of the weight that he claims, his profit percentage is 43. What is the percentage by which the trader adds weight when selling to the customer?
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Context:
A root of unity is a complex number that is a solution to $z^{n}=1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^{2}+a z+b=0$ for some integers $a$ and $b$.
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Context:
Define a $\emph{smart integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.
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Context:
Let  $m, n, a, k$  be positive integers and  $k>1$  such that the equality   $$ 5^m+63n+49=a^k $$  holds. Find the minimum value of  $k$ .
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Context:
How many parallelograms with sides of lengths 1 and 2 and angles of $60^{\circ}$ and $120^{\circ}$ can be placed inside a regular hexagon with a side length of 3?
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Context:
How many 9-digit numbers are there in the decimal system that are divisible by 11 and include each digit from 1 to 9 exactly once?
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Context:
7. In $\square A B C D$, $A B=4, B C=2, \angle D A B$ $=60^{\circ}$, a point $P$ on side $B C$ and a point $Q$ on the extension of $D C$ satisfy $S_{\triangle B P Q}=\frac{\sqrt{3}}{32}$. Then the minimum value of $\overrightarrow{A P} \cdot \overrightarrow{B Q}$ is $\qquad$
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Context:
Chidi is a small business owner in Rivers State, Nigeria, who runs a local grocery store. Every week, he buys 120 bags of rice at ₦8,500 per bag and 200 crates of eggs at ₦1,500 per crate. If Chidi aims to make a profit of 20% on the rice and 25% on the eggs, what should be the total selling price for all the rice and eggs for that week?
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Context:
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart?
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Context:
In a café, 55 Indians and Turks met, each of whom drank either tea or coffee. All Indians tell the truth when they drink tea and lie when they drink coffee, while all Turks do the opposite. When asked "Are you drinking coffee?" 44 people answered "yes", when asked "Are you a Turk?" 33 people answered "yes", and 22 people agreed with the statement "It is raining outside." How many Indians in the café are drinking tea?
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Context:
During a certain two-week period, 48 percent of the movies rented from a video store were comedies, 16 percent were action movies, and the remaining movies rented consisted of dramas, thrillers, and sci-fi movies. If the number of rented dramas was 3 times the number of action movies, the number of rented thrillers was 2 times the number of rented dramas, and the number of rented sci-fi movies was equal to the number of rented action movies, and there were a action movies rented, then how many comedies, in terms of a, were rented during that two-week period?
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Context:
How many distinct permutation cycles can be expressed for the elements $(a, b, c, d)$, considering cycles to be different if a cyclic shift results in a new cycle? For instance, $(a, b, c, d)$ and $(d, a, b, c)$ are considered different.
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Context:
已知$M$是抛物线${{x}^{2}}=4y$上一点，$F$为其焦点，点$A$在圆$C:{{\left( x+1 \right)2}}+{{\left( y-6 \right)}^{2}}=1$上，则$\left| MA \right|+\left| MF \right|$的最小值是__________．
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Context:
A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit?
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Context:
Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_1, y_2, \ldots, y_n$:
\[
(x+1)^2 + y_1^2 = (x+2)^2 + y_2^2 = \ldots = (x+n)^2 + y_n^2.
\]
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Context:
The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of $\Gamma$.
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Context:
Given a positive integer  $n$ , suppose that  $P(x,y)$  is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all  $x,y\in\{0,1,2,\dots,n\}$ } \] What is the minimum degree of  $P$ ?
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Context:
The side of an equilateral triangle is $a$. Determine the area of the part of the triangle that lies outside a circle with radius $a / 3$, whose center coincides with the center of the triangle.
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Context:
Let  $A_1$ ,  $\ldots$ ,  $A_{2022}$  be the vertices of a regular  $2022$ -gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)

In each round, Bob chooses a point in the plane (not necessarily among  $A_1, \ldots, A_{2022}$ ) and Alice responds truthfully with the color of that point. What is the smallest number  $Q$  for which Bob has a strategy to always determine the colors of points  $A_1, \ldots, A_{2022}$  in  $Q$  rounds?
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Context:
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircase from the 1st floor to the 2nd floor?
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Context:
On their school trip, Lisa, Robert, Claire, James, and Emily took photos at various locations. Lisa took 3 times as many photos as Claire, whereas Robert took 5/4 of the number of photos that Claire took. James, on the other hand, took 250% of the photos Robert took, and Emily took the average of the photos that Lisa and Claire took. Added to this, Paul joined them recently but he took only half the photos Emily took. Given that the combined total number of photos taken by all six students is 840, how many photos has Claire taken?
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Context:
I am preparing for a French exam, which includes recalling vocabulary. There are 800 words in total on the exam, and the exam grade is based on the percentage of these words that I recall correctly. If I randomly guess the words I do not learn, my guessing has a 5% chance of being correct. What is the least number of words I should learn to ensure I score at least $90\%$ on the exam?
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Context:
The cost price of some books is equal to the selling price of 16 books. The gain or loss percentage is 50%. How many books were bought at the cost price?
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Context:
The fractional equation $\dfrac{x-5}{x+2}=\dfrac{m}{x+2}$ has a root, determine the value of $m$.
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Context:
From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6, $a$, $b$, 9 is ____.
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Context:
The graph of the function $y=x^2$ ($x>0$) has a tangent line at the point $(a_k, a_k^2)$, which intersects the x-axis at point with an x-coordinate of $a_k + b_k$. Here, $b_k$ is a positive integer. If $a_1=16$, find the sum $a_1 + a_3 + a_5$.
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Context:
A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$.
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Context:
The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:

- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.

