Baklavas with nuts are laid out on the table in a row at the Nowruz celebration. Kosa and Kechel saw this and decided to play a game. Kosa eats one baklava from either the beginning or the end of the row in each move. Kechel either doesn't touch anything in each move or chooses the baklava he wants and just eats the nut on it. They agree that the first Kosa will start the game and make $20$ moves in each step, and the Kechel will only make $1$ move in each step. If the last baklava eaten by the Kosa is a nut, he wins the game. It is given that the number of baklavas is a multiple of $20.$ $A)$ If the number of baklavas is $400,$ prove that Kosa will win the game regardless of which strategy Kechel chooses. $B)$ Is it always true that no matter how many baklavas there are and what strategy Kechel chooses, Kosa will always win the game?

Let $r,s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that \[ \frac{\binom s0}{\binom tr} + \frac{\binom s1}{\binom{t}{r+1}} + \cdots + \frac{\binom ss}{\binom{t}{r+s}} = \frac{t+1}{(t+1-s)\binom{t-s}{r}}. \]

The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. After the competition, they queue in front of the restaurant according to the following rules. [list] [*]The Jury chooses the initial order of the contestants in the queue. [*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$. [list] [*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions. [*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends. [/list] [/list] [list=a] [*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices. [*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves. [/list]

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares. $(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard? $(b)$ When it is possible, find the minimum number of trominoes needed.

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

Humberto and Luciano use the break between classes to have fun with the following game: Humberto writes a list of distinct positive integers on a green sheet of paper and hands it to Luciano. Luciano then writes on a board all the possible sums, without repetitions, of one or more different numbers written on the green sheet. For example, if Humberto writes $1$, $3$ and $4$ on the green sheet, Luciano will write $1$, $3$, $4$, $5$, $7$ and $8$ on the board. (a) Let $n$ be a positive integer. Determine all positive integers $k$ such that Humberto can write a list of $n$ numbers on the green sheet in order to guarantee that Luciano will write exactly $k$ numbers on the board. (b) Luciano now decides to write a list of $m$ distinct positive integers on a yellow sheet of paper. Determine the smallest positive integer $m$ such that it is possible for Luciano to write this list so that, for any list that Humberto writes on the green sheet, with a maximum of $2023$ numbers, not all the numbers on the yellow sheet will be written on the board.

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

Suppose that $n\ge3$ is a natural number. Find the maximum value $k$ such that there are real numbers $a_1,a_2,...,a_n \in [0,1)$ (not necessarily distinct) that for every natural number like $j \le k$ , sum of some $a_i$-s is $j$. [i]Proposed by Navid Safaei [/i]

Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.