Context:
Let $a$, $b$, $c$ be positive real numbers for which:
\[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b.
\]
If $a+b+c = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
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Context:
Determine the number of zeros of the polynomial \( z^{17} + z - 1 \) that lie within the region \( |z| < 1 \).
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Context:
Through a point located inside a triangle, lines are drawn parallel to the sides of the triangle.

These lines divide the triangle into three smaller triangles and three quadrilaterals. Let \( a \), \( b \), and \( c \) be the heights of the three smaller triangles parallel to the sides of the original triangle. Find the height of the original triangle parallel to these.
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Context:
Determine the Picard group of the projective space over the integers, \( \operatorname{Pic}(\mathbb{P}^n_{\mathbb{Z}}) \).
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Context:
Two five-digit numbers were written on the wall. Pat added a one in front of one of these numbers, and Mat added a one after the other. This resulted in two six-digit numbers, one of which was three times larger than the other.

Which five-digit numbers were originally written on the wall?

(L. Hozová)

Hint. Which of the newly created numbers was larger?
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Context:
Calculate the area of the region enclosed by the polar curve \( r = \arctan(r) \) from \( \phi = 0 \) to \( \phi = \frac{\pi}{\sqrt{3}} \).
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Context:
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Ivan and Foma in front of the store, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait for the store to open for no more than 4 minutes.
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Context:
Each square of a chessboard is occupied by a knight. What is the minimum number of knights that need to be removed so that there is no knight left on the board that attacks exactly three other knights? (A knight attacks squares that are one square away horizontally and two squares vertically, or vice versa.)
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Context:
It is given that  $2^{333}$  is a 101-digit number whose first digit is 1.  How many of the numbers  $2^k$ ,  $1\le k\le 332$  have first digit 4?
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Context:
1. On the first day, $2^{n}$ schoolchildren played ping-pong in a "knockout" format: first, two played, then the winner played with the third, the winner of this pair played with the fourth, and so on until the last schoolchild (there are no draws in ping-pong). On the second day, the same schoolchildren played for the cup: first, they arbitrarily split into pairs and played, the losers were eliminated, and the winners again arbitrarily split into pairs and played, and so on. It turned out that the sets of pairs that played on the first and second days were the same (possibly with different winners). Find the largest possible value of $n$.

B. Frenkin
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Context:
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\).
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Context:
a dealer offers a cash discount of 15 % and still makes a profit of 36 % when he further allows 25 articles to be sold at the cost price of 20 articles to a particular sticky bargainer . how much percent above the cost price were his articles listed ?
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Context:
Point \( O \) is inside an equilateral triangle \( \triangle ABC \). Given that \( OB = 3 \), \( OA = 4 \), and \( OC = 5 \), find the difference in the areas of \( \triangle AOC \) and \( \triangle BOC \).
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Context:
A player starts a game with $i$ tokens, and there are $n$ tokens in total. The game lasts for a maximum of $t$ rounds. In each round, the player wins one token with probability $\frac{1}{3}$ or loses one token with probability $\frac{2}{3}$. The game ends when one of the following conditions is met: the player runs out of tokens, the player collects all $n$ tokens, or $t$ rounds are completed. At the end of the game, the player's prize is calculated as $prize = 2^{result}$, where $result$ is the number of tokens the player has at the end. Calculate the expected value of the prize, $E[prize]$. Provide your answer as a single numerical value.
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Context:
Problem 5. (20 points) A point is randomly thrown onto the segment $[8 ; 13]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-2 k-35\right) x^{2}+(3 k-9) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
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Context:
Calculate the mass of a bullet-shaped object formed by a cylinder with a curved top. The cylinder has a radius of \(a\) and is situated with its base on the \((x, y)\)-plane, with the origin at the center of the base. The curved top is described by the equation \(z = 4a - \frac{x^2}{a} + \frac{y^2}{a}\). The density of the object is given by \(D \cdot \left(\frac{\rho^2}{a^2} + 1 \right)\), where \(a\) and \(D\) are constants. Determine the mass of the object using appropriate integration techniques.
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Context:
Calculate the mass of the hemisphere with radius 1, centered along the z-axis such that the south pole is at the origin, using the mass function \( p(x,y,z) = 1-z \).
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Context:
It is known that
$$
\sqrt{9-8 \sin 50^{\circ}}=a+b \sin c^{\circ}
$$
for exactly one set of positive integers \((a, b, c)\), where \(0 < c < 90\). Find the value of \(\frac{b+c}{a}\).
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Context:
Given that \( n \) is a fixed positive integer, for \( -1 \leq x_{i} \leq 1 \) (where \( i = 1, 2, \cdots, 2n \)), find the maximum value of
\[
\sum_{1 \leq r < s \leq 2n} (s - r - n) x_{r} x_{s}.
\]
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Context:
Let \( p \) be a prime number and \( f(x) = ax^2 + bx + c \) be a quadratic polynomial with integer coefficients such that \( 0 < a, b, c \leq p \). Suppose \( f(x) \) is divisible by \( p \) for every positive integer \( x \). Determine all possible values of \( a + b + c \).
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Context:
What is the value of \( \sum_{1 \le i < j \le 10}(i+j) \) for odd \( i+j \) minus \( \sum_{1 \le i < j \le 10}(i+j) \) for even \( i+j \)?
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Context:
Given that the cubic equation \(x^3 + bx^2 + cx + 1 = 0\) has only one real root \(\alpha\), with \(b < c\), find the value of the expression:
\[2\tan^{-1}(\csc \alpha) + \tan^{-1}(2\sin \alpha\sec^2\alpha)\]
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Context:
Find the smallest natural number \( n \) such that in any set of \( n \) integers, there exist two numbers whose sum or difference is divisible by \( 2k + 1 \), where \( k \) is a natural number.
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Context:
In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?
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Context:
Evaluate the limit \(\lim\limits_{n \to \infty} \frac{\sqrt[n]{a_1a_2\cdots a_n}}{a_n},\) where \(a_n\) is the largest natural number \(k\) such that \(k! \leq \alpha^n\), and \(\alpha > 1\).
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Context:
Today is $12/14/24,$ which is of the form $ab/ac/bc$ for not necessarily distinct digits $a$, $b$, and $c$. Find the number of other dates in the $21$st century that can also be written in this form.
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Context:
Find the probability that $Y > X/3$ given that the joint pdf of $X$ and $Y$ is:

$$f(x, y) = \frac {1}{2\pi} e^{-\sqrt{x^2 + y^2}}\,; \quad x,y\in \mathbb{R}$$
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Context:
Evaluate the double integral \( \iint_E f(x,y) \, dx \, dy \) where \( f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if } 0<y<|x-1/2| \\ 0 &\text{else} \end{cases} \) and \( E=[0,1]\times[0,1] \).
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Context:
Given that $\cos \alpha =\dfrac{\sqrt{5}}{5}$ and $\sin (\alpha -\beta )=\dfrac{\sqrt{10}}{10}$, calculate the value of $\cos \beta$.
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Context:
Given that $A$, $B$, and $C$ are the three internal angles of $\triangle ABC$, if $\cos (B+C)= \frac {1}{2}$, then $A=$ _____ .
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Context:
Evaluate the line integral \( \int_C (z^2, xz, 2xy) \cdot dr \), where \( C \) is the curve formed by the intersection of the surface \( z = 1 - y^2, z \ge 0 \) and the plane \( 2x + 3z = 6 \), oriented anti-clockwise.
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Context:
Calculate the surface integral of the vector field \( \vec{F} = x\vec{i} + y^3\vec{j} + \vec{k} \) over the plane passing through the points \((0,0,2)\), \((0,1,0)\), and \((2,1,0)\). The equation of the plane is \( 2y + z = 2 \). Use the unit outward normal vector \( \hat{n} = \frac{2\vec{j} + \vec{k}}{\sqrt{5}} \). Compute the integral \( \int\int_{S} \vec{F} \cdot \hat{n} \, dS \).
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Context:
Find the smallest real number \( p\left( { \leq  1}\right) \) that satisfies the following
 condition.
 (Condition) For real numbers \( {x}_{1},{x}_{2},\ldots ,{x}_{2024},{y}_{1},{y}_{2},\ldots ,{y}_{2024} \) if