What is the sum of the whole sequence?
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Context:
A staircase-shaped brick consists of 12 unit cubes, has a width of 2, and has 3 steps. Find all positive integers \( n \) such that several of these bricks can form a cube with side length \( n \).
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Context:
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
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Context:
Let \( p(x) = 2x^3 - 3x^2 + 1 \). How many squares of integers are there among the numbers \( p(1), p(2), \ldots, p(2016) \)?
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Context:
Let $ ABCDE $ be a convex pentagon with $ AB || CE, BC || AD, AC || DE, \angle ABC=120^\circ, AB=3, BC=5, $ and $DE = 15. $ Given that the ratio between the area of triangle $ ABC $ and the area of triangle $ EBD $ is $ m/n, $ where $ m $ and $ n $ are relatively prime positive integers, find $ m+n. $
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Context:
The perimeter of triangle \(ABC\) is 1. Circle \(\omega\) is tangent to side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).
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Context:
A certain rectangle had its dimensions expressed in whole numbers of decimeters. Then, it changed its dimensions three times. First, one of its dimensions was doubled and the other was adjusted so that the area remained the same. Then, one dimension was increased by $1 \mathrm{dm}$ and the other decreased by $4 \mathrm{dm}$, keeping the area the same. Finally, its shorter dimension was reduced by $1 \mathrm{dm}$, while the longer dimension remained unchanged.

Determine the ratio of the lengths of the sides of the final rectangle.

(E. Novotná)
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Context:
Given a triangle \( ABC \). A circle with a radius \( R \) is tangent to lines \( AB \) and \( BC \) at points \( A \) and \( C \), and intersects the median \( BD \) at point \( L \), such that \( BL = \frac{5}{9} BD \).

Find the area of the triangle.
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Context:
A ball invites 2018 couples, each assigned to areas numbered $1, 2, \cdots, 2018$. The organizer specifies that at the $i$-th minute of the ball, the couple in area $s_i$ (if any) moves to area $r_i$, and the couple originally in area $r_i$ (if any) exits the ball. The relationship is given by:
$$
s_i \equiv i \pmod{2018}, \quad r_i \equiv 2i \pmod{2018},
$$
with $1 \leq s_i, r_i \leq 2018$. According to this rule, how many couples will still be dancing after $2018^2$ minutes? (Note: If $s_i = r_i$, the couple in area $s_i$ remains in the same area and does not exit the ball).
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Context:
设 $a_{1}, a_{2}, \cdots, a_{18}$ 都是非零实数。试求下式的最小可能值:
$$
S=\sum_{i=1}^{18} \sum_{j=1}^{18} \frac{\left(a_{i}+a_{j}\right)^{2}\left(1+a_{i}^{2} a_{j}^{2}\right)}{a_{i}^{2} a_{j}^{2}}
$$
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Context:
How many ternary relations are possible from a set with 4 elements to a set with 3 elements?
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Context:
A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x) + f'(x) < 1$ and $f(0) = 3$. Find the solution set for the inequality $f(x) > \frac{2}{e^{x}} + 1$, where $e$ is the base of the natural logarithm.
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Context:
Find the last two digits of \(\left[(\sqrt{29}+\sqrt{21})^{1984}\right]\).
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Context:
A man can row his boat with the stream at some speed and against the stream at 12 km/h. The man's rate is 2 km/h. What is the man's speed when he rows with the stream?
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Context:
A museum has 7 adjacent exhibition rooms. A guard starts walking to the first room, then to the second room... after reaching the seventh room, he turns back and goes to the sixth room, then to the fifth room... Every time he enters a room, he switches the light in that room on or off. If all 7 rooms are illuminated at the beginning, how many rooms will still be illuminated after he has walked through 100 rooms?
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Context:
若存在正整数 $k<n\left(n \geqslant 3, n \in \mathrm{~N}^{*}\right)$, 使得 $\mathrm{C}_{n}^{k-1}, \mathrm{C}_{n}^{k}, \mathrm{C}_{n}^{k+1}$ 构成等差数列，其中组合数 $\mathrm{C}_{n}^{k}=$ $\frac{n!}{k!(n-k)!}(n!=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n$, 约定 $0!=1)$ ，则称 $n$ 为理想数。不超过 2020 的理想数的个数是多少？
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Context:
A right triangle \(ABC\) \((\angle A = 90^\circ)\) has two squares \(BEFC\) and \(AMNC\) positioned such that points \(E\) and \(A\) lie on opposite sides of line \(BC\), and points \(M\) and \(B\) lie on opposite sides of line \(AC\). Find the distance between the centers of the squares if \(AB = a\), \(AC = b\).
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Context:
What is the annual interest rate when the continuous compound interest on a principal of Rs. 600 amounts to Rs. 160 in 4 years, given the amount is compounded continuously?
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Context:
In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.6 times the probability of earning a B. The probability of earning a D is 0.3 times the probability of earning a B. Assuming that all grades are A, B, C, or D, how many B's will there be in a mathematics class of 50 students?
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Context:
Let the sequence $\{ a_n \}_{n = -2}^\infty$ satisfy the following initial conditions and recursive formula:
- $a_{-1} = a_{-2} = 0$
- $a_0 = 1$

