4. A positive integer n is given.Find the smallest $k$ such that we can fill a $3*k$ gird with non-negative integers such that: $\newline$ $i$) Sum of the numbers in each column is $n$. $ii$) Each of the numbers $0,1,\dots,n$ appears at least once in each row.

Carnegie Corporation is trying to promote a new department director from its employees. Carnegie Corporation has $n$ employees each with some unknown real-valued score and wants to pick the $M$-th highest scoring employee to be the new department director. The corporation has a magical machine that, once a day, can be used to compare two employees to see which one has a higher score. Unfortunately, this machine has a magical consequence: after every $k$ uses of this machine, if a new department director has not been chosen by the end of the day, one random employee is fired and a new employee (who has not necessarily the same score) is hired. Assume no two employees have equal scores and scores don't change over time. Machines with a higher constant of $k$ will be more expensive, so management wants to minimize the value of $k$. Devise an algorithm which uses the minimum possible $k$ guaranteeing that, given any $1\leq M \leq n$, we can promote the $M$-th highest scoring employee as the new department director in finitely many days. [b]Scoring:[/b] An algorithm that solves the case for a certain $k$ in terms of $n\geq 2$ and some constant $c\in \mathbb{R}^+$ will be awarded: [list] [*] $10$ pts for any finite $k$ [*] $20$ pts for any $k\leq cn\log(n)$ for some constant $c>0$ [*] $40$ pts for any $k\leq cn$ for some constant $c>1$ [*] $70$ pts for $k\leq n$ [*] $85$ pts for $k\leq n-\lfloor\sqrt{n/2}-\tfrac{1}{2}\rfloor + 1$ [*] $100$ pts for $k\leq n-\lfloor\sqrt{n}-\tfrac{1}{2}\rfloor + 1$ [/list] [i]Proposed by David Tang[/i]

\[\int_{-1}^1 x^{2022}\cos\left(\tfrac \pi {12}-x\right)\sin\left(\tfrac \pi{12}+x\right)\,\mathrm dx\] [i]Proposed by Michael Duncan, Connor Gordon, and Vlad Oleksenko[/i]

Let $ABC$ be a triangle such that $|AB|=7$, $|BC|=8$, $|AC|=6$. Let $D$ be the midpoint of side $[BC]$. If the circle through $A$, $B$ and $D$ cuts $AC$ at $A$ and $E$, what is $|AE|$? $ \textbf{(A)}\ \dfrac 23 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac 32 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $

Let $n$ be a positive integer, and $a,b\ge 1$, $c>0$ arbitrary real numbers. Prove that \[\frac{(ab+c)^n-c}{(b+c)^n-c}\le a^n.\]

Choose a real number between $0$ and $10$, inclusive. If your number is less than the average of all numbers chosen, you will get your number's worth of points, but if your number is greater than or equal to the average, you will get $0$ points. For example, if the average of all numbers chosen is $1.2$, and you pick $1.6$, then you will receive $0$ points, but if you pick $0.5$, then you will receive $0.5$ points. Express your answer to the nearest thousandth. For example, $7.800$, $2.110$, and $0.234$ are valid responses, but $7.8$ and $0.2345$ are not. An invalid response will receive a score of zero.

[b]a)[/b] Determine if it's possible write $6$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers. [b]b)[/b] Determine if it's possible write $8$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers.

With $\sigma (n)$ we denote the sum of the positive divisors of the natural number $n$. Prove that there exist infinitely many natural numbers $n$, for which $n$ divides $2^{\sigma (n)} -1$.

Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?

For each integer $n \ge 3$ solve in real numbers the system of equations: $$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$

The sum of nine consecutive odd numbers is $2007$. Find the greatest of these nine numbers.

A point-sized cue ball is fired in a straight path from the center of a regular hexagonal billiards table of side length $1$. If it is not launched directly into a pocket but travels an integer distance before falling into one of the pockets (located in the corners), find the minimum distance that it could have traveled.

Let $x<y$ be positive real numbers such that $$\sqrt{x}+\sqrt{y}=4 \quad \text{and} \quad \sqrt{x+2}+\sqrt{y+2}=5.$$ Compute $x.$

Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.

The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.

There are $41$ letters on a circle, each letter is $A$ or $B$. It is allowed to replace $ABA$ by $B$ and conversely, as well as to replace $BAB$ by $A$ and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations? Maxim Didin

The sum of all of the interior angles of seven polygons is $180\times17$. Find the total number of sides of the polygons.

Let $A_1$, $A_2$, $\ldots$, $A_n$, $n\geq 2$ be $n$ finite sets with the properties i) $|A_i| \geq 2$, for all $1\leq i \leq n$; ii) $|A_i\cap A_j| \neq 1$, for all $1\leq i<j\leq n$. Prove that the elements of the set $\displaystyle \bigcup_{i=1}^n A_i$ can be colored with 2 colors, such that all the sets $A_i$ are bi-color, for all $1\leq i \leq n$.

Harry wants to label the points of a regular octagon with numbers $1,2,\ldots ,8$ and label the edges with $1,2,\ldots, 8$. There are special rules he must follow: If an edge is numbered even, then the sum of the numbers of its endpoints must also be even. If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd. Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If $N$ is the number of possible octagon labelings, find the remainder when $N$ is divided by $100$. [i]Proposed by Harry Kim[/i]

Find the largest possible real number $C$ such that for all pairs $(x, y)$ of real numbers with $x \ne y$ and $xy = 2$, $$\frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C.$$ Also determine for which pairs $(x, y)$ equality holds.

Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.

Find all natural numbers of $90$ digits that are multiples of $13$ and have the first $43$ digits equal to each other and nonzero, the last $43$ digits equal to each other, and the middle $4$ digits are $2, 0, 1, 0$, in that order.

[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

We call natural numbers $n$ and $k$ are similar if they are multiples of square of a number greater than $1$. Let $f(n)$ denote the number of numbers from $1$ to $n$ similar to $n$ (for example, $f(16)=4$, since the number $16$ is similar to $4$, $8$, $12$ and $16$). What integer values can the quotient $\frac{n}{f(n)}$ take?

On a trip from the United States to Canada, Isabella took $ d$ U.S. dollars. At the border she exchanged them all, receiving $ 10$ Canadian dollars for every $ 7$ U.S. dollars. After spending $ 60$ Canadian dollars, she had $ d$ Canadian dollars left. What is the sum of the digits of $ d$? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$