Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$. (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.

If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$? $\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$

Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$. Show that $$k\leq 6^{(n+1)/2}$$

Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

[b]a)[/b] show that every curve $f:I \longrightarrow S^2$ is homotop with a path with another curve in $S^2$ like $g$ such that Image of $g$, doesn't contain all of $S^2$. [b]b)[/b] conclude that $S^2$ is simple connected. [b]c)[/b] construct a topological space such that it's fundamental group is $\mathbb Z_2$.

Let $T$ be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points $(x, y, z)$ and $(u, v,w)$ are called [i]neighbors[/i] if $|x - u| + |y - v| + |z - w| = 1$. Show that there exists a subset $S$ of $T$ such that for each $p \in T$ , there is exactly one point of $S$ among $p$ and its [i]neighbors[/i].

The positive integer $m$ is a multiple of $101$, and the positive integer $n$ is a multiple of $63$. Their sum is $2018$. Find $m - n$.

For all reals $a, b, c,d $ prove the following inequality: $$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$

The motion of the particles of a fluid in the plane is specified by the following components of velocity $$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\;\; \frac{dy}{dt}=-x.$$ Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$, and justify your conclusion.

Determine for which positive integers $n$ there exists a positive integer $A$ such that • $A$ is divisible by $2022$, • the decimal expression of $A$ contains only digits $0$ and $7$, • the decimal expression of $A$ contains [i]exactly[/i] $n$ times the digit $7$.

A [i]cactus[/i] is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle. [i]Kevin Yang[/i]

We call a pair $(a,b)$ of positive integers, $a<391$, [i]pupusa[/i] if $$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$ Find the minimum value of $b$ across all [i]pupusa[/i] pairs. Fun Fact: OMCC 2017 was held in El Salvador. [i]Pupusa[/i] is their national dish. It is a corn tortilla filled with cheese, meat, etc.

A natural number $n$, which is at most $500$, has the property that when one chooses at at random among the numbers $1, 2, 3,...,499, 500$, then the probability is $\frac{1}{100}$ for $m$ to add up to $n$. Determine the largest possible value of $n$.

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that: $$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$

Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds \[ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. \]

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.

In a triangle $ ABC$, let $ M_{a}$, $ M_{b}$, $ M_{c}$ be the midpoints of the sides $ BC$, $ CA$, $ AB$, respectively, and $ T_{a}$, $ T_{b}$, $ T_{c}$ be the midpoints of the arcs $ BC$, $ CA$, $ AB$ of the circumcircle of $ ABC$, not containing the vertices $ A$, $ B$, $ C$, respectively. For $ i \in \left\{a, b, c\right\}$, let $ w_{i}$ be the circle with $ M_{i}T_{i}$ as diameter. Let $ p_{i}$ be the common external common tangent to the circles $ w_{j}$ and $ w_{k}$ (for all $ \left\{i, j, k\right\}= \left\{a, b, c\right\}$) such that $ w_{i}$ lies on the opposite side of $ p_{i}$ than $ w_{j}$ and $ w_{k}$ do. Prove that the lines $ p_{a}$, $ p_{b}$, $ p_{c}$ form a triangle similar to $ ABC$ and find the ratio of similitude. [i]Proposed by Tomas Jurik, Slovakia[/i]

In a tetrahedron, the six triangles determined by an edge of the tetrahedron and the midpoint of the opposite edge all have equal area. Prove that the tetrahedron is regular.

In the every cell of the board $5\times5$ there is one of the numbers: $-1$, $0$, $1$. It is true that in every $2 \times 2$ square there are three numbers summing up to $0$. Determine the maximal sum of all numbers in a board.

\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Find the least odd prime factor of $2019^8 + 1$.