 - \( 0 \leq  {x}_{1} \leq  {x}_{2} \leq  \cdots  \leq  {x}_{2024} \leq  1 \) ,
 - \({0 \leq  {y}_{1} \leq  {y}_{2} \leq  \cdots  \leq  {y}_{2024}} \leq  1 \) ,
 - \( \mathop{\sum }\limits_{{i = 1}}^{2024}{x}_{i} = \mathop{\sum }\limits_{{i = 1}}^{2024}{y}_{i} = {2024p} \) ,

then the inequality \( \mathop{\sum }\limits_{{i = 1}}^{{2024}}{x}_{i}\left( {{y}_{{2025} - i} - {y}_{{2024} - i}}\right)  \geq  1 - p \) holds.
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Context:
In the city of Fuchs, Ostap Bender organized a distribution of elephants to people. There were present 28 trades union members and 37 non-members. Ostap distributed the elephants so that the shares of all trades union members were equal, and the shares of non-members were also equal. It turned out that there existed only one such distribution (of all elephants). What greatest number of elephants could O. Bender have had?
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Context:
How many positive integers with at most $n$ digits are there whose square ends in the same $n$ digits (possibly padded with some leading zeros)?
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Context:
Transform the complex function \( f(x,y) = (x^3 - 3xy^2) + i(3x^2y - y^3) + iC \), where \( u(x,y) \) and \( v(x,y) \) are harmonic conjugates, into the form \( f(z) \) using the substitutions \( x = \frac{z + \overline{z}}{2} \) and \( y = \frac{z - \overline{z}}{2i} \). Express the function in terms of \( z \) only, ensuring that no \( \overline{z} \) terms remain.
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Context:
If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?
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Context:
Evaluate the limit: $$ \lim_{n \to \infty} \int_0^n \frac{dx}{1 + n^2 \cos^2 x} $$
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Context:
Consider two distinct positive integers  $a$  and  $b$  having integer arithmetic, geometric and harmonic means. Find the minimal value of  $|a-b|$ .
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Context:
4. Given 100 lines on a plane, let $T$ denote the set of right-angled triangles formed by some three of these lines. Find the maximum value of $|T|$. (Supplied by Zou Jin)
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Context:
The base three representation of $x$ is
\[ 12112211122211112222. \]
What is the first digit (on the left) of the base nine representation of $x$?
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Context:
Given that point $P$ is an intersection point of the ellipse $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyperbola, $e_{1}$, $e_{2}$ are the eccentricities of the ellipse and hyperbola respectively, and $\angle F_{1}PF_{2} = \frac{2\pi}{3}$, find the maximum value of $\frac{1}{e_{1}} + \frac{1}{e_{2}}$.
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Context:
Azar and Carl play a game of tic-tac-toe. Azar places an \(X\) in one of the boxes in a \(3\)-by-\(3\) array of boxes, then Carl places an \(O\) in one of the remaining boxes. After that, Azar places an \(X\) in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third \(O\). How many ways can the board look after the game is over?
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Context:
Given that I is the incenter of triangle ABC, AC = 2, BC = 3, and AB = 4. If vector AI is expressed as a linear combination of vectors AB and AC, find the value of x + y, where x and y are the coefficients of vectors AB and AC, respectively. Express your answer as a numerical value.
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Context:
Determine the number of finite dimensional Lie algebras of vector fields on the real line $\mathbf{R}$ up to diffeomorphisms.
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Context:
Given that each of the numbers n, n+1, n+2, n+3 is divisible by its sum of digits in its decimal representation, and the number in the ones column of n is 8, how many different values can the tens column of n have? Express your answer as a single integer.
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Context:
Find a basis for the image of the linear transformation $T:\mathbb{R}^4 \rightarrow \mathbb{R}^3$ defined by $T(a,b,c,d) = (a+b, c-d, a+c)$, given that a basis for the kernel of $T$ is $\{(1,-1,0,0), (0,0,1,-1)\}$.
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Context:
Determine the maximum value of the function \( g(u,v) = \frac{u\sin(\log u) - v\sin(\log v)}{v-u} \) for \( 0 < u < v \).
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Context:
Determine the monotonically decreasing interval of the function $f(x)=\frac{e^x}{x^2}$.
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Context:
Calculate the surface area of the segment of a sphere enclosed by a cylinder with center at (0, 2) and radius 2, using a double integral.
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Context:
A train 550 m long is running with a certain speed. In 29.997600191984645 seconds, it will pass a man who is running at 6 km/hr in the direction opposite to that in which the train is going. What is the speed of the train in km/hr?
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Context:
The ages of Patrick and Michael are in the ratio of 3 : 5, and the ages of another pair of people are in the same ratio. The sum of their ages is 196, and the difference between the ages of Patrick and the last person (Monica) is 64. What is the ratio of the ages of the second pair of people (Michael and Monica)?
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Context:
黑板上写着从1开始的n个连续正整数，擦去其中的一个数后，其余各数的平是36$\frac { 2 } { 5 }$，则擦去的数是
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Context:
Determine the maximum possible real value of the number k, such that
$$(a+b+c)\left(\frac{1}{a+b}+\frac{1}{c+b}+\frac{1}{a+c}-k\right) \geq k$$
for all real numbers $a, b, c \geq 0$ with $a+b+c=ab+bc+ca$.
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Context:
令 G 为三维空间中保持某个给定正方体的旋转构成的群. 设 G 的共轭类为 C_1, C_2, \cdots, C_n，满足 |C_1|\geq |C_2|\geq\cdots\geq|C_n|，求 |C_1|+|C_2|.
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Context:
设正整数 $a, b, c, d$ 满足 $a>b>c>d$, 且
$$
a b+b c+c a+d^{2} \mid(a+b)(b+c)(c+a)
$$
求 $a b+b c+c a+d^{2}$ 的素因子个数的最小值。(素因子的个数计重数，如 12 的素因子个数为 3 )。
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \left( \frac{\ln(n+1)^{n+1}}{\ln n^n} \right)^n$$
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Context:
A train of a certain length crosses an electric pole at a speed of 122 km/hr. It takes 4.425875438161669 seconds to cross the pole. What is the length of the train?
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Context:
Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)?
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Context:
$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is:
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Context:
Suppose there is an arithmetic sequence $\{a_n\}$ with common difference $d$ where $a_1=1$, and $a_1, a_2$, and $a_4$ form a geometric sequence. Let another sequence $b_n$ be defined as $b_n = \frac{1}{a_n^2 - 1}$. Find the sum of the first $n$ terms of the sequence $b_n$, denoted as $S_n$.
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Context:
Evaluate the integral: $$\int_0^\infty \frac{e^{2x}}{\left(e^{2x}\cdot\frac{1-x}{1+x}\right)^2+1}\frac{x^2}{(1+x)^2}dx$$
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Context:
Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of segment \( AB \) if \( BD \cdot BC = 5 \).
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Context:
If a positive integer can be expressed as the difference of squares of two consecutive even numbers, then this positive integer is called a "mysterious number". The sum of all "mysterious numbers" between 1 and 200 is     .
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Context:
Suppose  $\alpha,\beta,\gamma\in\{-2,3\}$  are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note:  $\mathbb{R}_{\ge0}$  is the set of nonnegative real numbers). What is the sum of the possible values of  $M$ ?
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Context:
Let $f(x)=x^3+7x^2+9x+10$. Which value of $p$ satisfies the statement
\[
f(a) \equiv f(b) \    (\text{mod } p) \Rightarrow a \equiv b \  (\text{mod } p)
\]
for every integer $a,b$?
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Context:
Determine the asymptotic behavior of the expression \( \frac{\Gamma(n+\alpha+1)}{\Gamma(n+2)} \) as \( n \to \infty \).
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Context:
Evaluate the integral \( \int_C \cot^2 z \, dz \) where \( C \) is the positively oriented circle \( |z| = 4 \).
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Context:
Evaluate the integral: \[ \int_{0}^{4\pi}\frac{x|\sin x|}{1 + |\cos x|} \, dx \]
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Context:
On the side  $ AB$  of a cyclic quadrilateral  $ ABCD$  there is a point  $ X$  such that diagonal  $ BD$  bisects  $ CX$  and diagonal  $ AC$  bisects  $ DX$ . What is the minimum possible value of  $ AB\over CD$ ?