For all non-negative integers $n$, the sequence satisfies:
$$n^2 = \sum_{k = 0}^n a_{n - k}a_{k - 1} + \sum_{k = 0}^n a_{n - k}a_{k - 2}$$

Given that $a_{2018}$ is a rational number, find the maximum integer $m$ such that $2^m$ divides the denominator of the reduced form of $a_{2018}$.
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Context:
In the 100-digit number 12345678901234...7890, all digits in odd positions were crossed out; in the resulting 50-digit number, all digits in odd positions were crossed out again, and so on. The crossing out continued as long as there were digits to cross out. Which digit was crossed out last?
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Context:
Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
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Context:
Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units.
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Context:
In a coastal village, 7 people go fishing every day, 8 people go fishing every other day, 3 people go fishing every three days, and the rest do not fish at all. Yesterday, 12 people went fishing, today 10 people are fishing. How many people will go fishing tomorrow?
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Context:
Determine the greatest possible value of \(\sum_{i=1}^{10} \cos(3x_i)\) for real numbers $x_1,x_2,\dots,x_{10}$ satisfying \(\sum_{i=1}^{10} \cos(x_i) = 0\).
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Context:
Find the sum of the two smallest possible values of $x^\circ$ (in degrees) that satisfy the following equation if $x$ is greater than $2017^\circ$:
\[
\cos^5(9x) + \cos^5(x) = 32\cos^5(5x)\cos^5(4x) + 5\cos^2(9x)\cos^2(x)(\cos(9x) + \cos(x)).
\]
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Context:
What is the smallest number of points that need to be marked inside a convex \( n \)-gon so that every triangle with vertices at the vertices of the \( n \)-gon contains at least one marked point?
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Context:
There are $2n$ cards, each numbered from 1 to $n$ (each number appearing exactly on two cards). The cards are placed face down on the table. A set of $n$ cards is called good if each number appears exactly once in the set. Baron Munchausen claims that he can point to 80 sets of $n$ cards, at least one of which is definitely good. For what maximum value of $n$ can the baron's words be true?
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Context:
Peyton puts 30 L of oil and 15 L of vinegar into a large empty can. He then adds 15 L of oil to create a new mixture. What percentage of the new mixture is oil?
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Context:
Dima has written the numbers $\frac{1}{80!}, \frac{1}{81!}, \ldots, \frac{1}{99!}$ on 20 infinite pieces of paper as decimal fractions. For example, the following is written on the last piece: \( \frac{1}{99!} = 0.\underbrace{00\ldots00}_{155\text{ zeros}}10715\ldots \). Sasha wants to cut a fragment of $N$ consecutive digits from one of these pieces without the comma. What is the maximal $N$ for which Sasha can do this so that Dima cannot guess from which piece Sasha has cut his fragment?
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Context:
John has saved up $5555_8$ dollars for a new laptop. A laptop he desires costs $1500_{10}$ dollars. How many dollars will he have left after purchasing the laptop if all his money is in base eight and the laptop price is in base ten?
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Context:
Consider the set $\Omega$ of all bijections $\phi: \mathbb{N} \cup \{0\} \rightarrow \mathbb{Q}$. We can visualize each $\phi \in \Omega$ as a "staircase" in the real plane, where for $x \in \mathbb{R}^+$, we have $x \mapsto \phi(\lfloor x \rfloor)$. The largest step in this staircase is given by $\sup_{n \geq 0} |\phi(n+1)-\phi(n)|$. What is the smallest possible value of this largest step among all bijections in $\Omega$? That is, what is \[\inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1)-\phi(n)|\ \ ?\]
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Context:
In the tetrahedron \( OABC \), \(\angle AOB = 45^\circ\), \(\angle AOC = \angle BOC = 30^\circ\). Find the cosine value of the dihedral angle \(\alpha\) between the planes \( AOC \) and \( BOC \).
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Context:
A person spends 1/3 of the money he has on clothes, 1/5 of the remaining on food, and 1/4 of the remaining on travel. He had Rs 999.9999999999998 with him in the beginning. How much money is he left with after spending on clothes, food, and travel?
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Context:
Let $x_1<x_2<x_3$ be the three real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.
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Context:
If it takes a machine some time to produce one item, the machine will produce 150 items in 2 hours. How long does it take for the machine to produce one item?
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Context:
In a town with a population of 10,000, the number of children under the age of 18 is 3,500, the number of adults from the age of 18 to 64 is 5,000, and the number of adults from the age of 65 and above is 1,500. The population of children increases annually at the rate of 10%, the population of working-age adults increases at the rate of 15%, and the population of adults above 65 increases at the rate of 5%. What will be the population of the town in two years?
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Context:
In $\triangle ABC$, $\overrightarrow{AB} \cdot \overrightarrow{AC} = 2\sqrt{3}$ and $\angle BAC = 30^\circ$. Let $M$ be a point inside $\triangle ABC$. Define $f(M) = (x, y, z)$, where $x$, $y$, and $z$ represent the areas of $\triangle MBC$, $\triangle MAC$, and $\triangle MAB$, respectively. If $f(M) = \left(x, y, \frac{1}{2}\right)$, find the minimum value of $\frac{1}{x} + \frac{4}{y}$.
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Context:
A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval [0, 1]. This process is repeated two independent times, obtaining two real numbers x and y. Calculate the probability that |x - y| > 1/2. Express your answer as a fraction in simplest form.
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Context:
A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period?
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Context:
Let  $k$  be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white.
	A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly  $k$  times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.
	Let  $m$  be the minimal  $k$  value such that Vera can guarantee that she wins no matter what Marco does. For  $k=m$ , let  $t$  be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most  $t$  turns. Compute  $100m + t$ .

