Given a triangle $ABC$ with $BC = 5$, $AC = 7$, and $AB = 8$, find the side length of the largest equilateral triangle $P QR$ such that $A, B, C$ lie on $QR$, $RP$, $P Q$ respectively.

$a_1,a_2,...,a_9$ are nonnegative reals with sum $1$. Define $S$ and $T$ as below: $$S=\min\{a_1,a_2\}+2\min\{a_2,a_3\}+...+9\min\{a_9,a_1\}$$ $$T=\max\{a_1,a_2\}+2\max\{a_2,a_3\}+...+9\max\{a_9,a_1\}$$ When $S$ reaches its maximum, find all possible values of $T$.

There are $30$ men with their $30$ wives sitting at a round table. Show that there always exist two men who are on the same distance from their wives. (The seats are arranged at vertices of a regular polygon.)

Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the[i] odd part[/i] of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the [i]odd part[/i] of $3a_k+1$. Find $a_n$.

A regular tetrahedron has two vertices on the body diagonal of a cube with side length $12$. The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron.

Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$. Show that $p<2n$.

Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.

Three parallel lines $\ell_1, \ell_2$ and $\ell_3$ of a plane are given such that the line $\ell_2$ has the same distance $a$ from $\ell_1$ and $\ell_3$. We put $5$ points $M_1, M_2, M_3,M_4$ and $M_5$ on the lines $\ell_1, \ell_2$ and $\ell_3$ in such a way that each line contains at least one point. Detennine the maximal number of isosceles triangles that are possible to be formed with vertices three of the points $M_1, M_2, M_3, M_4$ and $M_5$ in the following cases: (i) $M_1,M_2,M_3 \in \ell_2, M_4 \in \ell_1$ and $M_5 \in \ell_3$. (ii) $M_1,M_2 \in \ell_1, M_3,M_4 \in \ell_3$ and $M_5 \in \ell_2$.

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

A black pawn and a white pawn are placed on the first square and the last square of a $ 1\times n$ chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.

Let $ M,N,P,Q $ be points on the sides $ AB,BC,CD,DA $ (respectively) of a convex quadrilateral $ ABCD $ so that: $$ \frac{MA}{MB} =\frac{NB}{NC} =\frac{PD}{PC} =\frac{QA}{QD}\neq 1 $$ Show that the area of $ MNPQ $ is half the area of $ ABCD $ if and only if $ ABD $ and $ BCD $ have equal areas. [i]Petre Asaftei[/i]

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]

For which of the following expressions, there exists an integer $x$ such that the expression is divisble by $25$? $ \textbf{(A)}\ x^3-3x^2+8x-1 \\ \qquad\textbf{(B)}\ x^3+3x^2-2x+1 \\ \qquad\textbf{(C)}\ x^3+14x^2+3x-8 \\ \qquad\textbf{(D)}\ x^3-5x^2+x+1 \\ \qquad\textbf{(E)}\ \text{None of above} $

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c \\ x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \quad \text{for all } n \geq 1. \end{align*} Show that if $c$ is a positive integer, then $x_{n}$ is an integer for all $n \geq 1$. [i](South Africa)[/i]

Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$. Nguyễn Minh Hà

Find all the solutions of the equation \[\left(x+1\right)^y-x^z=1\] For $x,y,z$ integers greater than 1.

Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.

$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.

Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by $3$, then it is not possible to end the game with only one pawn left on the board.

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Determine all polynomials $P (x, y), Q(x, y)$ and $R(x, y)$ with real coefficients satisfying $$P (ux + vy, uy + vx) = Q(x, y)R(u, v)$$ for all real numbers $u, v, x$ and $y$.