Context:
The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers?
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Context:
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 have the digit one in their recording, 19 have the digit two in their recording, and 30 numbers have neither the digit one nor the digit two. How many numbers did Misha erase?
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Context:
Given the 5-adic number \(x = 2 + 1 \cdot 5 + 3 \cdot 5^2 + 2 \cdot 5^3 + \ldots \), find the first four digits of \(\frac{1}{x}\) in its 5-adic expansion.
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Context:
When observing an object from the same position, the maximum number of faces that can be seen at once is $\boxed{\text{answer}}$.
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Context:
Determine the dimension of the Clifford algebra $Cl_{2,1}$.
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Context:
How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}$?
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Context:
Given that $f(x)$ is an odd function that is increasing and defined on $\mathbb{R}$, and $\{x_n\}$ is an arithmetic sequence with a common difference of $2$, satisfying

$f(x_7) + f(x_8) = 0$, find the value of $x_{2017}$.
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Context:
A line that always passes through a fixed point is given by the equation $mx - ny - m = 0$, and it intersects with the parabola $y^2 = 4x$ at points $A$ and $B$. Find the number of different selections of distinct elements $m$ and $n$ from the set ${-3, -2, -1, 0, 1, 2, 3}$ such that $|AB| < 8$.
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Context:
On an $8 \times 8$ grid, 64 points are marked at the center of each square. What is the minimum number of lines needed to separate all of these points from each other?
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Context:
Find a function, other than $e^x$ and $\cos x$, whose derivative is a function of itself, similar to how $f'(x) = f(x)$ for $f(x) = e^x$ and $f'(x) = f(x+\pi/2)$ for $f(x) = \cos x$. Provide one example of such a function.
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Context:
Find the area of the surface S, which is the portion of the cylinder defined by \(x^2+y^2=a^2\) that is bounded below by the \(xy\)-plane and above by the cone \(z=\sqrt{(x-a)^2+y^2}\).
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Context:
正整数 $9+95+995+\cdots+\underbrace{99\cdots95}_{2016\text{个}9}$ 的十进制表示中数字 1 的个数为多少？
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Context:
Determine the smallest positive integer \( N \) such that for any initial arrangement of \( 2n \) balls, each colored with one of \( n \) colors (two balls per color), Phoe Wa Lone can always sort the balls into \( n \) boxes, each containing two balls of the same color, using at most \( N \) moves. A move consists of either reversing the order of two balls in a box or moving a top ball to an empty box or to a box with a ball of the same color.
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Context:
Solve the equation

$$
\sin ^{4} x+\cos ^{4} x=\sin ^{4} 2 x+\cos ^{4} 2 x
$$
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Context:
Given that $\triangle ABC$ is an inscribed triangle in the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and the side $AB$ passes through the point $P(1,0)$. Find the maximum area of $\triangle ABC$.
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Context:
Let \( a, b, c \) be non-negative real numbers satisfying
\[ a + b + c = 8 \]
and
\[ ab + bc + ca = 16.\]
Let \( m = \min \{ ab, bc, ca \} \). Find the maximum possible value of \( m \).
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Context:
A club with 75 members ordered fruit juices. Three-sevenths of them ordered lemon juice. One-fourth of the remaining members ordered mango juice, and the rest ordered orange juice. How many members ordered orange juice?
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Context:
The CMU Kiltie Band is attempting to crash a helicopter via grappling hook.  The helicopter starts parallel (angle $0$ degrees) to the ground.  Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x$ degrees with the ground.  Causing the helicopter to tilt to $90$ degrees or beyond will crash the helicopter.  Find the expected number of times the band must pull the hook in order to crash the helicopter.

[i]Proposed by Justin Hsieh[/i]
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Context:
Find another element of order 2 in the dihedral group \(D_{2p}\) besides the element \(b\), where \(p\) is a prime greater than 2.
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Context:
For any integer k > 1, the term "length of an integer" refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. If x and y are positive integers such that x > 1, y > 1, and the maximum possible sum of the length of x and the length of y is 16, what is the maximum value of x + 3y?
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Context:
14. Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14 th jump lands at the origin?
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Context:
Using $$\int_0^{\pi/2} \log\cos x\,\mathrm dx= -\dfrac{\pi}{2} \log 2$$, find $$I=\int_0^{\infty} \log{(x^2+1/x^2)}\,\frac{dx}{1+x^2}.$$
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Context:
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \geq 1$ and $a+b+c \leq 7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$
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Context:
Given that \( x, y, z \) are the side lengths of a triangle and \( x + y + z = 1 \). Find the minimum value of the real number \( \lambda \) such that the inequality
$$
\lambda(x y + y z + z x) \geqslant 3(\lambda + 1)x y z + 1
$$
always holds.
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Context:
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such that there are infinitely many positive integers $n$ for which $S(n) - S(n + a) = 2018$.
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Context:
Let \( S = \{1, 2, 3, 4, \ldots, 100000\} \). Find the least possible value of \( k \) such that any subset \( A \) of \( S \) with \( |A| = 2010 \) contains two distinct numbers \( a \) and \( b \) with \( |a - b| \leq k \).
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Context:
Each row of a  $24 \times 8$  table contains some permutation of the numbers  $1, 2, \cdots , 8.$  In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)*
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Context:
In how many ways can the natural numbers from 1 to 10 (each used exactly once) be arranged in a $2 \times 5$ table so that the sum of the numbers in each of the five columns is odd?
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Context:
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.
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Context:
Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\left\lfloor\frac{N}{3}\right\rfloor$.
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Context:
There are three piles of stones: the first has 10 stones, the second has 15 stones, and the third has 20 stones. Each turn, one is allowed to split any pile into two smaller piles. The player who cannot make a move loses. Who will win?
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Context:
Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.

$$
y = e^{x} + e, \ln \sqrt{3} \leq x \leq \ln \sqrt{15}
$$
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\boxed{2 + \frac{1}{2} \ln \left( \frac{9}{5} \right)}