*Proposed by Ashwin Sah*
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Context:
For which \( k \) do there exist \( k \) pairwise distinct primes \( p_{1}, p_{2}, \ldots, p_{k} \) such that

\[ p_{1}^{2} + p_{2}^{2} + \cdots + p_{k}^{2} = 2010? \]
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Context:
There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.
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Context:
In a tetrahedron $ABCD$ where all edges have a length of $1$, $E$ is the midpoint of $AD$, and $P$ is a moving point on the line $CE$. Find the minimum value of $|BP|+|DP|$.
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Context:
## 2. Megacube

From identical small cubes, one large cube is assembled. The length of the edge of the large cube is a hundred times larger than the length of the edge of each small cube. The large cube is placed on a table, and all its sides, except the one that touches the table, are painted red.

How many small cubes have exactly two red sides?

Result:
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Context:
Rebecca runs a hair salon. She charges $30 for haircuts, $40 for perms, $60 for dye jobs, and $80 for hair extensions. In addition, she needs to buy supplies for each service, which costs $5 for every haircut, $10 for a box of hair dye for every dye job, and $25 for hair extension materials per client. She offers a 10% discount to students and a 15% discount to seniors.

Today, she has five haircuts, three perms, two dye jobs, and one hair extension appointment scheduled. Among these clients, two haircuts are for students, one perm is for a senior, and one dye job is for a first-time visitor who receives a $5 discount. She also makes $75 in total tips. After paying her daily expenses of $45, how much money will she have in dollars at the end of the day?
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Context:
In triangle $ABC$, point $D$ is on side $AB$, $CD \perp BC$, $AC = 5\sqrt{3}$, $CD = 5$, $BD = 2AD$. Find the length of $AD$.
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Context:
The function $f(x) = x^2 + 2(a-1) + 3$ is increasing on the interval $[1, +\infty)$. Determine the range of real numbers for $a$.
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Context:
Find the smallest positive integer $n$ such that $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$.

(The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.)
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Context:
In $\triangle ABC$, let $\vec{m} = \vec{CB} - 3\vec{AC}$ and $\vec{n} = \vec{CB}$. If $\vec{m} \perp \vec{n}$, find the maximum value of $\sin A$.
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Context:
6. In space, there are 2017 points. The midpoints of the line segments connecting each pair of points are colored red. The minimum number of red points is
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Context:
9.1. At a round table, 2015 people are sitting, each of them is either a knight or a liar. Knights always tell the truth, and liars always lie. Each person was given a card with a number on it; all the numbers on the cards are different. After looking at their neighbors' cards, each person said: "My number is greater than the number of each of my two neighbors." After this, $k$ of the people said: "My number is less than the number of each of my two neighbors." For what largest $k$ could this have happened?

(O. Podlipsky)
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Context:
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
Five students $A, B, C, D$, and $E$ have participated in a competition. Before the competition, two people, $X$ and $Y$, guessed the rankings. $X$ thought the ranking would be $A, B, C, D, E$. $Y$ thought the ranking would be $D, A, E, C, B$. At the end, it was revealed that $X$ didn't guess any rankings of the participants correctly and didn't guess any orderings of pairs of consecutive participants correctly. $Y$ guessed the correct rankings of two participants and the correct ordering of two pairs of consecutive participants. Determine the rankings of the competition.
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Context:
Five lanterns of the same size but different colors (red, orange, yellow, green, purple) are connected from top to bottom and hung together end to end. It is required that red and purple are adjacent, orange and green are not adjacent, and orange cannot be at the top. The number of different ways to hang the lanterns is ______.
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Context:
Problem 10. Points $M, N$, and $K$ are located on the lateral edges $A A_{1}, B B_{1}$, and $C C_{1}$ of the triangular prism $A B C A_{1} B_{1} C_{1}$ such that $A M: A A_{1}=3: 7, B N: B B_{1}=2: 5, C K: C C_{1}=4: 9$. Point $P$ belongs to the prism. Find the maximum possible value of the volume of the pyramid $M N K P$, if the volume of the prism is 40.
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Context:
In a park there are 10,000 trees planted in a square grid pattern (100 rows and 100 trees per row). What is the maximum number of trees that can be cut down such that if you stand on any stump, no other stump is visible? (You can assume the trees are sufficiently thin.)
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Context:
Define the sequences \( a_n \), \( b_n \), \( c_n \) as follows: \( a_0 = k \), \( b_0 = 4 \), \( c_0 = 1 \). If \( a_n \) is even, then \( a_{n+1} = \frac{a_n}{2} \), \( b_{n+1} = 2b_n \), \( c_{n+1} = c_n \). If \( a_n \) is odd, then \( a_{n+1} = a_n - \frac{b_n}{2} - c_n \), \( b_{n+1} = b_n \), \( c_{n+1} = b_n + c_n \). Find the number of positive integers \( k < 1995 \) such that some \( a_n = 0 \).
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Context:
$\triangle GHI$ is inscribed inside $\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle GZC, \triangle HYD, \triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{YI} = \stackrel{\frown}{GZ},\ \stackrel{\frown}{XI} = \stackrel{\frown}{HZ},\ \stackrel{\frown}{XH} = \stackrel{\frown}{GY}$. The length of $GY$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
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Context:
Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?

