$\vartriangle ABC$ is right with $\angle C = 90^o$. The internal angle bisectors of $\angle A$ and $\angle B$ meet at point $D$, while the external angle bisectors of $\angle A$ and $\angle B$ meet at point $E$. Suppose that $AD = 1$ and $BD = 2$. The value of $DE^2$ can be expressed as $x+y \sqrt{z}$ for integers $x$, $y$, and $z$, where $z$ is greater than $1$ and not divisible by the square of any prime. Compute $100x + 10y + z$. Note: For a generic triangle $\vartriangle PQR$, if we let $Q'$ be the reflection of $Q$ over $P$, then the external angle bisector of $\angle P$ is the line that contains the internal angle bisector of $\angle Q'PR$.

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$ Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.

Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.

Find all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]

The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$. [i]Proposed by Nikolay Beluhov[/i] EDIT: It was the problem 3 (not 2), corrected the source title.

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

On the side $ CD $ of the square $ ABCD, $ consider $ E $ for which $ \angle ABE =60^{\circ } . $ On the line $ AB, $ take the point $ F $ distinct from $ B $ such that $ BE=BF $ and such that it is on the segment $ AB, $ or $ A $ is on $ BF. $ Moreover, $ M $ is the intersection of $ EF,AD. $ [b]a)[/b] Show that $ \angle BME =75^{\circ } . $ [b]b)[/b] If the bisector of $ \angle CBE $ intersects $ CD $ in $ N, $ show that $ BMN $ is equilateral.

Let $ABC$ be an equilateral triangle with the side length equals $a+ b+ c$. On the side $AB{}$ of the triangle $ABC$ points $C_1$ and $C_2$ are chosen, on the side $BC$ points $A_1$ and $A_2$, arc chosen, and on the side $CA$ points $B_1$ and $B_2$ are chosen such that $A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b, C_1C_2 = BA_1 = AB_2 = c$. Let the point $A^{’}$ be such that the triangle $A^{'} B_2C_1$ is equilateral, and the points $A$ and $A^{'}$ lie on different sides of the line $B_2C_1$. Similarly, the points $B^{’}$ and $C^{'}$ are constructed (the triangle $B^{'} C_2A_1$ is equilateral, and the points $B$ and $B^{’}$ lie on different sides of the line $C_2A_1$; the triangle $C^{'} A_2B_1$ is equilateral, and the points $C$ and $C^{'}$ lie on different sides of the line $A_2B_1$). Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.

Determine all triples \( (a, b, c) \) of positive integers such that: \[ a + b + c = abc. \]

Positive reals $x, y, z$ satisfy $$\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}$$ Do they all have to be equal? [i](Proposed by Oleksii Masalitin)[/i]

Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.

Let $\Gamma$ be a circumference and $A$ a point outside it. Let $B$ and $C$ be points in $\Gamma$ such that $AB$ and $AC$ are tangent to $\Gamma$. Let $P$ be a point in $\Gamma$. Let $D$, $E$ and $F$ be points in $BC$, $AC$ and $AB$ respectively, such that $PD \perp BC$, $PE \perp AC$, and $PF \perp AB$. Show that $PD^2 = PE \cdot PF$

Determine all pairs of real numbers $ a, b $ for which the polynomials $ x^4 + 2ax^2 + 4bx + a^2 $ and $ x^3 + ax - b $ have two different common real roots.

Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.

Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

Let $C_1$, $C_2$ and $C_3$ be three parallel chords of a circle on the same side of the center. The distance between $C_1$ and $C_2$ is the same as the distance between $C_2$ and $C_3$. The lengths of the chords are 20, 16, and 8. The radius of the circle is $\text{(A)} \ 12 \qquad \text{(B)} \ 4\sqrt{7} \qquad \text{(C)} \ \frac{5\sqrt{65}}{3} \qquad \text{(D)} \ \frac{5\sqrt{22}}{2} \qquad \text{(E)} \ \text{not uniquely determined}$

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

Let $a_0,a_1,a_2 \dots a_n, \dots$ be an infinite sequence of real numbers, defined by $$a_0 = c$$ $$a_{n+1} = {a_n}^2+\frac{a_n}{2}+c$$ for some real $c > 0$. Find all values of $c$ for which the sequence converges and the limit for those values.

Two perpendicular lines $p,q$ and a point $A\notin p\cup q$ are given in plane. Find locus of all points $X$ such that \[XA=\sqrt{|Xp|\cdot|Xq|\,},\] where $|Xp|$ denotes the distance of $X$ from $p.$

Of the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were renumbered and lined up in a circle. Then a certain number $m$ (possibly greater than $9$) and the direction of reference were determined. People began to be counted, starting from the first. Each one became a winner and was eliminated from the drawing, and counting, starting from the next, continued until four winners were identified. The first three prizes were awarded to three people who had numbers $2$, $7$ and $5$ in the original lineup (they were eliminated in that order). What number did the fourth winner of the competition have in the initial lineup?

In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$

An $m \times n$ grid of squares (with $m$ rows and $n$ columns) has some of its squares colored blue. The grid is called [i]fish-friendly[/i] if a fish can swim from the left edge of the grid to the right edge of the grid only moving through blue squares. In other words, there is a sequence of blue squares, each horizontally or vertically adjacent to the previous square, starting in the first column and ending in the last column. Prove that the number of fish-friendly $42 \times 49$ grids is at least $2^{2022}$.

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as $a, b, c$ such that $a >gcd(b, c)$, then $a$ gets replaced by $ a-gcd(b, c)$. The game ends if there is no way to denote the numbers such that $a >gcd(b, c)$. Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers. (Theresia Eisenkölbl)