Context:
An isosceles triangle can be divided into two isosceles triangles (not necessarily equal). Find the possible values for the smallest angle of such a triangle. Provide the smallest of these values in degrees, multiplied by 6006.
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Context:
Two people, A and B, start from the same point on a 300-meter circular track and run in opposite directions. A runs at 2 meters per second, and B runs at 4 meters per second. When they first meet, A turns around and runs back. When A and B meet again, B turns around and runs back. Following this pattern, after how many seconds will the two people meet at the starting point for the first time?
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Context:
Determine the remainder of the 81-digit integer \(\overline{111 \cdots 1}\) divided by 81.
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Context:
Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length \(\frac{1}{2}\) and edges parallel to those of the unit cube that contains all three points?
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Context:
Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees.
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Context:
Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.
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Context:
Given real numbers $x_1$, $x_2$, $y_1$, $y_2$ satisfying: $x_1^2+y_1^2=1$, $x_2^2+y_2^2=1$, $x_1x_2+y_1y_2=\frac{1}{2}$, find the maximum value of $\frac{|x_1+y_1-1|}{\sqrt{2}} + \frac{|x_2+y_2-1|}{\sqrt{2}}$.
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Context:
Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$,
$$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$
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Context:
Find a three-digit number that is divisible by 45, given that the difference between this number and its reversed version is 297.
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Context:
7. (10 points) There are 37 people standing in a line to count off. The first person reports 1, and each subsequent person reports a number that is 3 more than the previous person's number. During the counting, one person made a mistake and reported a number that was 3 less than the previous person's number. The sum of the numbers reported by these 37 people is exactly 2011. Then, the person who made the mistake is the $\qquad$th person to report a number.
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Context:
Find the smallest positive integer $n$ for which $315^2 - n^2$ evenly divides $315^3 - n^3$.
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Context:
Given a mall with four categories of food: grains, vegetable oils, animal products, and fruits and vegetables, with 40, 10, 20, and 20 varieties, respectively, calculate the total sample size if 6 types of animal products are sampled.
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Context:
If the function $f(x)=(a-2)x^{2}+(a-1)x+3$ is an even function, then the interval in which $f(x)$ is increasing is $\_\_\_\_\_\_.$
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Context:
Select four students from five students named A, B, C, D, and E to participate in a volunteer service activity. Each person is assigned one of four jobs: translator, tour guide, etiquette, or driver. Each job is taken by one person. A cannot drive, B cannot translate, but both can do the other three jobs, while C, D, and E are capable of doing all four jobs. The number of different arrangement schemes is ______.
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Context:
According to legend, the poet Li Bai from the Tang Dynasty went to buy wine, singing as he walked down the street. Each time he passed a store, the amount of his wine doubled, and each time he saw a flower, he drank 2 cups. After encountering four stores and flowers in total, he still had 2 cups of wine left at the end. How many cups of wine did he originally have?
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Context:
$P$是$ΔABC$内一点，引线段$APD$，$BPE$和$CPF$，使$D$在$BC$上，$E$在$AC$上，F在AB上.已知$AP = 6 , BP = 9 ,PD = 6 , PE = 3 , CF = 20$，求$ΔABC$的面积.
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Context:
The length of a red bus is four times the length of an orange car. The length of the orange car is also 3.5 times shorter than the length of a yellow bus. A green truck is parked between the red bus and the yellow bus, and the green truck is as long as two orange cars. If the red bus is 48 feet long, and the yellow bus and green truck are parked end to end next to the red bus, what length of the yellow bus may the red bus driver see?
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Context:
Andrew installed hardwood flooring in his house. His bedroom took eight wooden planks, his living room took twenty planks, his kitchen took eleven planks, and his dining room took thirteen planks. The guest bedroom took two fewer planks than Andrew’s bedroom, and each of his two hallways took four planks. The study took the same number of planks as the guest bedroom, plus three. He accidentally ruined three planks in each bedroom, two planks in the living room, and one plank in the study by dragging furniture over them and had to replace them. He ended up with seven leftover planks at the end. How many wooden planks did Andrew buy to start with?
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Context:
Source: 2017 Canadian Open Math Challenge, Problem B4
-----
Numbers  $a$ ,  $b$  and  $c$  form an arithmetic sequence if  $b - a = c - b$ . Let  $a$ ,  $b$ ,  $c$  be positive integers forming an arithmetic sequence with  $a < b < c$ . Let  $f(x) = ax2 + bx + c$ . Two distinct real numbers  $r$  and  $s$  satisfy  $f(r) = s$  and  $f(s) = r$ . If  $rs = 2017$ , determine the smallest possible value of  $a$ .
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Context:
Taylor and his friends decide to bring their pets together for a walk. Six of Taylor's friends join him. Three of his friends have twice as many pets as Taylor, who has 4 cats. One friend has 3 dogs and a bird, while another brings 5 dogs and 2 cats. The last friend brings a unique pet collection of 2 reptiles, 3 birds, and a cat. Calculate the total number of cats, dogs, birds, and reptiles the friends have together.
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Context:
For $f \in L^2([0, 1]) $ and $x \in [0, 1]$ we have the following $$ A(f)(x) = \int_0^1 (x - y) f(y)dy $$ Calculate the operator norm $\| A \|$.
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Context:
Find the splitting field of the polynomial $x^4 + x^3 + 1$ over the finite field $\mathbb{F}_{32}$. Determine the degree of the extension field over $\mathbb{F}_{32}$ that contains all the roots of the polynomial.
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Context:
 The deputies in a parliament were split into  $10$  fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation.

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Context:
A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?
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Context:
Given the vector space $V = M_{2\times 3}(\mathbb{R})$ and the bilinear form $f: V \times V \to \mathbb{R}$ defined by $f(X, Y) = \text{Tr}(X^T A Y)$, where $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, find the matrix representation of $f$ with respect to the standard basis of $M_{2\times 3}(\mathbb{R})$. What is the order of this matrix?
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Context:
Determine if the dimension of the symplectic group Sp(2n, \(\tilde{F}\)) over the algebraic closure of a finite field \(F\) is equal to \(2n^2 + n\).
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Context:
Determine whether the cardinality of the set of all real-valued functions is $2^{\mathfrak c}$, where $\mathfrak c$ is the cardinality of the real numbers.
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Context:
Let $n > 1$ be an odd number. Consider an $n \times n$ matrix $A$ defined by the following rule for its entries:
\[
a_{i, j} =
\begin{cases}
1, & \text{if } i - j \equiv \pm 2 \pmod n,\\
2, & \text{if } i = j,\\
0, & \text{otherwise}.
\end{cases}
\]
Calculate the determinant of the matrix $A$. \( \boxed{} \)
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Context:
You arrange $12$ biologists of $12$ different heights in three rows of $4$, with the same conditions on height as in part 10(a) for all three rows. How many different arrangements are possible?

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Context:
If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a
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Context:
Balls of $3$ colours — red, blue, and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.
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Context:
The product of six distinct integers is 36. Find the sum of these six integers.
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Context:
How many integer triples(x,y,z)are there such that
 \( x - y{z}^{2} \equiv  1\;\left( {\;\operatorname{mod}\;{13}}\right) \)
 \( {xz} + y \equiv  4{\;(\operatorname{mod}\;{13})} \)
 where \( 0 \leq  x < {13} \) , \( 0 \leq  y < {13} \) ,and \( 0 \leq  z < {13} \) ?

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Context:
Determine the packing density of the heptomino P on a square grid.
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Context:
Evaluate the sum \( \sum_{k=1}^{2023}(-1)^{^{\binom {k+3}4}}k \) and determine if the result is a perfect square.
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Context:
Determine whether the dimension of any open subset of \(\mathbb{R}^n\) is equal to \(n\).
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Context:
Find the limit of the following expression as $n$ approaches infinity, given that $b > 0$:
\[ \lim_{n \to \infty} \frac{\sum_{i=0}^{n/3-1}{\frac{(bn^3)^{i}}{i!}}}{e^{bn^3}} \]
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Context:
设函数 $f(x)=e^{x}-x^{2} e^{x}-a x-1$.
若对于任意的 $x_{1}, x_{2} \in(1,+\infty), x_{1} \neq x_{2}$, 都有 $\frac{f\left(x_{1}\right)-f\left(x_{2}\right)}{x_{1}-x_{2}}<x_{1}+x_{2}$ 恒成立, 求实数 $a$ 的取值范围.
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Context:
G20 (7-3, Czechoslovakia) A tetrahedron $ABCD$ is divided into two parts by a plane parallel to $AB$ and $CD$. It is known that the ratio of the distance from $AB$ to this plane and the distance from $CD$ to the plane is equal to $k$. Find the ratio of the volumes of the two parts into which the tetrahedron is divided.
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Context:
There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students.
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Context:
Let $(B_{t})_{t \geq 0}$ be a one-dimensional Brownian motion and $(\phi(t,0))_{t \geq 0}$ be its local time at $0$. Determine whether $\lim_{t \uparrow \infty}\phi(t,0)=\infty$ almost surely.
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Context:
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
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Context:
In triangle \( ABC \), angle \( A \) is \( 60^\circ \). A bisector \( AD \) is drawn, and the radius of the circumcircle of triangle \( ADC \) with center at point \( D \) is \( \frac{2\sqrt{3}}{3} \). Given that \( AB = 1 \), find the length of segment \( BM \), where \( M \) is the intersection point of segments \( AD \) and \( BO \).
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Context:
Given the Hamiltonian \( H = \frac{1}{2}(S_1^2+S_2^2+\beta S_3^2)+R_1 \) and the first integrals \( f_1 = R_1^2+R_2^2+R_3^2 \) and \( f_2 = R_1S_1+R_2S_2+R_3S_3 \), compute the Poisson bracket \( \{f_1, f_2\} \) using the relations \( \{S_i, S_j\} = \epsilon_{ijk}S_k \), \( \{R_i, S_j\} = \epsilon_{ijk}R_k \), and \( \{R_i, R_j\} = 0 \). Verify if \( \{f_1, f_2\} = 0 \).
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Context:
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of diﬀerent positive weights of chemicals that Gerome could measure.
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Context:
Let $\alpha$ and $\beta$ be the roots of the polynomials $x^2-2$ and $x^2+2$, respectively, over an algebraic closure of $\mathbb{F}_{2023}$. Determine the degree of $\alpha+\beta$ over $\mathbb{F}_{2023}$: is it $2$ or $4$?
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Context:
Find the integers \( x,y,z \) for which
 \( \frac{1}{x + \frac{1}{y + \frac{1}{z}}} = \frac{7}{17} \)
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Context:
Find the integrating factor for the differential equation: \((3x+\frac{6}{y})\mathrm{d}x+(\frac{x^2}{y}+\frac{3y}{x})\mathrm{d}y=0\). Determine if it is possible to find a common integrating factor for the entire equation.
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Context:
Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at points $A$, $B$, and $C$. Compute the area of triangle $ABC$.
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Context:
In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\circ$, what is the perimeter of the shaded region?
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Context:
The social activist and the neighborhood watch leader are planning a meeting to discuss strategies to reduce crime in the area. They have invited 15 community members, and each member will bring 2 friends. If the meeting space can hold a maximum of 50 people, how many additional people can the activist and the leader invite without exceeding the capacity?
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Context:
Derive the gradient of \( \beta = x^TA^Tx \) with respect to \( x \), where \( x \) is an \( n \times 1 \) vector and \( A \) is an \( n \times n \) matrix that does not depend on \( x \).
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Context:
Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \( n \) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \( n \) for which this is possible?
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Context:
There are $20$ balls in a bag. $a$ of them are red, $b$ of them are white, and $c$ of them are black. It is known that

$ \bullet$ if we double the white balls, the probability of drawing one red ball is $\dfrac 1{25}$ less than the probability of drawing one red ball at the beginning,
and

$ \bullet$ if we remove all red balls, the probability of drawing one white ball is $\dfrac 1{16}$ more than the probability of drawing one white ball at the beginning.