*Proposed by Michael Tang*
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Context:
A line passing through the right focus of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) with a slope of $2$ intersects the right branch of the hyperbola at two points. The range of the eccentricity of this hyperbola is ______.
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Context:
A total of $960$ people are randomly numbered from $1$ to $960$. Using systematic sampling, $32$ people are selected for a survey. Find the number of people to be selected from those with numbers falling within $[450,750]$.
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Context:
The six edges of a tetrahedron $EFGH$ measure $10, 14, 19, 28, 37$ and $42$ units. If the length of edge $EF$ is $42$, determine the length of edge $GH$.
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Context:
Find the inverse Laplace transform of $\ln\left( 1+\frac{w^2}{s^2}\right)$.
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Context:
Evaluate the following limit:

$$\lim_{n\rightarrow\infty} \int_{0}^{1} \frac{n x^{n-2}}{x^{2n} + x^n + 1} \mbox {d}x$$
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Context:
Let \( P \) be a point on the ellipse \(\frac{x^{2}}{100} + \frac{y^{2}}{36} = 1\), and let \( F_{1} \) and \( F_{2} \) be the foci. If the area of the triangle \(\triangle P F_{1} F_{2}\) is 36, then \(\angle F_{1} P F_{2} =\) ____ .
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Context:
A farmer owns a piece of land, from which 95 percent was cleared for planting. Out of the cleared land, 35 percent was planted with soybeans, 40 percent with wheat, 15 percent with potatoes, and 8 percent with vegetables. If the remaining 630 acres of cleared land was planted with corn, how many acres did the farmer own?
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Context:
Let \( a_{n} \) be the number written in base 10 with \( 2^{n} \) digits '9', for \( n \geq 0 \). Let \( b_{n} = a_{0} a_{1} \cdots a_{n} \) be the product of the first \( a_{k} \). What is the sum of the digits of \( b_{n} \)?
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Context:
Let $w, x, y$, and $z$ be positive real numbers such that $0 \neq \cos w \cos x \cos y \cos z$, $2 \pi =w+x+y+z$, $3 \tan w =k(1+\sec w)$, $4 \tan x =k(1+\sec x)$, $5 \tan y =k(1+\sec y)$, $6 \tan z =k(1+\sec z)$. Find $k$.
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Context:
In the production of a certain item, its weight \( X \) is subject to random fluctuations. The standard weight of the item is 30 g, its standard deviation is 0.7, and the random variable \( X \) follows a normal distribution. Find the probability that the weight of a randomly selected item is within the range from 28 to 31 g.
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Context:
A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is found that if the price of each shirt is reduced by $1, the store can sell an additional 2 shirts per day on average. If the store makes an average daily profit of $2000, what should be the selling price of each shirt?
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Context:
Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each side of the triangle.
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Context:
Compute
\[\frac{1}{2^3 - 2} + \frac{1}{3^3 - 3} + \frac{1}{4^3 - 4} + \dots + \frac{1}{100^3 - 100}.\]
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Context:
On a ship, it was decided to determine the depth of the ocean at their current location. The signal sent by the echo sounder was received on the ship after 5 seconds. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean.
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Context:
The number of integer points inside the triangle $OAB$ (where $O$ is the origin) formed by the line $y=2x$, the line $x=100$, and the x-axis is $\qquad$.
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Context:
若正数 \(a, b\) 满足 \(a+b=1\)，求 \(\frac{1}{a}+\frac{27}{b^{3}}\) 的最小值。原始答案的形式为\(\frac{k+m \sqrt{3}}{n}\)，请给出k+m+n的值。
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Context:
Of the 4,500 employees in Company X, 5/12 are clerical staff. If the number of clerical staff were reduced by 1/4 and 50 new non-clerical employees were hired, what percent of the total number of remaining employees would then be clerical staff?
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Context:
Let \( n \) be the decimal integer 111...1 (with 1998 digits). What is the 1000th digit after the decimal point of \( \sqrt{n} \)?
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Context:
Given the sequence \( x_{1}, x_{2}, \cdots, x_{1993} \) which satisfies \( \left| x_{1} - x_{2} \right| + \left| x_{2} - x_{3} \right| + \cdots + \left| x_{1992} - x_{1993} \right| = 1993 \), let \( y_{k} = \frac{x_{1} + x_{2} + \cdots + x_{k}}{k} \) for \( k=1, 2, \cdots, 1993 \). Find the maximum possible value of \( | y_{1} - y_{2} | + | y_{2} - y_{3} | + \cdots + | y_{1992} - y_{1993} | \).