*Proposed by S. Berlov*
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Context:
Find the value of the constant $c$ such that the random variable \( Y = \frac{c(X_1 - X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \), where \( X_1, X_2, X_3, \) and \( X_4 \) are independent standard normally distributed variables, follows a t-distribution.
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Context:
How many parallelograms with sides 1 and 2, and angles $60^{\circ}$ and $120^{\circ}$ can be placed at most inside a regular hexagon with side length 3?
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Context:
In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum.
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Context:
Given that for every \(0 \leq k \leq n\), \(a_k \geq 0\) and \(\sum_{k=0}^n \frac{a_k}{2^k} < \delta\), find an upper bound for \(\sum_{k=0}^n a_k\) that is less than \(n2^n\delta\).
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Context:
Determine the dimension of the symplectic group \( \text{Sp}(2n, F) \) when \( F \) is a finite field.
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Context:
Given that the difference between two 3-digit numbers $\overline{x y z}$ and $\overline{z y x}$ is $700-c$, where $x>z$. If $d$ is the greatest value of $x+z$, find the value of $d$.
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Context:
John took a test that was divided into four sections with varying numbers of questions, different success rates, and partial marks.

In the first section, there were 50 questions, and John answered 82% of the questions correctly. For the remaining 18% questions, he earned half the marks by getting partial answers correct.

The second section had 60 questions, and John was able to answer 75% of the questions correctly. For the remaining questions, he earned one-third of the marks for providing partially correct answers.

The third section had 30 questions, out of which John answered 90% correctly. For the remaining 10% questions, he didn't receive any marks.

The final section had 40 questions, and John managed to secure 95% of the correct answers. He earned half the marks for the rest of the questions by partially answering them.

If there is a negative marking scheme in which 0.25 marks are deducted for every wrong answer in sections two and three only, how many total marks will John obtain?
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Context:
Let set \( A = \{1, 2, 3, \cdots, 1997\} \). For any 999-element subset \( X \) of \( A \), if there exist \( x, y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a good subset. Find the largest natural number \( a \) (where \( a \in A \)) such that any 999-element subset containing \( a \) is a good subset.
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Context:
Determine the fair price of the option to regenerate your number after both numbers have been revealed in a game where each participant contributes 20 USD, and the participant with the higher number wins the entire 40 USD.
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Context:
A deck of cards has 54 cards with assigned points: Jokers = 0, Ace = 1, Jack = 11, Queen = 12, King = 13, and other cards receive their face values as points. What is the minimum number of cards you need to draw to ensure that among the drawn cards, there are always 2 cards whose point total equals 14? Express your answer as a single integer.
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Context:
On February 1, it was $16.2^{\circ} \mathrm{C}$ outside Jacinta's house at 3:00 p.m. On February 2, it was $-3.6^{\circ} \mathrm{C}$ outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, what was the rate at which the temperature decreased?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{1.8^{\circ} \mathrm{C} / \mathrm{h}}


Context:
Natural numbers, multiples of 3, are painted in two colors: red and blue, such that the sum of a blue and a red number is red, and the product of a blue and a red number is blue. In how many ways can the numbers be painted so that the number 546 is blue?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(PQR\), the angle \(QRP\) is \(60^\circ\). Find the distance between the points of tangency on side \(QR\) of the inscribed circle with radius 2 and the circle with radius 3 that is tangent to the extensions of sides \(PQ\) and \(PR\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
记 \(F(x, y)=(x-y)^{2}+\left(\frac{x}{3}+\frac{3}{y}\right)^{2}(y \neq 0)\)。求 \(F(x, y)\) 的最小值。答案的形式为 \(\frac{k}{m}\)，请给出 \(k + m\) 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a real number $\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and
\[ \alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \text{ for } n \ge 1. \]
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume $n$ is a positive integer. Considers sequences $a_{0}, a_{1}, \ldots, a_{n}$ for which $a_{i} \in \{1,2, \ldots, n\}$ for all $i$ and $a_{n}=a_{0}$.
Suppose $n$ is odd. Find the number of such sequences if $a_{i}-a_{i-1} \not\equiv i \pmod{n}$ for all $i=1,2, \ldots, n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of continuous functions \( f : [0,1] \rightarrow \mathbb{R}^2 \) such that \( f(0) = a \) and \( f(1) = b \), where \( a \) and \( b \) are distinct points in \( \mathbb{R}^2 \). Express your answer in terms of beth numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{dx}{3+2\cos x+\sin x}\) using the Weierstrass substitution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the moment of inertia about the $z$-axis of the domain $$E=\{(p,\theta,\phi)\,|\,0\le p\le2,0\le\theta\le2\pi,\frac{\pi}{3}\le\phi\le\frac{\pi}{2}\}$$ which is the domain between the cone $z=\sqrt{(x^2+y^2)/3}$ and under the half sphere $z=\sqrt{4-x^2-y^2}$. The density is constant $\rho(x,y,z)=k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the joint moment generating function of the time-to-failure random variables of two components, \( M_{X,Y}(s,t)=\frac{1}{(1-3s-2t+6ts)} \), calculate the probability that the machine consisting of these two components will fail after the third hour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The light glows for every 16 seconds. Between 1:57:58 am and another time, it glowed 310.5625 times. What was the ending time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a_{k} \) be the coefficient of \( x^{k} \) in the expansion of
\[ (x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\cdots+(x+1)^{99}. \]

Determine the value of \( \left\lfloor \frac{a_{4}}{a_{3}} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider fractions of the form \(\frac{n}{n+1}\), where \(n\) is a positive integer. If 1 is subtracted from both the numerator and the denominator, and the resultant fraction remains positive and is strictly less than \(\frac{\alpha}{7}\), determine \(\beta\), the number of these fractions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$n$ players participate in a chess tournament where each player plays a game with every other player. The winner of a game receives 1 point, the loser receives 0 points, and a draw awards each player 0.5 points. A tournament is said to have property $P(m)$ if, among any $m$ players, there is one player who has defeated the other $m-1$ players, and there is one player who has lost to the other $m-1$ players. For a given $m$ ($m \geq 4$), find the minimum value of $n$, denoted by $f(m)$, such that in any tournament with property $P(m)$, all $n$ players have different scores.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many multiples of 4 are there between 200 and 500?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set \(\bigcap_{x \in A} \bigcup_{n \in \Bbb N} \left(\frac{1}{n}, x-\frac{1}{n^2}\right)\) where \(A=\{ x \in \Bbb R \mid x > \pi\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设整数 $n>4,(x+2 \sqrt{y}-1)^{n}$ 的展开式中 $x^{n-4}$ 与 $x y$ 两项的系数相等, 则 $n$的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of $x$ that minimizes the expression $$x+[c+p(x)wy]\left[1-\frac{p(x)y}{y^{\max}} \right]$$ given that $w=0$. Assume all other variables are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose \( f(z) = \prod_{n=1}^{17}(z-\frac{\pi}{n}) \) for \( z \in \mathbb{C} \), and let \( \gamma(t) = e^{2it} \) for \( t \in [0, 2\pi] \). If \( \int_{\gamma} \frac{f'(z)}{f(z)}\, dz = \alpha \pi i \), find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The pentagon PQRST is divided into four triangles with equal perimeters. The triangle PQR is equilateral. PTU, SUT, and RSU are congruent isosceles triangles. Find the ratio of the perimeter of the pentagon PQRST to the perimeter of the triangle PQR. Express your answer as a ratio in simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer solutions to the Diophantine equation \(4x^3 - 3 = y^2\) for \(x > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider digits $A, B, C, D$, with $A \neq 0$, such that $ABCD = (CD)^2 - (AB)^2$. Compute the sum of all distinct possible values of $A + B + C + D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $A=\{1, 3, 2m-1\}$ and $B=\{3, m^2\}$; if $B \subseteq A$, find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matrix $M \in M_{5}(\mathbb{R})$ with the characteristic polynomial $(x-2)^5$, determine the largest possible dimension of the centralizer of $M$ in $M_{5}(\mathbb{R})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( I = \int_\gamma \left(4e^{(y-4x+3)^2}-y\right)dx + \left(-e^{(y-4x+3)^2}+2x\right)dy \), where \( \gamma \) is the curve along the parabola \( y = x^2 \) from \( (1,1) \) to \( (3,9) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the integral

$$ \mathcal {I} [y] = \int_{- \infty}^{\infty} \dfrac{1}{2} y'^2 + 1 - \cos y \; \mathrm d x $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2 \pi$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$, find the area expressed as $kd^2$ for some constant $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jake is retrieving the notes on a communal board and sorting them into different bins based on their colors and categories. The red notes are complaints, the blue notes are compliments, the green notes are suggestions, and the yellow notes are inquiries. The board has the following distribution of notes:

- Red notes are in 5 rows with 6 notes in each row, and there are 3 more red notes scattered around the board.
- Blue notes are in 4 rows with 7 notes in each row, and another 12 blue notes are placed in various corners of the board.
- Green notes are in 3 triangular arrangements, with each triangle having 1 green note at the top and additional green notes forming the triangle's base: the first triangle has a base of 4 notes, the second triangle has a base of 5 notes, and the third triangle has a base of 6 notes.
- Yellow notes are in 2 diagonal arrangements starting from the top corners of the board: the first diagonal has 5 yellow notes, and the second diagonal has 3 yellow notes. There are also 6 yellow notes in the center of the board, forming a hexagon.