Find $a,b,c$.
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Context:
Determine if the integral \( \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx \) can be evaluated without using complex analysis methods.
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Context:
At the beginning of a trip, the mileage odometer read $ 56200$ miles. The driver filled the gas tank with $ 6$ gallons of gasoline. During the trip, the driver filled his tank again with $ 12$ gallons of gasoline when the odometer read $ 56560$. At the end of the trip, the driver filled the tank again with $ 20$ gallons of gasoline. The odometer read $ 57060$. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?

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Context:
Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \left| \frac{1}{(1+\sin x)\ln(\sin x)} + \frac{1}{\cos^2x} \right| $$
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Context:
甲、乙、丙三个数的比是4,5：2,3：1,2，甲数比丙数大18，则甲、乙的平均数是多少．
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Context:
Determine the expression for the fourth mixed partial derivative of the probability generating function \( Q(x,y) \) in terms of the expected values of \( X \) and \( Y \).
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Context:
A set $S$ of natural numbers is called good, if for each element $x \in  S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a good set which is a subset of the set $A = \{1,2, 3,  ...,63\}$.
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Context:
In unit square \( {ABCD} \) ,diagonals \( \overline{AC} \) and \( \overline{BD} \) intersect at \( E \) . Let \( M \) be the midpoint of \( \overline{CD} \) ,with \( \overline{AM} \) intersecting \( \overline{BD} \) at \( F \) and \( \overline{BM} \) intersecting \( \overline{AC} \) at \( G \) . Find the area of quadrilateral \( {MFEG} \) .
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Context:
Determine the sign of the integral \( \iint_{|x|+|y| \leq 1} \ln(x^{2}+y^{2}) \,dx\,dy \) without explicitly computing it.
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Context:
Given a group $G$ and two subgroups $H$ and $K$ such that $H, K \subseteq G$, if for a fixed element $k \in K$, the conjugate $kHk^{-1}$ is a subset of $H$, does it necessarily follow that $kHk^{-1} = H$? Answer "yes" or "no."
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Context:
For a sequence of distinct real numbers $b_1, b_2, \ldots, b_k$ in the interval $[1, 2023]$, determine the minimum value of $k$ such that there must exist indices $i$ and $j$ with $i \neq j$ satisfying $0 < b_i - b_j < 1 + 3\sqrt[3]{b_i \cdot b_j}$. What is this minimum value of $k$?
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Context:
We call a time on a $12$ hour digital clock nice if the sum of the minutes digits is equal to the hour. For example, $10:55$, $3:12$ and $5:05$ are nice times. How many nice times occur during the course of one day? (We do not consider times of the form $00:\text{XX}$.)

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Context:
Task 3. Let $n \geq 2$ be a positive integer. Each cell of an $n \times n$ board is colored red or blue. We place dominoes on the board, each covering two cells. We call a domino plain if it lies on two red or two blue cells, and colorful if it lies on one red and one blue cell. Find the largest positive integer $k$ with the following property: no matter how the red/blue coloring of the board is done, it is always possible to place $k$ non-overlapping dominoes on the board that are either all plain or all colorful.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the matrix \( M = \begin{bmatrix} x & 0 \\ 0 & -\frac{1}{x} \end{bmatrix} \) cannot be expressed as a product of two involutory real matrices.
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Context:
聪聪计划做一个底面半径5cm，高12cm的圆柱形笔筒，全用人造革做，要用人造革多少cm**2．
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Context:
Determine the number of conjugacy classes in the quaternion group $Q$ and find the size of each class.
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Context:
A certain speed in m/s is equivalent to 1.1 kilometers per hour. What is the speed in m/s?
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Context:
For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always
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Context:
Let \( T: \mathbb{R}^4 \to \mathbb{R}^4 \) be a linear map such that the null space of \( T \) is \( \{(x,y,w,z) \in \mathbb{R}^4 : x+y+w+z=0\} \) and the rank of \( (T-4I_4) \) is 3. If the minimal polynomial of \( T \) is \( x(x-4)^{\alpha} \), find the value of \( \alpha \).
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Context:
Indecisive Andy starts out at the midpoint of the 1-unit-long segment $\overline{H T}$. He flips 2010 coins. On each flip, if the coin is heads, he moves halfway towards endpoint $H$, and if the coin is tails, he moves halfway towards endpoint $T$. After his 2010 moves, what is the expected distance between Andy and the midpoint of $\overline{H T}$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
 How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once.

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Context:
Find the number of non-singular $3 \times 3$ matrices with exactly four entries as $1$ and all other entries as $0$.
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Context:
In a single-elimination tournament consisting of $2^{9}=512$ teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy is then given the results of all 511 matches and must create a list of teams such that she can guarantee that the third-best team is on the list. What is the minimum possible length of Joy's list?
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Context:
Given that the function $y=\log_{2}(ax-1)$ is monotonically increasing in the interval $(1,2)$, the range of values for $a$ is.
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Context:
Consider a sample median $Y_n$ from $n$ independent and identically distributed samples with a population median $m$ and a probability density function $f$. Given the central limit theorem for the sample median:
\[ \sqrt{n}(Y_n - m) \xrightarrow{d} N(0, [2f(m)]^{-2}), \]
as $n \to \infty$, determine if the variances converge such that:
\[ nE[(Y_n-m)^2] \to [2f(m)]^{-2}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the intersection form for a smooth subvariety $\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1$, where $f\in H^0(\mathcal{O}(1,2))$. Determine the value of the intersection $\iota^*\mathcal{O}(2,3).\iota^*\mathcal{O}(2,3)$. Provide your answer as a single value or expression.
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Context:
Determine the signature (n+, n-, no) of the symmetric bilinear form \( s(f,g) = \int_{-1}^{1} f(x)g(x)(35x^4-30x^2+3)dx \) on the real vector space \( V = \mathbb{R}[X]_{\leq3} \) of polynomials of degree \( \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rhombus $ABCD$ has side length $2$ and $\angle B = 120$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?