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Context:
Given that vectors $\vec{a}$ and $\vec{b}$ are both non-zero, and that $\vec{a} + 3\vec{b}$ is perpendicular to $7\vec{a} - 5\vec{b}$, and $\vec{a} - 4\vec{b}$ is perpendicular to $7\vec{a} - 2\vec{b}$, find the angle between $\vec{a}$ and $\vec{b}$.
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Context:
A school has 200 teachers, 1200 male students, and 1000 female students. Now, using stratified sampling, a sample of size $n$ is drawn from all the teachers; given that the number of female students drawn is 80, then the value of $n$ is ______.
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Context:
Lagaan is levied on the 60 percent of the cultivated land. The revenue department collected a total of Rs. 3,44,000 through the lagaan from the village of Mutter. Mutter, a very rich farmer, paid a certain amount as lagaan. The percentage of total land of Mutter over the total taxable land of the village is 0.23255813953488372%. How much money did Mutter pay as lagaan?
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Context:
What is the minimum number of points that need to be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point inside?
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Context:
The h.c.f. of two numbers is 15 and the other two factors of their l.c.m. are 11 and a certain value. The larger of the two numbers is 225. What is the other factor of their l.c.m.?
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Context:
A regular pentagon can have the line segments forming its boundary extended to lines, resulting in an arrangement of lines that intersect at ten points. How many ways are there to choose five points from these ten such that no three of the points are collinear?
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Context:
Let \( n \) be the maximum number of bishops that can be placed on the squares of a \( 6 \times 6 \) chessboard such that no two bishops are attacking each other. Let \( k \) be the number of ways to put \( n \) bishops on a \( 6 \times 6 \) chessboard such that no two bishops are attacking each other. Find \( n + k \). (Two bishops are considered to be attacking each other if they lie on the same diagonal. Equivalently, if we label the squares with coordinates \((x, y)\), with \( 1 \leq x, y \leq 6 \), then the bishops on \((a, b)\) and \((c, d)\) are attacking each other if and only if \( |a - c| = |b - d| \).)
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Context:
Find the smallest positive integer  $n$  that is divisible by  $100$  and has exactly  $100$  divisors.
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Context:
A $3 \times 7$ rectangle is covered without overlap by 3 shapes of tiles: $2 \times 2$, $1\times4$, and $1\times1$, shown below. What is the minimum possible number of $1\times1$ tiles used?
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Context:
The polynomial $f(x)=x^{3}-3 x^{2}-4 x+4$ has three real roots $r_{1}, r_{2}$, and $r_{3}$. Let $g(x)=x^{3}+a x^{2}+b x+c$ be the polynomial which has roots $s_{1}, s_{2}$, and $s_{3}$, where $s_{1}=r_{1}+r_{2} z+r_{3} z^{2}$, $s_{2}=r_{1} z+r_{2} z^{2}+r_{3}, s_{3}=r_{1} z^{2}+r_{2}+r_{3} z$, and $z=\frac{-1+i \sqrt{3}}{2}$. Find the real part of the sum of the coefficients of $g(x)$.
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Context:
In the quadrilateral \(ABCD\), it is known that \(\angle ABD = \angle ACD = 45^\circ\), \(\angle BAC = 30^\circ\), and \(BC = 1\). Find \(AD\).
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Context:
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, find the value of $m$.
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Context:
Given that the function $f(x)$ satisfies $f(x)={f'}(2)e^{x-2}-f(0)x+\frac{1}{2}x^{2}$, find the interval where $f(x)$ is monotonically decreasing. Express your answer in interval notation.
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Context:
Given a cube, determine the number of pairs of skew lines that can be formed by one of its diagonals and the edges of the cube. Express your answer as a whole number.
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Context:
There are 15 rectangular sheets of paper. In each move, one of the sheets is chosen and cut with a straight line, not passing through its vertices, into two sheets. After 60 moves, it turned out that all the sheets are triangles or hexagons. How many hexagons are there?
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Context:
Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.
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Context:
[ Pythagorean Theorem (direct and inverse) ]  [ Isosceles, inscribed, and circumscribed trapezoids ]