Jake needs to place the notes into the correct bins. How many notes in total are put into the complaints, compliments, suggestions, and inquiries bins?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in this row?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work done by the force \( \vec{F} = \sin(x^2)\hat{x} + (3x-y)\hat{y} \) in moving a particle along the path from the origin \( O \) to point \( A = (3,0) \) and then to point \( B = (0,4) \) in straight lines.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ribamko A.V.

In the lower left corner of a $n \times n$ chessboard, there is a knight. It is known that the minimum number of moves it takes for the knight to reach the upper right corner is equal to the minimum number of moves it takes to reach the lower right corner. Find $n$.

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


Context:
How many distinct ordered triples of real numbers $(a, b, c)$ exist such that $a, b, c$ are the roots of the cubic equation $x^3 + ax^2 + bx + c = 0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest \( n \) so that the number of integers less than or equal to \( n \) and divisible by 3 equals the number divisible by 5 or 7 (or both).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 19-digit binary string contains all possible 4-digit binary strings exactly once. If this string starts with 1111, what are the last four digits?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of an integer $B$ such that for every 9 distinct natural numbers with the sum of $2023$, there must exist a sum of 4 of these numbers that is greater than or equal to $B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to color each rational number either white or red such that for any two distinct rational numbers \( x \) and \( y \), if they satisfy any of the following conditions: (a) \( xy = 1 \), (b) \( x + y = 0 \), or (c) \( x + y = 1 \), then \( x \) and \( y \) are painted different colors. How many such colorings, called "sanferminera" colorings, exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let point O be located inside a regular tetrahedron ABCD. G is the centroid of the tetrahedron. The line OG intersects the faces of the tetrahedron at points A1, B1, C1, D1. Find the value of (A1O/A1G) + (B1O/B1G) + (C1O/C1G) + (D1O/D1G). Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). If
\[
a=\frac{\sqrt{6+\sqrt{6+\cdots+\sqrt{6}}}}{2016 \text{ nested radicals}}, \\
b=\frac{\sqrt[3]{6+\sqrt[3]{6+\cdots+\sqrt[3]{6}}}}{2017 \text{ nested radicals}},
\]
then find \([a+b]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the congruence class of \( \sigma(m^2) \) modulo 4 that would make the equation \( \frac{2D(m^2)s(m^2)}{D(p^k)s(p^k)} = (\gcd(m^2, \sigma(m^2)))^2 \) invalid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. The original answer for \(\tfrac{AB}{BC}\) is in the form \(k + \sqrt{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $\frac{b_{36}}{b_{25}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school there are 1200 students. Each student must join exactly  $k$  clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seven sticks with lengths 2, 3, 5, 7, 11, 13 and 17 inches are placed in a box. Three of the sticks are randomly selected. What is the probability that a triangle can be formed by joining the endpoints of the sticks? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若整数 a,b,c 满足：0\leq a\leq 10, 0\leq b\leq 10, 0\leq c\leq 10，10\leq a+b+c\leq 20，则满足条件的有序数组 (a,b,c) 共有__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola $y^2=2px$ ($p>0$) and a point $M(1, m)$ on it, the distance from $M$ to its focus is 5. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the graph of a power function passes through the points $(2, 16)$ and $\left( \frac{1}{2}, m \right)$, then $m = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the ring $R[[X]]$, where $R$ is a Noetherian ring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given four points \( A, B, C, \) and \( D \) in space that are not coplanar, each pair of points is connected by an edge with a probability of \( \frac{1}{2} \). The events of whether any two pairs of points are connected are mutually independent. What is the probability that \( A \) and \( B \) can be connected by a spatial path (composed of one edge or a series of edges)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle and let $P$ be a point in its interior. Suppose $ \angle B A P = 10 ^ { \circ } , \angle A B P = 20 ^ { \circ } , \angle P C A = 30 ^ { \circ } $ and $ \angle P A C = 40 ^ { \circ } $. Find $ \angle P B C $.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the elliptic curve \(E\) over \(\mathbb{F}_{11}\) defined by \(y^2 = x^3 + 5\) and the function \(f(x,y)=\frac{2x+y-3}{3x-y+2}\), calculate \(f(\mathcal{O})\) using projective coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At an athletic facility with 5 tennis courts, doubles teams use a court for a duration that follows an exponential distribution with a mean of 40 minutes. If a team arrives and finds all courts occupied with k other teams waiting, what is the expected waiting time for them to get a court?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( {a}_{1},{a}_{2},\ldots ,{a}_{1000} \) be positive integers whose sum is \( S \) . If \( {a}_{n} \) ! divides \( n \)
 for each \( n = 1,2,\ldots ,{1000} \) , compute the maximum possible value of \( S \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ be a 28-digit palindromic number that is a multiple of 13. All the digits of $a$ are 1, except for the 13th, 14th, 15th, and 16th digits, which are $A$, $B$, $C$, and $D$ respectively. Determine the minimum possible value of $A + B + C + D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. Find the sum of the $10$ least values of $n$ such that $S_n$ is divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the product of the number of balls in each urn when K balls are uniformly distributed into S urns. For example, if K = 8 and S = 3, one possible distribution is (4, 2, 2), resulting in a product of 16. Determine the expected value for K = 8 and S = 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
9-8. All digits in the notation of 6-digit natural numbers $a$ and $b$ are even, and in the notation of any number between them, there is an odd digit. Find the largest possible value of the difference $b-a$.

---

Translation:
9-8. All digits in the notation of 6-digit natural numbers $a$ and $b$ are even, and in the notation of any number between them, there is an odd digit. Find the largest possible value of the difference $b-a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_C{\omega} \) where \( \omega=\frac{dx \wedge dy}{x^2+y^2} \) and \( C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - \text{{(0,0)}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the distance from a point $(x_{0}, y_{0})$ on the parabola $x^{2}=28y$ to the focus is $3$ times the distance from the point to the $x$-axis, then $y_{0}=\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define $g_n(x) = 3x^ne^{-x^2}-1$. For $n>1$, let $w_n$ be a real number such that $g_n(w_n)=0$. Determine the value of $\lim\limits_{n\to\infty}\frac{w_n^2}{n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ring homomorphisms from the ring of real numbers \((\mathbb{R}, +, \times)\) to itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9.

For example, if \( n = 2024 \) and we choose \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 3 \), then \( m = 4177 \) is an alero of \( n \), but if we choose the digits \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 6 \), then we don't obtain an alero of \( n \), because \( 4 + 6 \) exceeds 9.

Find the smallest \( n \) which is a multiple of \( 2024 \) that has an alero which is also a multiple of \( 2024 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the following expression in the finite field $\mathbb{F}_{2^4}$ using the irreducible polynomial $f(x) = x^4 + x + 1$:

$$\left( \frac{\alpha^{10} + \alpha^{15}}{\alpha^8+\alpha^5} \right)^2 + \frac{\alpha^{10} + \alpha^{15}}{\alpha^8+\alpha^5} + \alpha^8 + \alpha^5 + \alpha^6$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The high-speed train "Sapsan," approaching a railway station at a speed of \( v = 216 \) km/h, emits a warning sound signal lasting \( \Delta t = 5 \) seconds when it is half a kilometer away from the station. What will be the duration of the signal \( \Delta t_{1} \) from the perspective of passengers standing on the platform? The speed of sound in the air is \( c = 340 \) m/s.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$. A transversal is a set $\{\omega_1,\ldots,\omega_k\}$ with $\omega_j\in\Omega_j$ for each $j$. For a transversal $\tau=\{\omega_1,\ldots,\omega_k\}$ and subset $H\subseteq G$, define
\[H(\tau) = \bigcup_{h\in H} \{\omega_1^h,\ldots,\omega_k^h\}. \]
Let $N(G)$ be the smallest integer such that for some transversal $\tau$ and some $H\subseteq G$ with $|H|=N$, we have $H(\tau)=\Omega$. Given $n=|\Omega|$ and $m=\max_{i=1}^k |\Omega_i|$, determine $N(G)$. It is known that $N(G)\ge m$ and $N(G)\le m\log n$. What is $N(G)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We see a rod of length $a$ on top of a tower of height $b$. At what distance from the tower do we see this rod at the maximum angle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $a, b \in \mathbf{N}^{*}, a+b \leq 2024$, 使得 $a b^{2}+b+7$ 整除 $a^{2} b+a+b$ 的解的 $(a, b)$ 有 $\qquad$组.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute
\[\sum_{i=1}^{1000} g(i).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $k$ and $n \ge 3$ be two natural numbers. How many strings in $\{1,...,n\}^k$ contain at least one occurrence of $1$ and $2$, or at least one occurrence of $2$ and $3$ or at least one occurrence of $1$ and $3$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the positive integers \( m \) and \( n \) satisfy \( \frac{(m+n)!}{n!}=5040 \), then the value of \( m!n \) is ________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$.
Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process.
Find $a,\ b,\ c,\ d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the intrinsic dimension of the set \( M = \{ (x,y,z) \in \mathbb{R}^3 : (a,b,c,d) \in \mathbb{R}^4\} \) given the parameterized functions:
\[ x(a,b,c,d) = ac \]
\[ y(a,b,c,d) = bc + ad \]
\[ z(a,b,c,d) = bd \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
K2) Let $n$ be a natural number. A volleyball team consisting of $n$ women and $n$ men lines up for a game. Each team member occupies one of the positions $1,2, \ldots, 2 n$, whereby exactly positions 1 and $n+1$ are outside the court. During the game, all team members rotate, switching from position $i$ to position $i+1$ (or from $2 n$ to 1). How many possibilities are there for the starting line-up so that there are always at least $n-1$ women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 250 m long train is running at a speed of 55 km/hr. It crossed a platform of a certain length in 35.99712023038157 seconds. What is the length of the platform?
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Context:
Let  $a,b, c$  denote the real numbers such that  $1 \le  a, b, c\le  2$ .
Consider  $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$ .
Determine the largest possible value of  $T$ .
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Context:
Joe has written 5 questions of different difficulties for a test, numbered 1 through 5. He wants to make sure that problem \(i\) is harder than problem \(j\) whenever \(i - j \geq 3\). In how many ways can he order the problems for his test?
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Context:
The parameter of a square is equal to the perimeter of a rectangle of some length and breadth 10 cm. The circumference of a semicircle whose diameter is equal to the side of the square is 21.99 cm (rounded to two decimal places). What is the length of the rectangle?
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Context:
Given the line $l$: $y=2x$, a point $P$ on it forms a circle $C$: $(x-8)^{2}+(y-1)^{2}=2$. The tangents $l_{1}$, $l_{2}$ from point $P$ to the circle are symmetric about line $l$. Find the distance from point $P$ to the center of circle $C$.
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Context:
I5.2 If $1-\frac{4}{x}+\frac{4}{x^{2}}=0, b=\frac{a}{x}$, find $b$.
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Context:
Determine the number of different Euclidean divisions possible for a given pair of elements in the Gaussian integers, $\mathbb{Z}[i]$. Provide your answer as a single integer.
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Context:
David has a store where he sells cell phones. When he takes inventory at the end of the day, he notices that he has 10 Samsung cell phones but he started the day with 14 Samsung cell phones. Then he notices that he has 5 iPhones left, but started with 8. His staff then told him that in the afternoon they realized that 2 Samsung cell phones were damaged and 1 iPhone had a manufacturing defect in the screen, so they threw these out during the day.

During the day, there are two types of discounts offered on each phone: a 10% discount on Samsung cell phones and a 15% discount on iPhones. The tax rates applied on the final price were different for Samsung and iPhones at 12% and 10%, respectively.

Calculate the total revenue generated from the sales, considering discounts and taxes, if the retail price of a Samsung cell phone is $800, and an iPhone is $1,000.
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Context:
Evaluate the integral \( \oint_C \frac{dz}{z-2} \) where \( C \) is the square with vertices at \( \pm 2 \pm 2i \).
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Context:
How many four-digit numbers are there in which no two adjacent digits are the same?
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Context:
Find all prime numbers $p$ such that the number of distinct positive factors of $p^2 + 2543$ is less than 16.
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Context:
Determine the genus of the complete 4-partite graph $K_{4,2,2,2}$. Provide your answer as a single integer.
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Context:
Calculate the order of the image of the homomorphism $\varphi: \mathbb{Z}^2 \to S_{15}$ defined by $\varphi(1,0)=(2,7,3)(3,11,5)(12,13)$ and $\varphi(0,1)=(14,15)(1,4,6,8,9,10)$.
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Context:
An ant is 40 ft away from its nest, which is a large wall. Due to rain, its scent trail is erased, and it can travel a maximum of 280 ft more before starving. Determine the shortest path the ant can take to return to the nest, considering the nest as a point on a wall and the ant's starting point as a point on a circle with a radius of 40 ft.
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Context:
If an integer is divisible by $6$ and the sum of its last two digits is $15$, then what is the product of its last two digits?
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Context:
A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection.
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Context:
Determine the cardinality of the set $S$, which consists of all infinite increasing sequences of binary strings. An infinite increasing sequence of binary strings is a series of finite binary strings $\{w_1, w_2, \dots\}$ where each string $w_i$ is a prefix of the next, i.e., $w_i$ is a substring at the beginning of $w_{i+1}$. For example, the sequence "101", "$10100$", "$101001$", "$1010010111$", \dots belongs to $S$. What is the cardinality of $S$?
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Context:
Given natural numbers \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \) such that \( a_{1} + a_{2} + a_{3} = b_{1} + b_{2} + b_{3} \), \( a_{1}a_{2}a_{3} = b_{1}b_{2}b_{3} \), and \( a_{1}a_{2} + a_{1}a_{3} + a_{2}a_{3} \neq b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3} \). Find the minimal value of the largest number among \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \).
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Context:
Let \(\mathcal{F}\) be the family of all nonempty finite subsets of \(\mathbb{N} \cup \{0\}\). Find all positive real numbers \(a\) for which the series

\[
\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A} a^{k}}
\]

is convergent.
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Context:
Determine the irreducible representations of the tensor product $\varphi_2 \otimes \varphi_3$ for the $SU(2)$ group, where $\varphi_n$ denotes the standard irreducible representation with highest weight $n$.
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\boxed{\varphi_1 \oplus \varphi_3 \oplus \varphi_5}


Context:
An elementary school teacher is making Halloween goodie bags for her class. She wants the bags to be personalized, so she surveys her students asking whether they'd like a vampire-themed bag or a pumpkin-themed bag. Of her 25 students, 11 indicate they want the vampire-themed bag and 14 indicate they want the pumpkin-themed bag. The store the teacher shops at sells packs of 5 of each theme at a price of $3 per package, as well as individual bags of each theme at a price of $1 each. Additionally, the store offers a "Buy 3 packages, Get 1 package Free" discount to encourage bulk purchase. What is the minimum amount of money the teacher can spend on the bags if she buys every student the theme they requested while taking advantage of the discount offer?
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Context:
Evaluate the integral \( \iint_D (3x^3 + 3x^2y + x + y) \, dA \) over the region \( D \) bounded by the curves \( y = x^3 - 4 \), \( y = x^3 + 4 \), \( x + y = 6 \), and \( x + y = 0 \), using a suitable change of variables.
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Context:
A father wants to distribute his estate to his children: first giving 1000 yuan and $\frac{1}{10}$ of the remaining estate to the eldest child, then 2000 yuan and $\frac{1}{10}$ of the remaining estate to the second child, next 3000 yuan and $\frac{1}{10}$ of the remaining estate to the third child, and so on. It turns out that each child received the same amount of property. How many children does the father have?
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Context:
Find the number of matrices in the group $SL_3(\mathbb{Z}_2)$ by using the method of counting bases.
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Context:
A wall contains three switches $A$, $B$, and $C$, each of which powers a light when flipped on. Every $20$ seconds, switch $A$ is turned on and then immediately turned off again. The same occurs for switch $B$ every $21$ seconds and switch $C$ every $22$ seconds. At time $t = 0$, all three switches are simultaneously on. Let $t = T > 0$ be the earliest time that all three switches are once again simultaneously on. Compute the number of times $t > 0$ before $T$ when at least two switches are simultaneously on.
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Context:
In a contest with \( n \) questions taken by \( m \) contestants, each question is worth a certain positive number of points, and no partial credits are given. After grading, it was observed that by reassigning the scores of the questions, any desired ranking of the contestants could be achieved. What is the largest possible value of \( m \)?
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Context:
Evaluate the integral \( \int_{0}^{2\pi} e^{\sin(t)} \sin(\cos(t) + 5t) \, dt \) using complex analysis.
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Context:
设正整数$a, b, c, d$满足$a>b>c>d$，且
$$ab+bc+ca+d^{2}\mid(a+b)(b+c)(c+a)$$
求$ab+bc+ca+d^{2}$的素因子个数的最小值。（素因子的个数计重数，如12的素因子个数为3）。
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Context:
Find the greatest common factor (GCF) of the polynomials \( P(x) = x^{50} - 2x^2 - 1 \) and \( Q(x) = x^{48} - 3x^2 - 4 \).
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Context:
Calculate the limit $$\lim_{n\rightarrow\infty} \sum_{k=1}^{n}\cos\left( \frac{2\pi k}{2n+1} \right).$$
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Context:
Given four points \( A, B, C, D \) on the surface of a sphere with a radius of 2, and that \( AB = CD = 2 \), determine the maximum volume of the tetrahedron \( ABCD \).
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Context:
Given ${(2x+\frac{a}{x})}^{5}$, the coefficient of $x$ in its expansion is $80$. Find the value of $a$. Express your answer as a numerical value, including both positive and negative possibilities.
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Context:
Let  $S$  be the set of all lattice points  $(x, y)$  in the plane satisfying  $|x|+|y|\le 10$ . Let  $P_1,P_2,\ldots,P_{2013}$  be a sequence of 2013 (not necessarily distinct) points such that for every point  $Q$  in  $S$ , there exists at least one index  $i$  such that  $1\le i\le 2013$  and  $P_i = Q$ . Suppose that the minimum possible value of  $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$  can be expressed in the form  $a+b\sqrt{c}$ , where  $a,b,c$  are positive integers and  $c$  is not divisible by the square of any prime. Find  $a+b+c$ . (A *lattice point* is a point with all integer coordinates.)
<details><summary>Clarifications</summary>