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Context:
Let  $a=1+10^{-4}$ . Consider some  $2023\times 2023$  matrix with each entry a real in  $[1,a]$ . Let  $x_i$  be the sum of the elements of the  $i$ -th row and  $y_i$  be the sum of the elements of the  $i$ -th column for each integer  $i\in [1,n]$ . Find the maximum possible value of  $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$  (the answer may be expressed in terms of  $a$ ).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: $$ \lim _{n \to \infty} \frac{1}{n}\sum_{k=0}^{n-1}\left[\frac{k+1}{n}-\frac{1}{2}\right]\log(2n-2k-1) .$$
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Context:
In a diagram with 10 different natural numbers labeled as \(a, b, c, \ldots, k\), each number pointed to by an arrow equals the sum of the two numbers at the starting points of those arrows. Determine the minimum possible value of \(d\) in this arrangement.
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Context:
Determine whether the integral \( \int_\pi^{+\infty}\frac{\sin t}{t}\mathrm dt \) is positive, negative, or zero.
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Context:
The segments of two lines, enclosed between two parallel planes, are in the ratio of \( 5:9 \), and the acute angles between these lines and one of the planes are in the ratio of \( 2:1 \), respectively. Find the cosine of the smaller angle.
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Context:
Evaluate the limit of the function \( f(x) = \frac{\tan(\pi x)}{2x - 1} + \frac{2}{\pi(2x - 1)^2} \) as \( x \to \frac{1}{2} \). Determine whether the limit exists or if the function diverges at this point.
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Context:
设 $a_{1}, a_{2}, a_{3}, a_{4}$ 为四个有理数, 使得: $\left\{a_{i} a_{j} \mid 1 \leq i<j \leq 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}$, 求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的值。原始答案为\pm \frac{m}{n}格式，请给出m + n的值。
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Context:
A marksman shoots at a target located $d$ meters away. An observer standing $a$ meters from the marksman and $b$ meters from the target (as shown in Figure 13.10) hears the sound of the shot and the sound of the bullet hitting the target simultaneously. Determine the speed of the bullet, given that the speed of sound is $v$ meters per second.
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Context:
The capacity of a tank is 32 gallons. If a company bought 728 gallons of oil, how much oil is in the tank that is not full?
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Context:
Determine the number of equivalence relations on the set \( \{1,2,3,4,5,6,7\} \) that include the pairs \( \{(6,4),(4,7),(3,3),(5,1)\} \) but exclude the pairs \( \{(1,2),(6,1),(2,4)\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the endomorphism \( \phi: C_b(\mathbb{R}) \to C_b(\mathbb{R}) \) defined by \( \phi(f)(x) = x^2 f(x) \). Determine whether this endomorphism has an infinite norm, where the norm of a function \( f \) is defined as \( \|f\| = \sup_{x \in \mathbb{R}} |f(x)| \).
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Context:
已知双曲线$\frac{{{x}^{2}}}{{{a2}}}-\frac{{{y}^{2}}}{{{b2}}}=1\left( a>0,b>0 \right)$的左、右焦点分别为${{F}_{1}}$，${{F2}}$，点$A$是双曲线左支上的一点，若直线$A{{F}_{1}}$与直线$y=\frac{b}{a}x$平行且$\Delta A{{F1}}{{F}_{2}}$的周长为$9a$，则双曲线的离心率为______.
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Context:
The numbers $1, 2, \dots ,N$ are arranged in a circle where $N \geq  2$. If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of $N$?

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Context:
A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number of passengers that the tram could have?
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Context:
In a convex 2011-gon, what is the maximum number of diagonals that can be drawn such that each newly drawn diagonal intersects at most one of the previously drawn diagonals at an interior point?
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Context:
A function $f$ satisfies, for all nonnegative integers $x$ and $y$: - $f(0, x)=f(x, 0)=x$ - If $x \geq y \geq 0, f(x, y)=f(x-y, y)+1$ - If $y \geq x \geq 0, f(x, y)=f(x, y-x)+1$ Find the maximum value of $f$ over $0 \leq x, y \leq 100$.
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Context:
Roman the Tavernmaster has $20 worth of gold coins. He sells 3 gold coins to Dorothy. After she pays him, he has $12. How many gold coins does Roman have left?
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Context:
Given the function \( f(x) = \sin x + \cos x \), you are allowed to write the derivative of any previously written function, as well as the sum and product of any two previously written functions. This process can be repeated multiple times. Determine the constant \( c \) such that a function equal to \( c \) for all real \( x \) appears on the board.
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Context:
Two players, Bilion and Trilion, play a game where they take turns placing $1, $2, $5, or $10 bills into a pile, starting with Bilion. The game ends when the pile totals exactly $1,000,000. The player who makes the last move wins the $1,000,000 in the pile. If a move causes the pile to exceed $1,000,000, the player who made that move loses, and the other player wins the amount in the pile. Assuming both players play optimally, how many dollars will the winning player gain?
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Context:
Determine the isomorphism class of the quotient group \( \mathbb{Z}^3 / M \), where \( M \) is the subgroup of \( \mathbb{Z}^3 \) generated by the vectors \((13,9,2), (29,21,5), (2,2,2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse function of \( f(x) = \frac{\ln(x)}{x} \) for \( x > e \).
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Context:
Determine the value of \( x \) such that the Taylor series expansion \( e^{\left(\frac{d}{dx}\right)} = 1 + \frac{d}{dx} + \frac{1}{2} \left(\frac{d}{dx}\right)^2 + \cdots \) applied to a function \( f(x) \) at \( x = 0 \) results in \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the rotation of a set of $N$ points $p_1, \\dots, p_N$ with $p_1$ stationary, as a rotation matrix. Given $N-1$ vectors $\vec{v}_i$ from $p_1$ to $p_i$ and their new positions $p'_i$ with vectors $\vec{v}'_i$, express the rotation matrix using the data ($p, p', \vec{v}, \vec{v}'$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_C(z^2+4iz)\sin\left(\frac{3}{z+2i}\right)\,dz \) where \(C\) is the circle \(|z| = 3\) oriented in the counterclockwise direction.
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Context:
Calculate the acute angle between the surfaces given by the equations $xy^2z = 3x + z^2$ and $3x^2 - y^2 + 2z = 1$ at the point $(1, -2, 1)$. Express your answer in degrees or radians.
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Context:
Evaluate the integral \( \int_0^a \int_0^a \frac{dx \, dy}{(x^2+y^2+a^2)^{\frac{3}{2}}} \).
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Context:
Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$?

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Context:
Let the operation $\#$ be defined as $\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$, and $c$. Define a new operation $\oplus$ by $\oplus(a, b, c, d) = \#(a, b + d, c) - \#(a, b, c)$. What is the value of $\oplus(2, 4, 1, 3)$?
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Context:
Determine if the quotient module \( \mathbb{Z}^3 / \mathbb{Z} \cdot (2,0,-1) \) is a free module. If it is free, find its rank.
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Context:
Ryan initially had 120 balls. There were three times as many red balls as blue balls, and the number of green balls was equal to the sum of red and blue balls. The rest of the balls were yellow. However, 10 balls have been lost, so Ryan now has 110 balls in total. If there are 15 blue balls, how many yellow balls are there after taking into account the lost balls?
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Context:
One computer in a lab is programmed to back up data at the turn of the minute every five minutes. Another computer is programmed to back up data at the turn of the minute every two minutes. Find the number of times in twenty-four hours that the two computers back up data at the same time.

(Assume that the computers do not back up at the start of the 24-hour period.)
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Context:
If altitude $CD$ is $\sqrt3$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?

[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(3)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$30^{\circ}$",A-(0.05,0.4),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$D$",D,NE);
draw((0,.1)--(.1,.1)--(.1,0)); draw(D + .1*dir(210)--D + sqrt(2)*.1*dir(165)--D+.1*dir(120));
[/asy]
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Context:
In a metric space, determine whether the limit of an open ball centered at a point x with radius r, as r approaches 0, is equal to the set {x} or the empty set.
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Context:
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$?
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Context:
Let triangle $\vartriangle ABC$ be acute, and let point $M$ be the midpoint of $\overline{BC}$. Let $E$ be on line segment $\overline{AB}$ such that $\overline{AE} \perp \overline{EC}$. Then, suppose $T$ is a point on the other side of $\overleftrightarrow{BC}$ as $A$ is such that $\angle BTM = \angle ABC$ and $\angle TCA = \angle BMT$. If $AT = 14$, $AM = 9,$ and $\frac{AE}{AC} =\frac27$ , compute $BC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter $6.25$. Find the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The sides of the triangle are 5, 5, and 6.
}


Context:
A point \( P \) is chosen uniformly at random inside a square of side length 2. If \( P_{1}, P_{2}, P_{3}, \) and \( P_{4} \) are the reflections of \( P \) over each of the four sides of the square, find the expected value of the area of quadrilateral \( P_{1} P_{2} P_{3} P_{4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a \(5 \times 5\) table, numbers (not necessarily integers) are placed such that each number is three times smaller than the number in the cell to its right and twice larger than the number in the cell below it. Find the number in the central cell, given that the sum of all the numbers in the table is 341.
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Context:
Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$

Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.
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Context:
Example 8 (2004 Jiangsu Summer Camp) Let $S=\frac{1}{\sqrt{1}+\sqrt{2}}+$ $\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\frac{1}{\sqrt{2003}+\sqrt{2004}}$, find the integer part of $S$.