In an isosceles trapezoid \(ABCD\), the base \(AD\) is 4, and the base \(BC\) is 3. Points \(M\) and \(N\) lie on the diagonal \(BD\), with point \(M\) located between points \(B\) and \(N\), and segments \(AM\) and \(CN\) are perpendicular to the diagonal \(BD\). Find \(CN\) if \(BM: DN = 2: 3\).
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Context:
You have a collection of small wooden blocks that are rectangular solids measuring $3$×$4$×$6$. Each of the six faces of each block is to be painted a solid color, and you have three colors of paint to use. Find the
number of distinguishable ways you could paint the blocks. (Two blocks are distinguishable if you cannot
rotate one block so that it looks identical to the other block.)
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Context:
Find the maximum value of the function $$f(x)=\int_{1/2}^1e^{|t-x|}dt$$ for $1/2 \leq x \leq 1$, without using the method of splitting the integral.
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Context:
Max's house has 4 bedrooms, each with dimensions 15 feet by 12 feet by 9 feet. Each bedroom has walls that occupy 80 square feet in area, excluding doorways and windows. Calculate the total square feet of walls to be painted.
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Context:
Given the linear programming problem (LPP): Minimize $x+y+z-\alpha t$ subject to $2x - y + z = 6$, $-x + y + t = 3$, and $x, y, z, t \geq 0$. If $(3, 0, 0, \beta)$ is an optimal solution, what is the maximum value of $\beta - \alpha$?
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Context:
Find the largest positive integer $k$ such that there exists a set $A \subseteq \{1, 2, \ldots, 100\}$ with $k$ elements. For any $a, b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.
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Context:
The region consisting of all points in three-dimensional space within 4 units of line segment $\overline{CD}$, plus a cone with the same height as $\overline{CD}$ and a base radius of 4 units, has a total volume of $448\pi$. Find the length of $\textit{CD}$.
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Context:
For how many ordered triplets $(a,b,c)$ of positive integers less than $10$ is the product $a\times b\times c$ divisible by $20$?
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Context:
The solution set of the inequality $\tan x > -1$ is __________.
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Context:
Let \( g \) be a natural number, and suppose \( g^{4} + g^{3} + g^{2} + g + 1 \) is a perfect square. Find the sum of all such \( g \).
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Context:
The base of pyramid \( T A B C D \) is a trapezoid \( A B C D \) with \( B C \parallel A D \). The distances from points \( A \) and \( B \) to the plane \( T C D \) are \( r_1 \) and \( r_2 \) respectively. The area of triangle \( T C D \) is \( S \). Find the volume of pyramid \( T A B C D \).
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Context:
Find the smallest composite number that has no prime factors less than 15.
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Context:
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2 + kx + 2014$ and $x^2 + kx - 2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.
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Context:
Let  $S$  be a set of size  $11$ . A random  $12$ -tuple  $(s_1, s_2, . . . , s_{12})$  of elements of  $S$  is chosen uniformly at random. Moreover, let  $\pi : S \to S$  be a permutation of  $S$  chosen uniformly at random. The probability that  $s_{i+1}\ne \pi (s_i)$  for all  $1 \le i \le 12$  (where  $s_{13} = s_1$ ) can be written as  $\frac{a}{b}$  where  $a$  and  $b$  are relatively prime positive integers. Compute  $a$ .
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Context:
In a baseball league, there are a total of 10 teams. Each team plays 5 games a week for 6 months, and must play each other team at least once in the season. How many total games were played during the complete season?
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Context:
Let $|\vec{a}|=|\vec{b}|=\sqrt{2}$. If the function $f(x)=|\vec{a}+x\vec{b}|$ $(x\in\mathbb{R})$ has a minimum value of $1$, then $\vec{a}\cdot\vec{b}=$ \_\_\_\_\_\_.
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Context:
8. (1987 American Invitational Mathematics Examination) For a given sequence of distinct real numbers $r_{1}, r_{2}, r_{3}, \cdots, r_{n}$, an operation consists of comparing the second term with the first term, and swapping them if and only if the second term is smaller; this process continues until the last term is compared with its new predecessor, and they are swapped if and only if the last term is smaller. For example, the following diagram shows how the sequence $1, 9, 8, 7$ is transformed into the sequence $1, 8, 7, 9$ through one operation. The two numbers being compared at each step are marked with a “-” below them.
$$
\begin{array}{cccc}
\underline{1} & 9 & 8 & 7 \\
1 & \underline{9} & 8 & 7 \\
1 & 8 & \underline{9} & 7 \\
1 & 8 & \underline{7} & 9 \\
\hline
\end{array}
$$

Clearly, any given sequence can be transformed into a non-decreasing sequence through one or more such operations. Now assume $n=40$, and $r_{1}, r_{2}, \cdots, r_{40}$ are distinct and randomly arranged. Let $\frac{p}{q}$ (in lowest terms) be the probability that the 20th term $\left(r_{20}\right)$ is moved to the 30th position through one operation. Find the value of $p+q$.
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Context:
Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$.
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Context:
In a modified SHORT BINGO, a $5\times5$ card uses a WILD square in the middle and places 24 other numbers in the remaining squares. Numbers are placed as follows: 5 distinct numbers from $1-15$ in the first column, 5 distinct numbers from $16-30$ in the second column, 4 distinct numbers from $31-40$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $41-55$ in the fourth column, and 5 distinct numbers from $56-70$ in the last column. Determine the number of distinct possibilities for the values in the second column of this SHORT BINGO card.
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Context:
Consider an article with a cost price that is 65% of the marked price. After allowing an initial discount of 12%, the store offers two more successive discounts of 5% and 7%. Calculate the total gain percentage after all three discounts are applied.
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Context:
Anna thinks of an integer. It is not a multiple of three. It is not a perfect square. The sum of its digits is a prime number. What could be the integer that Anna is thinking of?
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Context:
What is the ones digit of the \[222{,}222-22{,}222-2{,}222-222-22-2?\]
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Context:
10.1. Find the smallest solution of the inequality

