The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

Let $n$ be a natural number. A sequence is $k-$complete if it contains all residues modulo $n^k$. Let $Q(x)$ be a polynomial with integer coefficients. For $k\ge 2$, define $Q^k(x)=Q(Q^{k-1}(x))$, where $Q^1(x)=Q(x)$. Show that if $$0,Q(0),Q^2(0),Q^3(0),\ldots $$is $2018-$complete, then it is $k-$complete for all positive integers $k$. [i]Proposed by Ma Zhao Yu[/i]

Given that the expression $\frac{20^{21}}{20^{20}} +\frac{20^{20}}{20^{21}}$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Ada Tsui[/i]

Is it true that for every $n\ge 2021$ there exist $n$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal? (O. Rudenko)

In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$.

Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

$P$ is a convex polyhedron such that: [b](1)[/b] every vertex belongs to exactly $3$ faces. [b](1)[/b] For every natural number $n$, there are even number of faces with $n$ vertices. An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible) Show that the number of times the ant turns left is the same as the number of times the ant turn right.

The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

Is the number $$\sin \frac{\pi}{18} \sin \frac{3\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18} \sin \frac{9\pi}{18}$$ rational?

Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.

Let $ABC$ be a triangle with circumcirle $\Gamma$, and let $M$ and $N$ be the respective midpoints of the minor arcs $AB$ and $AC$ of $\Gamma$. Let $P$ and $Q$ be points such that $AB=BP$, $AC=CQ$, and $P$, $B$, $C$, $Q$ lie on $BC$ in that order. Prove that $PM$ and $QN$ meet at a point on $\Gamma$. [i]Proposed by Victor Domínguez[/i]

Each of the $50$ students in a class sent greeting cards to $25$ of the others. Prove that there exist two students who greeted each other.

Nils has a telephone number with eight different digits. He has made $28$ cards with statements of the type “The digit $a$ occurs earlier than the digit $b$ in my telephone number” – one for each pair of digits appearing in his number. How many cards can Nils show you without revealing his number?

Fix positive integers $m$ and $n$. Suppose that $a_1, a_2, \dots, a_m$ are reals, and that pairwise distinct vectors $v_1, \dots, v_m\in \mathbb{R}^n$ satisfy $$\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0$$ for $i=1,2,\dots,m$. Prove that $$\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.$$

Let $ n \geq 3 $ be an integer and let \begin{align*} x_1,x_2, \ldots, x_n \end{align*} be $ n $ distinct integers. Prove that \begin{align*} (x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6. \end{align*}

Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum?

Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ xf(xy) + xyf(x) \ge f(x^2)f(y) + x^2y \] holds for all $x,y \in \mathbb{R}$.

Four lighthouses are located at points $ A$, $ B$, $ C$, and $ D$. The lighthouse at $ A$ is $ 5$ kilometers from the lighthouse at $ B$, the lighthouse at $ B$ is $ 12$ kilometers from the lighthouse at $ C$, and the lighthouse at $ A$ is $ 13$ kilometers from the lighthouse at $ C$. To an observer at $ A$, the angle determined by the lights at $ B$ and $ D$ and the angle determined by the lights at $ C$ and $ D$ are equal. To an observer at $ C$, the angle determined by the lights at $ A$ and $ B$ and the angle determined by the lights at $ D$ and $ B$ are equal. The number of kilometers from $ A$ to $ D$ is given by $ \displaystyle\frac{p\sqrt{r}}{q}$, where $ p$, $ q$, and $ r$ are relatively prime positive integers, and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$,

$ABCDE$ is convex pentagon. $m(\widehat{B})=m(\widehat{D})=90^\circ$, $m(\widehat{C})=120^\circ$, $|AB|=2$, $|BC|=|CD|=\sqrt3$, and $|ED|=1$. $|AE|=?$ $ \textbf{(A)}\ \frac{3\sqrt3}{2} \qquad\textbf{(B)}\ \frac{2\sqrt3}{3} \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \sqrt3 - 1 \qquad\textbf{(E)}\ \sqrt3 $

(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords. Is it necessary that two of these chords are of equal length? (b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords. Prove that among these $10$ chords there are two chords of equal length. (VV Proizvolov, Moscow)

The positive integer $k$ and the set $A$ of distinct integers from $1$ to $3k$ inclusively are such that there are no distinct $a$, $b$, $c$ in $A$ satisfying $2b = a + c$. The numbers from $A$ in the interval $[1, k]$ will be called [i]small[/i]; those in $[k + 1, 2k]$ - [i]medium[/i] and those in $[2k + 1, 3k]$ - [i]large[/i]. It is always true that there are [b]no[/b] positive integers $x$ and $d$ such that if $x$, $x + d$, and $x + 2d$ are divided by $3k$ then the remainders belong to $A$ and those of $x$ and $x + d$ are different and are: a) small? $\hspace{1.5px}$ b) medium? $\hspace{1.5px}$ c) large? ([i]In this problem we assume that if a multiple of $3k$ is divided by $3k$ then the remainder is $3k$ rather than $0$[/i].)

A [i]palindrome [/i] is a natural number that works in the decimal system forwards and backwards read is the same size (e.g. $1129211$ or $7337$). Determine all pairs $(m, n)$ of natural numbers, such that $$(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})$$ is a palindrome.