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Context:
Let $F$ be a field and consider the field extensions $L = F(xz, yz, xy)$ and $K = F(x, y, z)$. Determine if the degree $[K:L]$ of the extension $K$ over $L$ is finite, and if so, compute its value.
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Context:
Let \( \mathbf{A} \) be an \( m \times n \) matrix and \( \mathbf{B} \) be a symmetric \( n \times n \) matrix, where \( m < n \). Given that \( \operatorname{Rank}(\mathbf{A}) = \operatorname{Rank}(\mathbf{B}) = m \), determine the rank of the matrix product \( \mathbf{ABA}^\top \). Is it true that \( \operatorname{Rank}(\mathbf{ABA}^\top) = m \)?
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Context:
Given $a > 0$ and $a, b, x \in \mathbb{R}^n$, if for any $x > 0$, whenever $a^T x \leq 1$ we have $b^T x \leq 1$, does it imply that $a \geq b$?
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Context:
Ali and Veli go hunting. The probability that each will successfully hit a duck is 1/2 on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. Find the probability that Veli hits more ducks than Ali. Express your answer as a simplified fraction.
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Context:
Find the Euler characteristic of the surface defined by \( N = \{(x,y,z) \in \mathbb{R}^3 \mid x^4 + y^4 + z^2 = 1\} \).
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Context:
Given the relation $x = Ky$ where $K$ is a fixed scalar, and a function $f$ with its partial derivative $f'(x) = \frac{\partial f}{\partial x}(Ky)$, find the expression for $\frac{\partial f}{\partial y}(x)$ in terms of $x$.
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Context:
Given that \(\mathbf{m}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}\) and \(\mathbf{n}=(\sqrt{2}-\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the usual unit vectors along the \(x\)-axis and the \(y\)-axis respectively, and \(\theta \in (\pi, 2\pi)\). If the length or magnitude of the vector \(\mathbf{m} + \mathbf{n}\) is given by \(|\mathbf{m} + \mathbf{n}| = \frac{8 \sqrt{2}}{5}\), find the value of \(5 \cos \left(\frac{\theta}{2} + \frac{\pi}{8}\right) + 5\).
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Context:
The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers?
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Context:
Calculate the intersection multiplicity of the curves defined by $f=y^2-x^3$ and $g=y^3-x^7$ at the point $(0,0)$. Provide your answer as a single integer.
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Context:
What fraction of the area of a regular hexagon of side length 1 is within distance $\frac{1}{2}$ of at least one of the vertices?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the quadratic extension $\mathbb{Q}(\sqrt{p})$, where $p$ is a prime number not congruent to 1 modulo 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the range of values for the real number $a$ such that the inequality $2_x \ln x \geqslant -x^2 + ax - 3$ holds for all $x \in (0, +\infty)$. Express your answer as an interval or a specific numerical value.
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Context:
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, resulting in numbers a, b, c, d. Find the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} equals 4. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer is called fancy if it can be expressed in the form $$2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}$$ where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
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Context:
Consider the sequence: 1, 1, 2, 1, 2, 3, 1, 2, 3, _, 1, 2, 3. What is the tenth number in the sequence?
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Context:
In an M/G/1 queue with Poisson arrivals at rate \( \lambda = 1 \) and service time being the sum of two exponentially distributed variables with rates \( \mu_1 = 1 \) and \( \mu_2 = 2 \), calculate the probability that the waiting time \( W_q \) for an average customer is zero, i.e., find \( P(W_q = 0) \).
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Context:
Provide an example of a polynomial \( f(x) \) with integer coefficients that factors over \( \mathbb{Z}/n\mathbb{Z} \) for some integer \( n \), but has no roots in \( \mathbb{Z} \).
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Context:
Two particles move along the edges of equilateral $\triangle ABC$ in the direction $A\Rightarrow B\Rightarrow C\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$?
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Context:
Evaluate the Riemann-Stieltjes integral: \[ \int_{0}^{2}x^2d\left\lfloor 2x\right\rfloor \]
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Context:
Evaluate \( \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f'(z)}{f(z)} \, dz \), where \( f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} \).
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Context:
Determine the integers $a$ and $b$ such that $a^{4}+(a+b)^{4}+b^{4}$ is a perfect square.
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Context:
In how many ways can 13 identical balls be distributed into 6 different cells such that no cell contains exactly 3 balls?
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Context:
Evaluate the complex integral \(\int_0^{2\pi} e^{2ie^{it}}e^{-2it} dt\) using elementary methods.
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Context:
Subtract the number of positive multiples of $3$ that are less than $20$ from the number of positive multiples of $6$ that are less than $20$.
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Context:
In a bank with 1000 customers, the number of withdrawal requests per month follows a Poisson distribution with an average of 60 requests. Each withdrawal amount is exponentially distributed with an average of 700 euros. Calculate the probability that the total amount withdrawn in a given month exceeds 50,000 euros.
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Context:
Find the value of \(a\) such that the solution space of the system represented by the augmented matrix
\[\begin{pmatrix} 1 & 4 & -3 & 2 & 0\\ 2 & 7 & -4 & 4 & 0\\ -1 & a & 5 & -2 & 0\\ 3 & 10 & -5 & a^2 + 4a + 1 & 0 \end{pmatrix}\]
is two-dimensional.
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Context:
Determine the number of real roots of the function \( g(x) = \left(x^2-2\right)^2 \sinh (x) - \left(4-x^2\right)^2 \sinh (2x) - \left(\frac{x}{3}\right)^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the invariant factors of the finitely generated Abelian group $M$ given by the generators $x, y, z$ and the relations $4x + y + 2z = 0$, $5x + 2y + z = 0$, and $6y - 6z = 0$. Provide your answer as a list of invariant factors.
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Context:
Determine the order of the point \( M(0,1) \) on the curve defined by the polynomial \( f(x,y) = (x^2-1)^2 - y^2(3-2y) \).
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Context:
Let $G$ be an abelian group and $M$ be a sequence of $m$ elements from $G$. If $S$ and $F$ are subsets of $G$ with $|S|=n$ and $|F|=n-m$, determine if it is possible to find a sequence of elements from $S$ such that the sum of each element in $M$ with the corresponding element in the sequence is distinct and not in $F$. Answer with 'yes' or 'no'.
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Context:
Define $ a \circledast b = a + b-2ab $. Calculate the value of
$$A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014}$$
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Context:
If each of the three operation signs, $+$, $\text{--}$, $\times$, is used exactly ONCE in one of the blanks in the expression

$$5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3$$

then the value of the result could equal

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Context:
Find the number of integer solutions to the equation \( x + y + z = 100 \) such that \( 3 \leq x \leq y \leq z \leq 60 \).
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Context:
Evaluate the expression:
\[
\left(\frac{\left(z^{2 / p}+z^{2 / q}\right)^{2}-4 z^{2 / p+2 / q}}{\left(z^{1 / p}-z^{1 / q}\right)^{2}+4 z^{1 / p+1 / q}}\right)^{1 / 2}
\]
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\boxed{\left| z^{1 / p} - z^{1 / q} \right|}


Context:
Let $ABCD$ be a regular tetrahedron of side length $12$. Select points $E,F,G,H$ on $AC,BC,BD,AD$, respectively, such that $AE=BF=BG=AH=3$. Compute the area of quadrilateral $EFGH$.
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Context:
Find the limit $\lim_{n\to\infty} \left(\sin \frac{x}{2} \cdot \sin \frac{x}{4} \cdot \sin \frac{x}{8} \cdots \sin \frac{x}{2^n}\right)$.
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Context:
Evaluate the integral \( \int_{\mathbb{R}^1} f(x) \, dx \), where \( f(x) = |(A-x) \cap B| \) and \( A, B \) are measurable subsets of \( \mathbb{R}^1 \). Here, \( |\cdot| \) denotes the measure.
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Context:
Find a value of $n$ such that there are 100 consecutive composite numbers.
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Context:
Determine the order of growth of the sum \( \sum_{i=0}^{k-1}n^{(2^{i+1}-1)/2^{i+1}} \) as \( k \to \infty \), where \( n \) is a fixed number.
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Context:
Find the minimum value of \( \tan^2 x + \frac{\tan^2 y}{4} + \frac{\tan^2 z}{9} \) given that \( \tan x + 2\tan y + 3\tan z = 40 \) and \( x, y, z \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence defined by \( a_1 = e^i \) and \( a_{n+1} = a_n + e^{a_n} \) for all \( n \geq 1 \). Determine if the imaginary part of the sequence \( \Im(\lim_{n \to \infty} a_n) \) converges, and if so, find the value of this limit.
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Context:
Compute the integral \( \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\left(\frac{9x}{2}\right)}{\sin\left(\frac{x}{2}\right)}dx \).
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Context:
For a fixed positive integer $m$, determine if the inequality \( H_m = \liminf_{n \to \infty} (p_{n + m} - p_n) > p_{m + 2} \) holds for sufficiently large $n$. Answer "yes" or "no".
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Context:
Compute the integral \( \int_{\mathbb{R}} \int_{\mathbb{R}} \chi_S(x,y)\, d c(x)\, d \lambda(x) \) where \( S = \{ (t,t) \in \mathbb{R}^2, t \in [0,1]\} \), \( c \) is the counting measure, and \( \lambda \) is the Lebesgue measure.
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Context:
Evaluate the integral \( \int_{S} (x + 2y - z) \, dV \), where \( S \) is the tetrahedron in \( \mathbb{R}^3 \) with vertices \((0,0,0), (1,2,3), (0,1,2), (-1, 1, 1)\). Use a suitable linear transformation as a change of variables to simplify the integration.
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Context:
In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a  $12 \times 12$  board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.
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Context:
Given the polynomial \( \prod_{k=1}^{99} (x+k) = \sum_{k=0}^{99} a_k \cdot x^k \), and the expression \( M= \frac{\sum_{k=1}^{99} k\cdot a_k}{100!}= \sum_{i=1}^{99} \frac{1}{r_i} \) where \( r_i < r_j \) for all \( i < j \), find the value of \( S = \left\lfloor \sum_{k=1}^{99} \frac{1}{\sqrt{r_k-1}} \right\rfloor \).
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Context:
Evaluate the contour integral \( I = \frac{1}{2\pi i} \int_{\gamma} z^7 \cos \left(\frac{1}{z^2}\right) dz \). The function \( \cos \left(\frac{1}{z^2}\right) \) is not analytic at \( z=0 \) and does not have a pole there. Determine if the integral is zero based on the Cauchy Integral Theorem.
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Context:
There are 7 seats in a row, and 4 people are to be seated such that exactly two empty seats are adjacent, and person A and person B cannot sit next to each other. How many different seating arrangements are there? (Answer with a number).
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Context:
3. In how many ways can you partition the set $\{1,2, \ldots, 12\}$ into six mutually disjoint two-element sets in such a way that the two elements in any set are coprime?