- $k = 2013$ , i.e. the problem should read, ``... there exists at least one index  $i$  such that  $1\le i\le 2013$  ...''.  An earlier version of the test read  $1 \le i \le k$ .
</details>

*Anderson Wang*
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Context:
Find the number of ordered triples of complex numbers $(a, b, c)$ for which $a^3 - b$, $b^3 - c$, and $c^3 - a$ are all rational numbers, and the following equation holds:

\[ a^2(a^4 + 1) + b^2(b^4 + 1) + c^2(c^4 + 1) = 2\left[(a^3)b + (b^3)c + (c^3)a\right] \]
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Context:
Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3.
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Context:
Evaluate the surface integral \( \iint_{S} \vec{F} \cdot d\vec{S} \) to find the flux of the vector field \( \vec{F}(x,y,z) = y\mathbf{j} - z\mathbf{k} \) across the surface \( S \). The surface \( S \) consists of the paraboloid \( y = x^2 + z^2, 0 \leq y \leq 1, \) and the disk \( x^2 + z^2 \leq 1, y=1 \). Use the positive (outward) orientation for closed surfaces.
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Context:
Determine the rank of the symmetric bilinear form \( f(A,B) = \operatorname{tr}(A) \operatorname{tr}(B) - n\operatorname{tr}(AB) \) for square matrices \( A \) and \( B \) of size \( n \).
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Context:
Find the index of the module $\mathbb{Z}[\sqrt{7}]$ as a submodule of $\mathbb{Z}\left[\frac{1+\sqrt{7}}{2}\right]$. Provide your answer as a single integer.
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Context:
Calculate the value of the integral ratio:

$$\frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx}$$
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Context:
Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \[(ax+by)^3 + (cx+dy)^3 = x^3 + y^3\] holds for all complex numbers $x, y$.
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Context:
If the zero of the function $f(x) = -|x - 5| + 2x - 1$ lies in the interval $(k, k + 1)$, then the integer $k = \_\_\_\_\_\_\_\_$.
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Context:
Evaluate the infinite series \( \sum_{n=1}^{\infty}{\frac{\prod_{r=0}^{n-1}{(\pi-r})}{2^nn!}} \) and express it in the form \( \left(\frac{A}{B}\right)^{\pi}-C \), where \(A\), \(B\), and \(C\) are coprime integers. Find the value of \(A+B-C\).
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Context:
Two people start on opposite vertices of an octahedron. Each turn, they move to a random adjacent vertex. What is the probability that the person who moves first will land on the vertex occupied by the other person and win the game?
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Context:
Find the gradient of the quadratic form $z^TX^TXz$ with respect to the vector $z$, where $X$ is a matrix in $\mathbb{R}^{m \times n}$ and $z$ is a vector in $\mathbb{R}^{n}$.
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Context:
In a Cartesian coordinate system, you can move from point \((a,b)\) to \((a+1,b)\), \((a,b+1)\), or \((a+1,b+1)\). However, you must ensure that no right angles are formed when drawing lines during the move. Determine the number of ways to move from \((0,0)\) to \((5,5)\) under these conditions.
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Context:
令 $f(x)=[x]+[2 x]+[3 x]+[4 x]+[5 x]$, 当 $x$ 取遍 $[0,50]$ 中的实数时, $f(x)$ 可取到 $\qquad$个不同的整数值.
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Context:
在长方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中, $A B=$ $A A_{1}=2, A D=2 \sqrt{3}, M$ 为平面 $B A_{1} C_{1}$ 内一点.则 $\overrightarrow{M A} \cdot \overrightarrow{M C}$ 的最小值为。原始答案为 \frac{m}{n} 的格式，请给出 m + n 的值。
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Context:
Given a random sample $Y_1, Y_2, \ldots, Y_{10}$ from a normal distribution $Y \sim N(\mu, \sigma^2)$, consider the statistic:
\[ U^2 = \frac{1}{9} \sum_{i=1}^{10} (Y_i - \bar{Y})^2 \]
where $U^2$ is the empirical variance. Determine the distribution of the statistic \( \frac{9U^2}{\sigma^2} \).
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Context:
Let $A_1A_2A_3\cdots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $1+ \dfrac {\tan A}{\tan B}+ \dfrac {2c}{b}=0$, then $A=$ ______.
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Context:
Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements.
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Context:
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
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Context:
42. Factorize the number $989 \cdot 1001 \cdot 1007 + 320$ into prime factors.
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Context:
In the number $2016^{* * * *} 02 * *$, each of the six asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?
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Context:
Let us define a staircase of height \( n \) as a figure consisting of all the cells of an \( n \times n \) square that lie not above the diagonal. How many different ways are there to partition a staircase of height \( n \) into several rectangles, with sides along the grid lines, and areas pairwise distinct?
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Context:
Let \(ABCD\) be a quadrilateral inscribed in the unit circle such that \(\angle BAD\) is 30 degrees. Let \(m\) denote the minimum value of \(CP+PQ+CQ\), where \(P\) and \(Q\) may be any points lying along rays \(AB\) and \(AD\), respectively. Determine the maximum value of \(m\).
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Context:
Calculate the residue of the function \( f(z) = \frac{\sin z}{z^3} \) at \( z = 0 \) using the formula for residues of poles of higher order.
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Context:
Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters).
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Context:
Find the negation of the proposition "∀x > 0, x + 1/x ≥ 2". Express your answer as a logical statement with the correct quantifier and inequality.
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Context:
Given the function y = (x - 16)(x - 14)(x + 14)(x + 16), find the minimum value of y. Express your answer as a single integer.
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Context:
Given that the solution set of the inequality $ax^2 + bx + 3 > 0$ regarding $x$ is $(-1, \frac{1}{2})$, where $a$ and $b$ are constants, find the solution set of the inequality $3x^2 + bx + a < 0$. Express your answer in interval notation.
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Context:
Anastasia's tablecloth has a regular pattern. The tablecloth can be divided into 25 equal squares, with half of each of the 16 outer squares colored black. Find the percentage of Anastasia's tablecloth that is black. Express your answer as a percentage.
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Context:
Given that \( T_1 \) is an \( \alpha \times 100\% \) lower confidence limit for \( \theta \) and \( T_2 \) is an \( \alpha \times 100\% \) upper confidence limit for \( \theta \), with the condition \( P(T_1 < T_2) = 1 \), find a \( (2\alpha - 1) \times 100\% \) confidence interval for \( \theta \).
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Context:
Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\triangle ABC$. If the length of segment $AX$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.
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Context:
What is the percentage of 12,356 that equals 12.356000000000002?
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Context:
Let  $n$  be a positive integer. There are  $n$  soldiers stationed on the  $n$ th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of  $n$ , required to eliminate all the soldiers?
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Context:
In a 500 m race, two contestants A and B have a certain speed ratio. A has a start of 300 m and wins by 100 m. What is the ratio of the speeds of A and B?
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Context:
Two identical cubes each have an edge length of $a$. The diagonal of the first cube coincides with the diagonal of the second cube. A vertex of the second cube is positioned at the center of the first cube, and the second cube is rotated by $60^{\circ}$ around the diagonal relative to the first cube. Calculate the volume of the intersection of these two cubes.
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Context:
Evaluate the limit:

$$\lim_{n\to \infty} \sum_{i=1}^n \frac{1}{\frac{n}{2-1}+i}$$

where $a=1$ and $b=2$. Express your answer in terms of natural logarithms.
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Context:
Evaluate the integral \( \int_0^{2\pi}\int_0^R\frac{rx}{\sqrt{R^2-x^2-y^2}}\,dx\,dy \) by converting to polar coordinates.
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Context:
Evaluate the rational map \( \hat{\phi} : \mathbb{P}^2 \setminus \{(1:0:0),(0:1:0),(0:0:1)\} \to \mathbb{P}^2 \) given by \( \phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2}) \) at the point \((1:1:0)\).
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Context:
In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
A class went on a camping trip and walked along the road from the school to the base, then immediately returned to the school by the same route. If the distance traveled each day increased by 1 kilometer, and it took 4 days to go and 3 days to return, calculate the distance from the school to the base. Express your answer in kilometers as a whole number.
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Context:
Compute the radius of convergence of the power series \( \sum_{n \ge 0} f_n(z) \), where each function \( f_n: \mathbb{C} \rightarrow \mathbb{C} \) is defined by:
\[ f_n(z) = \frac{1}{p_n}[z(1-z)]^{4^n} \]
and \( p_n \) is the largest absolute value of a coefficient in the expansion of \([z(1-z)]^{4^n}\).
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Context:
There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.
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Context:
Given that $M = Gab^2c^3d^4$ where $a, b, c, d$ are distinct prime numbers, calculate the probability that a pair of numbers randomly chosen from the set of numbers of the form $Ga^ib^jc^kd^l$ (where $0 \leq i \leq 1$, $0 \leq j \leq 2$, $0 \leq k \leq 3$, $0 \leq l \leq 4$) will have a greatest common divisor of $G$ and a least common multiple of $M$. Express your answer as a fraction.
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Context:
Extract the coefficient of $u^{n-1}$ from the generating function $A = 2^{n-1}[u^{n-1}](\frac{1}{1-u})^n$ to determine the average number of nodes with an even number of children in a plane planted tree with $n$ nodes.
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2} \frac{(4 \sqrt{2-x}-\sqrt{3 x+2}) d x}{(\sqrt{3 x+2}+4 \sqrt{2-x})(3 x+2)^{2}}
$$
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Context:
There are 4 boxes, and each pair of boxes is weighed once, resulting in weights of 22, 23, 27, 29, and 30 kg for the 5 measurements, respectively. Calculate the difference in weight, in kilograms, between the heaviest box and the lightest one.
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Context:
In how many rearrangements of the numbers1,2,3,4,5,6,7,8,9do the numbers form a \( {hill} \) , that is, the numbers form
 an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in 129876543
 or 258976431?
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Context:
Five friends - Kristina, Nadya, Marina, Liza, and Galya - gather in the park every day after buying ice cream from the shop around the corner. One day, they had a conversation.

Kristina: There were five people in front of me.

Marina: I was the first in line!

Liza: There was no one after me.

Nadya: I was standing next to Marina.

Galya: There was only one person after me.

The girls are friends, so they do not lie to each other. How many people were between Kristina and Nadya?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the hyperbolic plane $\mathbb{H}$, consider two horocycles at points $u$ and $v$ with radii $h$ and $k$, respectively. Find the distance between these horocycles along the geodesic connecting $u$ and $v$ using a Möbius transformation that maps the geodesic to the real axis.
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Context:
What is the tens digit of the five-digit integer N, which is as close as possible to 30000, using the digits 2, 3, 5, 7, and 8 exactly once?
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Context:
A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find
 the number of5-digit happy integers.
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Context:
6. We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ fields. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board such that the cut goes along the edge $k$. Calculate the last digit of the sum we get when we add all $N(k)$, where $k$ is an inner edge.
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Context:
Hannah ran 9 kilometers on Monday at a pace of 6 minutes per kilometer. On Wednesday, she ran 4816 meters at a pace of 5.5 minutes per kilometer and on Friday, she ran 2095 meters at a pace of 7 minutes per kilometer. Calculate how much longer, in minutes, she ran on Monday compared to the combined time for Wednesday and Friday.
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Context:
How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set?
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Context:
In an isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\) (\(AD > BC\)), the lateral side is 40 cm, and the angle \(\angle BAC\) is \(45^\circ\). Let \(O\) be the center of the circle circumscribed around \(ABCD\). It turns out that point \(O\) lies on the line connecting point \(D\) to the midpoint of side \(AB\). Find the length of the base \(AD\) of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{20(\sqrt{6} + \sqrt{2}) \text{ cm}}


Context:
What is the maximum number of subsets of  $S = {1, 2, . . . , 2n}$  such that no one is contained in another and no two cover whole  $S$ ?

*Proposed by Fedor Petrov*
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Context:
George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\frac{1}{6}$ each, and the probability of rolling an 8 is $\frac{1}{5}$. What is the expected value of the number shown when this die is rolled? Express your answer as a decimal.
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Context:
The figure below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles. Find the degree measure of one of the larger angles.

Note: The figure is a nonagon (9-sided polygon).
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Context:
How many of the first $1000$ positive integers can be expressed in the form
\[\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
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Context:
Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion. If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce 11 times before returning to $(2,0)$ for the first time. If $\cot \theta=a-\sqrt{b}$ for positive integers $a$ and $b$, compute $100 a+b$.
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Context:
A train 250 m in length crosses a telegraph post in 4 seconds. What is the speed of the train?
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Context:
Kayla rolls four fair $6$-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Problem condition

Find the derivative.
$y=6 \arcsin \frac{\sqrt{x}}{2}-\frac{6+x}{2} \cdot \sqrt{x(4-x)}$
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Context:
A cube with side length \( n \) (\( n \geq 3 \)) is divided into unit cubes by partitions. What is the minimum number of partitions between unit cubes that need to be removed so that it is possible to reach the boundary of the cube from any unit cube?
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Context:
正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 的 9 条棱长都相等, $P$ 是 $C C_{1}$ 的中点, 二面角 $B-A_{1} P-B_{1}=eta$ ，则 $\beta$ 的原始答案为 \frac{\sqrt{m}}{n} 格式，请给出 m + n 的值。
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Context:
Find the maximum value of the expression $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$, given that $a, b, c$ are positive real numbers with the constraint $3a + 4b + 5c = 12$.
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Context:
Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \[(2^{a+b}+8)(3^a+3^b) \leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\] holds. Compute the smallest possible value of $128v^2+w^2$.
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Context:
Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\left(m-1\right)y+m-6=0$. Find the value of $m$.
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Context:
设m是一个给定的正整数，则使得$m ^ { 2 } + n$能整除$n ^ { 2 } + m$的最大正整数n是·
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Context:
5. There is an unlimited number of chips in six colors. What is the smallest number of chips that need to be arranged in a row so that for any two different colors, there are two adjacent chips of these colors in the row.
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Context:
In a $9 \times 2004$ grid, each integer from 1 to 2004 is filled exactly 9 times. Additionally, within each column, no integer appears more than 3 times. Determine the minimum possible sum of the numbers in the first row of the grid.
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Context:
Determine the smallest positive constant \( c \) such that for all sufficiently large \( N \), the inequality \( r(N) \geq cN^{1/2} \) holds, where \( r(N) \) is the least number of quadratic polynomials with non-negative coefficients needed to cover the interval \([1, N]\).
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Context:
在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面, 其球心距为 13, 若作一平面与这二球面相切, 且与圆柱面交成一个椭圆, 则这个椭圆的长轴长与短轴长之和是 $\qquad$.
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Context:
Determine the radius of convergence for the infinite series $$ 1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty} a_n (x-3)^n, $$ where the series is not centered at $x=0$.
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Context:
What is the interval of validity for the Maclaurin series representation of $\ln(1-8x^5)$, which is given by $\sum_{n=0}^{\infty} \frac{(-1)^{n-1}}{n} (-8x^5)^n$?
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Context:
Given the universe $U=\{(x,y) | x,y \in \mathbb{R}\}$, let set $M=\{(x,y) | \frac{y-3}{x-2} \neq 1\}$ and set $N=\{(x,y) | y \neq x+1\}$. Calculate the complement of $M \cup N$ with respect to $U$, $\complement_U (M \cup N)$. Express your answer as a set of points, including coordinates.
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Context:
How many ways are there to arrange the numbers $\{ 1,2,3,4,5,6,7,8 \}$ in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.
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Context:
9. Given the ellipse $\frac{x^{2}}{1 \mathrm{~h}}+\frac{y^{2}}{4}=1$ with its left and right foci being $F_{1}$ and $F_{2}$, respectively, point $P$ lies on the line $l: x-\sqrt{3} y+8$ $+2 \sqrt{3}=0$. When $\angle F_{1} P F_{2}$ is maximized, the ratio $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given sets $A = \{|a + 1|, 3, 5\}$ and $B = \{2a + 1, a^2 + 2a, a^2 + 2a - 1\}$, when $A \cap B = \{2, 3\}$, find $A \cup B$.
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Context:
Find the radius of convergence of the power series $$ \sum_{n=1}^\infty \frac{x^{n^2}}{2^{n-1} n^n}. $$
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Context:
A particular coin has a $\frac{1}{3}$ chance of landing on heads (H), $\frac{1}{3}$ chance of landing on tails (T), and $\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT?
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Context:
At Rachelle's school, an A counts as 4 points, a B as 3 points, a C as 2 points, and a D as 1 point. Her GPA for the four classes she is taking is calculated as the total sum of points divided by 4. She is certain to get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a \frac{1}{6} chance of getting an A in English, and a \frac{1}{4} chance of getting a B. In History, she has a \frac{1}{4} chance of getting an A, and a \frac{1}{3} chance of getting a B, independently of her grades in English. What is the probability that Rachelle will achieve a GPA of at least 3.5? Provide your answer in the form of \frac{k}{m} and give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \(f(x)\) where any point on the curve can be written as \(\left(a, a(1-f'(a)\ln{a})\right)\) for \(a > 0\), find the expression for \(x^{f(x)}\).
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Context:
[ Median to the hypotenuse ] [ Pythagorean Theorem (direct and inverse) ]

