Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

The diameter of a circle of radius $R$ is divided into $4$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.

Let $AMNB$ be a quadrilateral inscribed in a semicircle of diameter $AB = x$. Denote $AM = a$, $MN = b$, $NB = c$. Prove that $x^3- (a^2 + b^2 + c^2)x -2abc = 0$.

For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0,1\}$, such that $A^2$ is the matrix of all ones?

Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element and there exists infinitely many positive integer $m$ such that $\{m,m+2018\}\subset A$? Please prove your conclusion.

Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that \[(x+yt+z^2)(u+vt+wt^2)=1\]

In the triangle $ABC$, the length of the altitude from $A$ to $BC$ is equal to $1$. $D$ is the midpoint of $AC$. What are the possible lengths of $BD$?

Prove that for all $ x, y \in \[0, 1\] $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.

A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then (1) all tangent points of these faces to the sphere would coincide with one point, $H$, and (2) the vertices of the faces would lie on a circle centered at $H$.

For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that \[ \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} ) \le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, . \]

The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that: \[PX || AC \ , \ PY ||AB \] Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $s$ is a positive integer. Prove that: a) ${F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})$ and ${F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).$ b) $k({2^s}) = {3.2^{s - 1}}.$

Given are $n$ distinct natural numbers. For any two of them, the one is obtained from the other by permuting its digits (zero cannot be put in the first place). Find the largest $n$ such that it is possible all these numbers to be divisible by the smallest of them?

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3); draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0)); label("$A$", (.5,.5)); label("$M$", (7/6, 1/6)); label("$N$", (1/3, 4/3));[/asy] [i]Proposed by Aaron Lin[/i]

For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]. [i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.

What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\] $\textbf{(A) } \frac{5}{8} \qquad\textbf{(B) } \frac{11}{7} \qquad\textbf{(C) } \frac{8}{5} \qquad\textbf{(D) } \frac{18}{11} \qquad\textbf{(E) } \frac{15}{8}$

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

Given odd prime $p$. Sequence ${x_n}$ is defined by $x_{n+2}= 4x_{n+1}-x_n$. Choose $x_0,x_1$ such that for every random positive integer $k$, there exists $i\in \mathbb N$ such that $4p^2-8p+1|x_i - (2p)^k$.

For every algebraic integer $\alpha$ define its positive degree $\text{deg}^+(\alpha)$ to be the minimal $k\in\mathbb{N}$ for which there exists a $k\times k$ matrix with non-negative integer entries with eigenvalue $\alpha$. Prove that for any $n\in\mathbb{N}$, every algebraic integer $\alpha$ with degree $n$ satisfies $\text{deg}^+(\alpha)\le 2n$.

Three schools $A, B$ and $C$ , each with five players denoted $a_{i}, b_{i}, c_{i}$ respectively, take part in a chess tournament. The tournament is held following the rules: (i) Players from each school have matches in order with respect to indices, and defeated players are eliminated; the first match is between $a_{1}$ and $b_{1}$. (ii) If $y_{j}\in Y$ defeats $x_{i}\in X$ , his next opponent should be from the remaining school if not all of its players are eliminated; otherwise his next oponent is $x_{i+1}$ . The tournament is over when two schools are completely eliminated. (iii) When $x_{i}$ wins a match, its school wins $10^{i-1}$ points. At the end of the tournament, schools $A, B, C$ scored $P_{A}, P_{B}, P_{C}$ respectively. Find the remainder of the number of possible triples $(P_{A}, P_{B}, P_{C})$ upon division by $8.$