(Martin Panák)

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Context:
Let \( G \) be a finite abelian group such that for any \( x \in G \), \( x + x = 0 \), i.e., \( G = \mathbb{Z}_{2}^{\oplus k} \) for some \( k \in \mathbb{N} \). Consider a symmetrical bilinear form \((\cdot, \cdot): G \times G \to \mathbb{Z}_{2}\). Given the conditions:
\[(a, m) = 0,\]
\[(a, p) = 1,\]
\[(b, m) = 1,\]
\[(b, p) = 0,\]
determine if it is true that \((a, b) = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \( \int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx \).
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Context:
Determine whether the number 7 is prime, irreducible, or neither in the ring \( \mathbb{Z}_{21} \).
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Context:
Bart wrote the number 1015 as a sum of numbers using only the digit 7. He used a 7 a total of 10 times, including using the number 77 three times. Now he wants to write the number 2023 as a sum of numbers using only the digit 7, using a 7 a total of 19 times. How many times will the number 77 occur in the sum?
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Context:
Given the set \( M = \left\{ x \ | \ 5 - |2x - 3| \in \mathbf{N}^{*} \right\} \), the number of all non-empty proper subsets of \( M \) is?
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Context:
Evaluate the double integral \( \int\int_R \sqrt{xy - y^2} \, dx \, dy \), where \( R \) is the triangular region with vertices at \((0,0)\), \((10,1)\), and \((1,1)\).
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Context:
Use Stokes' theorem to compute the line integral \(\oint_{\gamma} \mathbf{u} \cdot \,\mathrm{d}\mathbf{r}\), where \(\gamma\) is the boundary of the triangle in the \(xy\)-plane with vertices at \((0,0,0)\), \((1,0,0)\), and \((0,1,0)\), traced out clockwise. The vector field is \(\mathbf{u} = (x \cos(x) \sin(x) + y, y - z, x + z^3)\). Assume the triangle is oriented with its normal vector \(\mathbf{N} = (0, 0, 1)\).
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Context:
Solve the equation: $$\sqrt{\frac{\sqrt{x!}\times \sqrt{x!!}\times \sqrt{x!!!}}{\sqrt{\left( x-2 \right)!+x}}}=12$$
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Context:
Let $f(x)=x^{2021}+15x^{2020}+8x+9$ have roots $a_i$ where $i=1,2,\cdots , 2021$. Let $p(x)$ be a polynomial of the sam degree such that $p \left(a_i + \frac{1}{a_i}+1 \right)=0$ for every $1\leq i \leq 2021$. If $\frac{3p(0)}{4p(1)}=\frac{m}{n}$ where $m,n \in \mathbb{Z}$, $n>0$ and $\gcd(m,n)=1$. Then find $m+n$.
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Context:
Determine whether a coffee mug with a tube running through its middle is topologically equivalent to a 2-fold torus, a 3-fold torus, or another shape.
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Context:
Suppose a right circular cylinder has a radius of $r = 5$ units and a height of $h = 12$ units. If the diagonal across the circular top and the height measures $13$ units, what is the radius of the cylinder?
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Context:
Let the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \).
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Context:
Calculate the expected value \( \mathbb{E}\big|X_P-X_Q\big|^2 \) for points \( P=(X_P,Y_P) \) and \( Q=(X_Q,Y_Q) \) chosen independently from the square with vertices \((-1,0), (0,-1), (1,0), (0,1)\) using geometric probability.
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Context:
8. There are two inlet pipes, A and C, and two outlet pipes, B and D, in a reservoir. To fill the reservoir, it takes 3 hours for pipe A alone and 5 hours for pipe C alone. To drain the reservoir, it takes 4 hours for pipe B alone and 6 hours for pipe D alone. Currently, the reservoir is $\frac{1}{6}$ full. If the pipes are opened in the order of A, B, C, D, and each pipe is opened for one hour per day in this cycle, how long will it take for the water to start overflowing from the reservoir?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the function \( f(z) = e^{2z} - 2ze^z - 1 \) within the annulus defined by \( 1 < |z| < 2 \).
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Context:
What is the smallest positive integer \( n \) such that \( x^2 + 7 \) can be expressed as the sum of squares of \( n \) polynomials with rational coefficients, \( f_1(x)^2 + f_2(x)^2 + \cdots + f_n(x)^2 \)?
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Context:
Solve the recurrence relation for the time complexity: \[ T(n) = T(\sqrt{n}) + n^2 \]
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Context:
Bob's Rice ID number consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any number of times. The ID number satisfies the following divisibility conditions:

- The number formed by the first two digits is divisible by $2$.
- The number formed by the first three digits is divisible by $3$.
- The number formed by the first four digits is divisible by $4$.
- The number formed by the first five digits is divisible by $5$.
- The entire six-digit ID number is divisible by $6$.

One example of such an ID number is $123252$. Determine how many different possibilities exist for Bob's ID number.
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Context:
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

