Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$. $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$. $CP$ and $AD$ intersects at $Q$, $BP$ and $AD$ intersects at $R$. Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$. Show that $RS=RT$

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

How many pairs of positive integers $(a,b)$ with $a\leq b$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{6}$?

$3.$ Let $S$ be the set of all positive integers from $1$ to $100$ included. Two players play a game. The first player removes any $k$ numbers he wants, from $S$. The second player's goal is to pick $k$ different numbers, such that their sum is $100$. Which player has the winning strategy if : $a)$ $k=9$? $b)$ $k=8$?

Assume that, for a certain school, it is true that [list]I: Some students are not honest II: All fraternity members are honest[/list] A necessary conclusion is: $\textbf{(A)}\ \text{Some students are fraternity members} \qquad\\ \textbf{(B)}\ \text{Some fraternity members are not students} \qquad\\ \textbf{(C)}\ \text{Some students are not fraternity members} \qquad\\ \textbf{(D)}\ \text{No fraternity member is a student} \qquad\\ \textbf{(E)}\ \text{No student is a fraternity member} $

A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point. (A. Zaslavsky)

Zaineb makes a large necklace from beads labeled $290, 291, \ldots, 2023$. She uses each bead exactly once, arranging the beads in the necklace any order she likes. Prove that no matter how the beads are arranged, there must be three beads in a row whose labels are the side lengths of a triangle.

Let $S=\{1,2,\cdots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S$, say $(A_1, A_2, A_3)$, such that $A_1\cup A_2\cup A_3=S$. Determine, in terms of $n$, \[ \sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|\]

There are $ n $ natural numbers ($ n\geq 2 $) whose sum is equal to their product. Prove that this common value does not exceed $2n$.

In a board of $2021 \times 2021$ grids, we pick $k$ unit squares such that every picked square shares vertice(s) with at most $1$ other picked square. Determine the maximum of $k$.

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.

Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.

Consider a natural number $n \ge 3$. A convex polygon with $n$ sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than $\frac{8}{n^2}$.

Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold: - If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$ - If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$. What is the maximum possible number of elements in $S$? $ \textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675 \qquad $

Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2n,2n)$ consists of $4n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0),(2n,0),(2n,2n),(0,2n)$ into 2 regions with even areas is $$\frac{{4n \choose 2n} + {2n \choose n}}{2}$$