Outside the right triangle $A B C$ on its legs $A C$ and $B C$, squares $A C D E$ and $B C F G$ are constructed.

The extension of the altitude $C H$ of triangle $A B C$ intersects the line $D F$ at point $K$. Find $H K$, if the legs are equal to 2 and 3.
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Context:
Tyler has two calculators, both of which initially display zero. The first calculator has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.

A positive integer $n$ is called \textit{ambivalent} if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
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Context:
Given a 1998-digit number where each pair of adjacent digits forms a two-digit number divisible by 17 or 23, and the last digit of the number is 1. What is the first digit?
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Context:
Smith operates a booth where customers arrive according to a Poisson process with a rate of $\lambda = 1$ per minute. He will close the shop the first time $a$ minutes have passed without a customer arriving. Let $X$ be the time until Smith closes his shop. Calculate the expected value $E(X)$. \( \text{Express your answer in terms of } a. \)
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Context:
A math teacher is 176 cm tall. The heights of his grandfather, father, and son are 173 cm, 170 cm, and 182 cm, respectively. Since the son's height is related to the father's height, the teacher uses the method of linear regression analysis to predict his grandson's height as ______ cm.
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Context:
Let \( f(z) \) be an analytic function in the open unit disc \( D \), continuous in the closed unit disc \( \bar{D} \), with \( |f(z)| \leq C \). Suppose \( f(z)/z^n \) can be extended to be analytic in the unit disc. Find a bound for \( |f(z_0)| \) given \( |z_0| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of colors needed to paint an annulus divided into 14 regions, such that no two adjacent regions share the same color? Provide your answer as a single integer.
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Context:
Given the system of equations for real numbers $a, b, c, d, e$:
\[ ac = 1, \]
\[ ad + bc = -1, \]
\[ ae + bd = 1, \]
\[ be = -6, \]
find the value of $a + b + c + d + e$. Provide your answer as a single number.
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Context:
The income and expenditure of a person are in the ratio 10 : 4. If the income of the person is Rs. 19000, and he pays taxes at a rate of 15% on his income, invests 10% of his remaining income on long-term investments, and spends 20% of the leftover amount on short-term investments, find his total savings.
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Context:
A plane is tiled with squares, each colored with one of 10 colors such that any two adjacent squares have different colors. Determine the minimum number of distinct color pairs that must appear across the edges of the squares.
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Context:
Let $f(x) = x^3 + 3x - 1$ have roots $a, b, c$. Given that \(\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\gcd(m, n) = 1\), find \(100m + n\).
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Context:
In $\triangle ABC$, let $BC = a$, $CA = b$, and $AB = c$. If $2a^2 + b^2 + c^2 = 4$, find the maximum area of $\triangle ABC$.
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Context:
In an isosceles right triangle \(\triangle ABC\), point \(E\) is on side \(BC\) such that \(BE = 3CE\). Points \(D\), \(A\), and \(F\) are collinear. In \(\triangle DBE\), let \(h_1\) be the altitude on side \(BE\), and in \(\triangle FEC\), let \(h_2\) be the altitude on side \(EC\). Given that \(3h_1 + h_2 = 3\) cm and the sum of the areas of \(\triangle DBE\) and \(\triangle FEC\) is 6 square centimeters, find the area of \(\triangle ABC\) in square centimeters.
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Context:
We have a five-digit positive integer  $N$ . We select every pair of digits of  $N$  (and keep them in order) to obtain the  $\tbinom52 = 10$  numbers  $33$ ,  $37$ ,  $37$ ,  $37$ ,  $38$ ,  $73$ ,  $77$ ,  $78$ ,  $83$ ,  $87$ .  Find  $N$ .
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Context:
In a room, there are some people, 2/5 of them are women. If 1/4 of the people are married, the maximum number of women in the room who could be unmarried is 40. How many people are in the room?
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Context:
There exist complex numbers $z_1, z_2, \dots, z_{10}$ which satisfy the condition: $$|z_k i^k + z_{k+1} i^{k+1}| = |z_{k+1} i^k + z_k i^{k+1}|$$ for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. Given that $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1| + |z_2| + \cdots + |z_{10}|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. How many positive integers $n \leq 1000$ satisfy the condition that $\left[\frac{998}{n}\right]+\left[\frac{999}{n}\right]+\left[\frac{1000}{n}\right]$ is not divisible by 3?
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Context:
Determine the number of pairs of positive constants \(a\) and \(b\) such that \(P(a < X < b) = 0.95\), where \(X\) follows a chi-squared distribution with \(n\) degrees of freedom.
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Context:
Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
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Context:
Determine the genus of the complete 4-partite graph $K_{4,2,2,2}$, where the vertex set is partitioned into 4 disjoint parts of sizes 4, 2, 2, and 2, and any two vertices are adjacent if and only if they belong to different parts.
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Context:
$A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What is the percentage of the flowers that are carnations? Provide your answer as a percentage.
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Context:
There are 10 seats in each of 10 rows of a theatre and all the seats are numbered. Two friends are buying tickets independently. What is the probability that they will occupy adjacent seats? Express your answer as a simplified fraction.
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Context:
On the sides $\mathrm{AB}$ and $\mathrm{AC}$ of a right triangle $\mathrm{ABC}\left(\angle B C A=90^{\circ}\right)$, right triangles $\mathrm{ABT}$ and $\mathrm{ACK}$ are constructed externally such that $\angle A T B=\angle A K C=90^{\circ}$ and $\angle A B T=\angle A C K=60^{\circ}$. A point $\mathrm{M}$ is chosen on the side $\mathrm{BC}$ such that $\mathrm{BM}=\mathrm{MC}$. Determine the measure of angle $\angle KMT$ in degrees.
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Context:
The number $125$ can be expressed as a sum of pairwise coprime integers, each greater than $1$. Determine the maximum number of terms possible in such a sum.
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Context:
Determine the order of the quotient group $G/G'$ for the group $G$ defined by the presentation:
\[ G = \langle a, b \mid a^{2^{n-1}} = 1, b^2 = a^{2^{n-2}}, b^{-1}ab = a^{-1} \rangle. \]
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Context:
Five people are arranged in a row, where person A is not next to person B, and person A is also not next to person C. Find the number of different arrangements. Express your answer as a whole number.
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Context:
If three different numbers are selected from 2, 3, 4, 5, 6 to be $a$, $b$, $c$ such that $N = abc + ab + bc + a - b - c$ reaches its maximum value, then this maximum value is.
Please reason step by step, and put your final answer within \boxed{}.