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Context:
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively. Suppose $AD, BE, CF$ are concurrent at $P$. If $\frac{PF}{PC} = \frac{2}{3}$, $\frac{PE}{PB} = \frac{2}{7}$ and $\frac{PD}{PA} = \frac{m}{n}$, where $m, n$ are positive integers with $\gcd(m, n) = 1$, find $m + n$.
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Context:
Given that $a_{1}, a_{2}, b_{1}, b_{2}, \cdots, b_{k}$ are vectors in the plane that are pairwise non-parallel, with $\left|a_{1}-a_{2}\right|=1$, and $\left|a_{i}-b_{j}\right| \in\{1,2,3\} (i=1,2; j=1,2, \cdots, k)$, determine the maximum value of $k$.
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Context:
a train of length 250 m crosses a bridge of length 150 m in 32 seconds . what is the speed of train ?
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Context:
$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $
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Context:
Find the number of integers $n$ such that $2000 \leq n \leq 2010$ and $7$ divides $\left\lfloor \frac{\sqrt{2}}{4}(1+\sqrt{2})^n \right\rfloor$. Provide your answer as a single integer.
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Context:
Find the spectrum of the operator \( T \) on the Banach space \( \ell^1 \), where \( \|x_n\|=\sum_{n=1}^\infty |x_n| \) and \( T\{x_n\}=\{x_2,x_1, x_4, x_3, x_6, x_5, x_8,x_7, \ldots \} \).
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Context:
Evaluate the integral \( \int_{C_r} \frac{100z^{100}+99z^{99}}{z^{100}+z^{99}+1} \, dz \) using the Cauchy integral theorem, where \(C_r\) is a contour that encloses all the roots of the denominator.
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Context:
Two cars start from the opposite places of a main road, 150 km apart. The first car runs for some distance and takes a right turn and then runs 15 km. It then turns left and then runs for the same distance as before and then takes the direction back to reach the main road. In the meantime, due to a minor breakdown, the other car has run only 62 km along the main road. The distance between the two cars at this point is 38 km. How far did the first car run before taking the right turn?
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Context:
Let $X_i \sim N(0,1)$ for $0 \leq i \leq n$ be independent standard normal distributions. Define $Y_n := \frac{\sqrt{n} X_0}{\sqrt{(\sum_{i=1}^n X_i^2)}}$. Determine the limiting distribution function for $Y_n$ as $n$ approaches infinity.
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Context:
Determine the asymptotic lower bound of the sum \( S = \frac{3}{2} + \sum_{k=3}^{n} \frac{n!}{k(n-k)!n^k} \).
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Context:
Compute the rank of the $\mathbb{Q}$ vector space $\text{Hom}_R(M,R/(x^2))$, where $R=\mathbb{Q}[x]$ and $M$ is the cokernel of the map $R^2 \rightarrow R^2$ given by the matrix \(\begin{bmatrix} x-1 & 0 \\ 1-x & x^2 \end{bmatrix}\).
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Context:
Let \( S = 2^3 + 3^4 + 5^4 + 7^4 + \cdots + 17497^4 \) be the sum of the fourth powers of the first 2014 prime numbers. Find the remainder when \( S \) is divided by 240.
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Context:
The positive divisors of a positive integer \( n \) are ordered as \( 1 = d_1 < d_2 < \cdots < d_k = n \). Suppose \( d_7^2 + d_{15}^2 = d_{16}^2 \). Find all possible values of \( d_{17} \).
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Context:
Find the derivative of \(\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)\) with respect to \(\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)\) at \(x=0\).
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Context:
The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one?
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Context:
Does there exist a function that has an infinite number of non-zero derivatives, starts at negative infinity, and approaches zero as \( x \) goes to infinity?
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Context:
Let $G$ be a group where $a$ and $b$ are non-identity elements such that $ab = b^2a$. If the subgroup of $G$ generated by $a$ has order 3, determine the order of the subgroup of $G$ generated by $b$.
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Context:
 Find three positive integers $a, b, c$ whose sum is $407$, and whose product (when written in base $10$) ends in six $0$'s.
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Context:
Consider the set of integers \( \mathbb{Z} \) with the usual less-than relation. Determine whether the well-founded part of this set, \( W(\prec) \), is the empty set.
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Context:
Determine the minimum capacity required for each of 10 boxes such that in 90% of the games, no box overflows. In each of the 50 trials, 4 boxes are selected at random, and one ball is placed into each selected box. There are 200 balls in total. What is the minimum capacity of each box?
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Context:
A spherical shell of mass $M$ and radius $R$ is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle $\theta$ with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is $g$. The moment of inertia of a thin shell of radius $r$ and mass $m$ about the center of mass is $I = \frac{2}{3}mr^2$; the momentof inertia of a solid sphere of radius r and mass m about the center of mass is $I = \frac{2}{5}mr^2$.
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Context:
Suppose $g(x)$ is a differentiable, real-valued, periodic function with period $a$ such that for all $u$, $\int_u^{u+a} g(x)\,dx=0$. Determine whether it is true that $\int_u^{u+a}g(x+g(x))\,dx=0$ for all $u$.
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Context:
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and two lines $l_1: y = \frac{1}{2}x$ and $l_2: y = -\frac{1}{2}x$, parallel lines to $l_1$ and $l_2$ pass through a point $P$ on the ellipse and meet $l_1$ and $l_2$ at points $M$ and $N$, respectively. If $|MN|$ is a constant value, then the value of $\sqrt{\frac{a}{b}}$ is ______.
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Context:
# Problem 7. Option 1.

Given a triangle with sides 6, 8, and 10. Find the length of the shortest segment connecting points on the sides of the triangle and dividing it into two equal areas.
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Context:
Given are the sequences
\[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...);  (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\]
of positive real numbers. For each integer $n$ the following inequalities hold:
\[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\]
\[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\]
\[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\]
Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a real-valued function \( f\left( {x,y}\right) \) of two positive real variables \( x \) and \( y \) ,define \( f \) to be linearly bounded if and only if there
 exists a positive number \( K \) such that \( \left| {f\left( {x,y}\right) }\right|  < K\left( {x + y}\right) \) for all positive \( x \) and \( y \) . Find necessary and sufficient conditic
 on the real numbers \( \alpha \) and \( \beta \) such that \( {x}^{\alpha }{y}^{\beta } \) is linearly bounded.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\alpha + \beta=1, \alpha,\beta \geq 0}


Context:
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let's define a point on the plane $(u, v)$ as lying between the parabolas \( y = f(x) \) and \( y = g(x) \) if \( f(u) \leq v \leq g(u) \). Find the smallest real number \( p \) for which the following statement is true: any line segment whose endpoints and midpoint lie between the parabolas \( y = x^2 \) and \( y = x^2 + 1 \), lies entirely between the parabolas \( y = x^2 \) and \( y = x^2 + p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a card game with 101 cards, each card has a unique positive integer from 1 to 101 written on it. You win if the last card you turn over is the highest in the deck. What is the optimal number of cards to turn over from the first half of the deck to maximize your probability of winning?
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Context:
Find the limit of the sequence defined by the recurrence relation \(x_{n+1} = x_n(2 - ax_n)\), where \(a\) is a positive real number.
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Context:
13. (10 points) A class of 30 students goes to the movies, and their student IDs are $1, 2, \cdots, 30$; the movie tickets they hold are exactly the 1st, 2nd, ..., 30th seats in a row. Now, the movie tickets are to be distributed to these students according to the following requirements: for any two students, A and B, if A's student ID can be divided by B's student ID, then A's movie ticket number can also be divided by B's movie ticket number. How many different ways are there to distribute the movie tickets? $\qquad$
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Context:
Given an isosceles triangle \( ABC \) with \( AB = BC \). Points \( K \), \( L \), and \( M \) are marked on the sides \( AB \), \( BC \), and \( AC \) respectively such that \(\angle AKM = 90^\circ\), \(\angle BLK = 90^\circ\), and \( KM = KL \). What is the measure of angle \( CML \)?
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Context:
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:

$\qquad\ (1) \; f(1) = 1.$

$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$

$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$

$\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$

Find the sum of all possible values of $f(2023)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a surface in \(\mathbb{R}^3\) with the first fundamental form given by \(ds^2=E(u)du^2+G(v)dv^2\). Determine whether the Gaussian curvature of this surface vanishes.
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Context:
During the van ride from the Grand Canyon to the beach, Michael asks his dad about the costs of renewable energy resources. "How much more does it really cost for a family like ours to switch entirely to renewable energy?"

Jerry explains, "Part of that depends on where the family lives. In the Western states, solar energy pays off more than it does where we live in the Southeast. But as technology gets better, costs of producing more photovoltaic power go down, so in just a few years more people will have reasonably inexpensive options for switching to clearner power sources. Even now most families could switch to biomass for between $\$200$ and $\$1000$ per year. The energy comes from sawdust, switchgrass, and even landfill gas. We pay that premium ourselves, but some families operate on a tighter budget, or don't understand the alternatives yet."

"Ew, landfill gas!" Alexis complains mockingly.

Wanting to save her own energy, Alexis decides to take a nap. She falls asleep and dreams of walking around a $2-\text{D}$ coordinate grid, looking for a wormhole that she believes will transport her to the beach (bypassing the time spent in the family van). In her dream, Alexis finds herself holding a device that she recognizes as a $\textit{tricorder}$ from one of the old $\textit{Star Trek}$ t.v. series. The tricorder has a button labeled "wormhole" and when Alexis presses the button, a computerized voice from the tricorder announces, "You are at the origin. Distance to the wormhole is $2400$ units. Your wormhole distance allotment is $\textit{two}$."'

Unsure as to how to reach, Alexis begins walking forward. As she walks, the tricorder displays at all times her distance from her starting point at the origin. When Alexis is $2400$ units from the origin, she again presses the "wormhole" buttom. The same computerized voice as before begins, "Distance to the origin is $2400$ units. Distance to the wormhole is $3840$ units. Your wormhole distance allotment is $\textit{two}$."

Alexis begins to feel disoriented. She wonders what is means that her $\textit{wormhole distance allotment is two}$, and why that number didn't change as she pushed the button. She puts her hat down to mark her position, then wanders aroud a bit. The tricorder shows her two readings as she walks. The first she recognizes as her distance to the origin. The second reading clearly indicates her distance from the point where her hat lies - where she last pressed the button that gave her distance to the wormhole.