$$
\frac{-\log _{2}(120-2 x \sqrt{32-2 x})^{2}+\left|\log _{2} \frac{120-2 x \sqrt{32-2 x}}{\left(x^{2}-2 x+8\right)^{3}}\right|}{5 \log _{7}(71-2 x \sqrt{32-2 x})-2 \log _{2}(120-2 x \sqrt{32-2 x})} \geqslant 0
$$
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Context:
Given vectors $\vec{a}, \vec{b}, \vec{c}$ satisfy $|\vec{a}| = |\vec{b}| = \vec{a} \cdot \vec{b} = 2$, and $(\vec{a} - \vec{c}) \cdot (\vec{b} - 2\vec{c}) = 0$, find the minimum value of $|\vec{b} - \vec{c}|$.
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Context:
Evaluate the limit: $$\lim_{n\to\infty} \frac{\Gamma (n+1,n)}{n \Gamma (n)}-e^{-n}$$
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Context:
In a football tournament, there were 15 teams, each playing against every other team exactly once. A win earned a team 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it was found that six teams had each scored at least \(N\) points. What is the maximum possible integer value of \(N\)?
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Context:
Compute the number of positive real numbers \( x \) that satisfy
$$
\left(3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x\right)^{16}=2022 x^{13} .
$$
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Context:
A man saves a certain portion of his income during a year and spends the remaining portion on his personal expenses. Next year his income increases by 35%, but his savings increase by some percentage. His total expenditure in 2 years is double his expenditure in the 1st year. What percentage increase in his savings occurred if he saved 35% of his income in the first year?
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Context:
In a tetrahedron V-ABC with edge length 10, point O is the center of the base ABC. Segment MN has a length of 2, with one endpoint M on segment VO and the other endpoint N inside face ABC. If point T is the midpoint of segment MN, then the area of the trajectory formed by point T is __________.
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Context:
How many integer pairs \((x, y)\) satisfy the equation \(x^2 + y^2 = 9999(x - y)\)?
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Context:
$已知函数f(x)=\left(\frac{1}{x}+a\right)\ln(1+x).$
$若函数f(x)在(0，+\infty )单调递增，求a的取值范围.$
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Context:
How many distinct normal subgroups H exist in the free group of rank 2, denoted as $F_2$, such that when $F_2$ is quotiented by $H$, it is isomorphic to the Klein four-group, $V_4$?
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Context:
某地的水果店老板记录了过去100天内A类水果的日需求量x(单位：箱)，整理得到如下表所示的数据.

|x  |22 |23 |24 |25 |26 |
|-- |-- |-- |-- |-- |-- |
|频数|20 |20 |30 |18 |12 |

其中每箱A类水果的进货价为50元，出售价为100元，如果当天卖不完，就将剩下的A类水果以20元每箱的价格出售给果汁加工企业.
已知该水果店在这100天中每天A类水果的进货量均为24箱，求这100天卖出A类水果所获得的日平均利润.
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Context:
There are $n$ mathematicians attending a conference. Each mathematician has exactly 3 friends (friendship is mutual). If they are seated around a circular table such that each person has their friends sitting next to them on both sides, the number of people at the table is at least 7. Find the minimum possible value of $n$.
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Context:
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{b}|=2|\overrightarrow{a}|=1$, and $\overrightarrow{a}$ is perpendicular to $(\overrightarrow{a}+\overrightarrow{b})$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.
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Context:
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=\frac{\pi}{6}, a_{n+1}=\arctan \left(\sec a_{n}\right)\left(n \in \mathbb{N}^{*}\right)$. 求正整数 $m$, 使得

$$
\sin a_{1} \cdot \sin a_{2} \cdots \sin a_{m}=\frac{1}{100}
$$
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Context:
Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed into a higher-numbered bin than the orange ball.
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Context:
Given \(a > 0\), \(b > 0\), two lines \(l_1:(a-1)x+y-1=0\) and \(l_2:x+2by+1=0\), and \(l_1\) is perpendicular to \(l_2\), find the minimum value of \(\dfrac{2}{a}+\dfrac{1}{b}\). Express your answer as a single number.
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Context:
Find the surface area of the intersection of the paraboloid $x^2+y^2-z=0$ and the plane $z=2$.
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Context:
3 [ Constructions on a Projection Drawing ]

In a regular quadrilateral pyramid with a lateral edge of 20, the angle between the lateral edges lying in the same face is $\frac{\pi}{3}$. A line perpendicular to one of the lateral edges and passing through a point on this edge intersects the height of the pyramid. Find the length of the segment of this line that lies inside the pyramid, if the point of intersection of this line with the height divides the height in the ratio 3:7, counting from the vertex.
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Context:
Let real numbers \( s \) and \( t \) satisfy \( 19s^2 + 99s + 1 = 0 \) and \( t^2 + 99t + 19 = 0 \), respectively, with \( st \neq 1 \). Find the value of \(\frac{st + 4s + 1}{t}\).
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Context:
Evaluate the double integral $$\int_{0}^{1}\mathrm dy\!\int_{0}^{1}\frac{x-y}{(x+y)^3}\mathrm dx.$$
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Context:
Let \( p = 2^{24036583} - 1 \), the largest prime currently known. For how many positive integers \( c \) do the quadratics \( \pm x^2 \pm p x \pm c \) all have rational roots?
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Context:
Problem 2. On a plane, 100 points are marked. It turns out that on two different lines a and b, there are 40 marked points each. What is the maximum number of marked points that can lie on a line that does not coincide with a and b?
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