Alexis picks up her hat and begins walking around. Eventually Alexis finds herself at a spot $2400$ units from the origin and $3840$ units from where she last pressed the button. Feeling hopeful, Alexis presses the tricorder's wormhole button again. Nothing happens. She presses it again, and again nothing happens. "Oh," she thinks, "my wormhole allotment was $\textit{two}$, and I used it up already!"

Despair fills poor Alexis who isn't sure what a wormhole looks like or how she's supposed to find it. Then she takes matters into her own hands. Alexis sits down and scribbles some notes and realizes where the wormhole must be. Alexis gets up and runs straight from her "third position" to the wormhole. As she gets closer, she sees the wormhole, which looks oddly like a huge scoop of icecream. Alexis runs into the wormhole, then wakes up.

How many units did Alexis run from her third position to the wormhole?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \(I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx\) using complex analysis techniques.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"

Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to flip it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.

If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins $\textit{ten gold coins.}$ What is the expected number of gold coins Jason wins at this game?

Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 4. Find the number of all natural numbers $n, 4 \leq n \leq$ 1023, such that their binary representations do not contain three consecutive equal digits.

Emil Kolev

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Context:
Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Reduce the lambda calculus expression $$(\lambda x. (\lambda y. x \ y) \ x) \ (\lambda z.p)$$ to its simplest form.
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Context:
Determine the value of $a$ such that the expression $n^4 + a$ can be factored into two expressions with integer coefficients for every natural number $n$.
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Context:
On the island of Misfortune, there live knights who always tell the truth and liars who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle and each said, "Both of my neighbors are liars". How many different values can $N$ take?

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Context:
Determine the range of the function \( \frac{f(x)}{x} \) as \( x \to \infty \), where \( f(x) = 2x - \sigma(x) \) and \( \sigma(x) \) is the sum-of-divisors function.
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Context:
Compute the homology group $H_2(\mathbb{S}^3/G; \mathbb{Z})$ for a finite subgroup $G$ of $SO(4)$ acting freely on $\mathbb{S}^3$. Is this homology group zero?
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Context:
Consider the set of solutions of the equation
 \( {x}^{2} + {y}^{3} = {z}^{2}. \)
 in positive integers. Is it finite or infinite?
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Context:
Determine the range of values for \( k \) such that the inequality \[ \int_0^x \frac {dt}{\sqrt {(3 + t^2)^3}} \geq k\int _0^x \frac {dt}{\sqrt {3 + t^2}} \] holds for all \( 0 \leq x \leq 1 \). You may use \( \ln 3 = 1.10 \) if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，求 $n$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( G = \mathbb{Z}_{p^2} \times \mathbb{Z}_{p^2} \), where \( p \) is a prime number. Determine the number of distinct subgroups of order \( p^2 \) in \( G \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence for the integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \). Consider how the parameter \( \alpha \) affects the convergence.
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Context:
How many positive integers $n \leq 100$ satisfy $\lfloor n\pi \rfloor = \lfloor (n-1)\pi \rfloor + 3$? Here $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$; for example, $\lfloor \pi \rfloor = 3$.
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Context:
Let \( x, y, z, w \) be four consecutive vertices of a regular \( A \)-gon. If the length of the line segment \( xy \) is 2 and the area of the quadrilateral \( xyzw \) is \( a + \sqrt{b} \), find the value of \( B = 2^a \cdot 3^b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four numbers, 2613, 2243, 1503, and 985, when divided by the same positive integer, yield the same remainder (which is not zero). Find the divisor and the remainder.
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Context:
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Before Harry Potter died, he decided to bury his wand in one of eight possible locations (uniformly at random). A squad of Death Eaters decided to go hunting for the wand. They know the eight locations but have poor vision, so even if they're at the correct location they only have a  $50\%$  chance of seeing the wand. They also get tired easily, so they can only check three different locations a day. At least they have one thing going for them: they're clever. Assuming they strategize optimally, what is the expected number of days it will take for them to find the wand?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using 6 different colors to paint the 4 cells in the picture, with each cell painted in one color, and requiring that at most 3 colors are used and no two adjacent cells have the same color, how many different coloring methods are there? (Answer using a number.)
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Context:
For which $n$-gons can a regular $n$-gon be placed on a lined sheet of paper such that all vertices lie on the lines?

(The lines are parallel and evenly spaced.)
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Context:
Find the residue of the function \( f(z) = z^{n-1} e^{\frac{pz}{2}} e^{\frac{p}{2z}} \) at the essential singularity \( z = 0 \), where \( n \in \mathbb{N} \) and \( p \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pedestrian and a cyclist are moving in one direction on a highway, while a cart and a car are moving in the opposite direction. All participants are moving at constant speeds (each with their own speed). The cyclist first overtook the pedestrian, then after some time met the cart, and then after the same amount of time met the car. The car first met the cyclist, then after some time met the pedestrian, and then after the same amount of time overtook the cart. The cyclist overtook the pedestrian at 10 o'clock, and the pedestrian met the car at 11 o'clock. When did the pedestrian meet the cart?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A police car is moving down a narrow street at 90 km/h, with a blue light on top rotating counterclockwise at one rotation per second. The buildings are 3 meters from the car on the right. Calculate the speed of the light beam on the wall when it is 6 meters ahead of its source.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of unique Tsuro tiles that can exist. Each tile has 8 entry points, and each point is connected to exactly one other point. Consider the uniqueness of tiles with respect to rotations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( I = \iint_R |xy|\,dx\,dy \) over the region \( R \) enclosed by the curves \( x^2+4y^2 = 1 \) and \( x^2+y^2 = 1 \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit \( \lim_{{n \to \infty}} x_n \) for the convergent sequence \( \{x_n\} \) defined by the equation \( x_n^3 - (n^2 + 3n)x_n - 3n^2 = 0 \) for all \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $4\times 4$ grid, there are $8$ blue tiles and $8$ white tiles. Determine the number of distinct ways to arrange the tiles in the grid, considering rotations and reflections as equivalent.
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Context:
Find the base with the smallest cardinality for the topological space (X, \(\tau\)), where X = \(\mathbb{R}^2\) and \(\tau = \{\emptyset, \mathbb{R}^2, \{(x, y) \in \mathbb{R}^2 \mid x^2+y^2<r \}, r \in \mathbb{R}, r>0\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a given positive integer $k$, consider a game played on an $n \times n$ board where $n \geq k$. The pigeon and the seagull take turns, with the pigeon going first. The pigeon chooses $m$ grids to lay an egg in each, and the seagull chooses a $k \times k$ subgrid to eat all the eggs inside it. The pigeon wins if every grid on the board eventually contains an egg; otherwise, the seagull wins. Determine all values of $m$ for which the pigeon can guarantee a win.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$.
How many times does the digit $1$ occur in its decimal representation?
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Context:
Let \( X \sim N(0,1) \). Determine the distribution of \( \Phi(X) = \int_{-\infty}^X {\exp\left(-{z^2}{/2}\right) \over \sqrt{2\pi}} \, \mathrm{d}z \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the series \(\sum_{n=0}^\infty (z^{2^{n+1}} - z^{2^n})\) for \(|z| < 1\).
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Context:
Evaluate the limit of the sequence of integrals: \[ \lim _{n \rightarrow \infty} \int_{-\infty}^\infty \frac{dx}{n(e^{x^2}-1) +1/n} \]
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Context:
What is the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?
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Context:
Consider an urn with $k$ balls, numbered from $1$ to $k$. A step involves drawing one ball uniformly at random, removing all balls with labels greater than the drawn ball's number, and returning the drawn ball to the urn. The process continues until the ball labeled $1$ is drawn. What is the expected number of steps needed to draw the ball with label $1$?
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Context:
Find the rational term for $b=11$ in the series \( \sum_{n=0}^\infty \frac{1}{n^2+bn+c} \).
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Context:
Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red?
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Context:
Determine the smallest order greater than 1 for which two orthogonal complete Latin squares exist. A Latin square of order $n$ is an $n \times n$ array filled with $n$ distinct symbols $\{s_1, \ldots, s_n\}$, such that each symbol appears exactly once in each row and each column. Two Latin squares, denoted $L_{ij}$ and $G_{ij}$, are orthogonal if each of the $n^2$ pairs $(L_{ij}, G_{ij})$ is distinct. A Latin square is complete if each of the $n(n-1)$ pairs $(L_{ij}, L_{i+1,j})$ and $(L_{ij}, L_{i,j+1})$ is distinct.
Please reason step by step, and put your final answer within \boxed{}.