Let $ABC$ be an acute-angled triangle, and $M$ be the midpoint of the minor arc $BC$ of its circumcircle. A circle $\omega$ touches the side $AB, AC$ at points $P, Q$ respectively and passes through $M$. Prove that $BP + CQ = PQ$.

Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.

Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ? (E. Barabanov)

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

A rectangle $ABCD$ is drawn on the cross-lined paper with its sides laying on the lines, and $|AD|$ is $k$ times more than $|AB|$ ($k$ is an integer). All the shortest paths from $A$ to $C$ coming along the lines are considered. Prove that the number of those with the first link on $[AD]$ is $k$ times more then of those with the first link on $[AB]$.

Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that (i) $$\frac{3}{4}\le xy+yz+zx<1$$ (ii) $$xyz\le \frac{1}{8}$$ Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.

For some positive numbers $A, B, C$ and $D$, the system of equations $$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$ has $m$ solutions, while the system of equations $$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$ has $n$ solutions. If $m > n > 1$, find $m$ and $n$. ( G Galperin)

Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which: [list] [*] only finitely many distinct labels occur, and [*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$. [/list] [i]Proposed by Ricky Liu[/i]

Let $ABCDEF$ be a regular hexagon with center $O$ and side length $1.$ Point $X$ is placed in the interior of the hexagon such that $\angle BXC = \angle AXE = 90^\circ.$ Compute all possible values of $OX.$

Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ [i]Proposed by A. Golovanov[/i]

Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and \[ P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}.\] Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}$.

Suppose $a$ and $b$ are real numbers such that the polynomial $x^3+ax^2+bx+15$ has a factor of $x^2-2$. Find the value of $a^2b^2$.

The values of $ k$ for which the equation $ 2x^2\minus{}kx\plus{}x\plus{}8\equal{}0$ will have real and equal roots are: $ \textbf{(A)}\ 9 \text{ and }\minus{}7 \qquad \textbf{(B)}\ \text{only }\minus{}7 \qquad \textbf{(C)}\ \text{9 and 7} \\ \textbf{(D)}\ \minus{}9 \text{ and }\minus{}7 \qquad \textbf{(E)}\ \text{only 9}$

A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.

The equation $x|x|+px+q=0$ is given. Which of the following is not true? $\text{(A)}$It has at most three real roots. $\text{(B)}$It has at least one real root. $\text{(C)}$Only if $p^2-4q\geq0 $,it has real roots. $\text{(D)}$If $p<0$ and $q>0$, it has three real roots.

Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$? [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. How many of the chords of the circle pass through $P$ and have integer length? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 29 $

Caitlin and Donal play a game called [i]Basketball Shoot-Out[/i]. The game consists of $10$ rounds. In each round, Caitlin and Donal both throw a ball simultaneously at each other's basket. If a player's ball falls into the basket, that player scores one point; otherwise, they score zero points. The scoreboard shows the complete sequence of points scored by each player in each of the $10$ rounds of the game. It turns out that Caitlin has scored at least as many points in total as Donal after every round of the game. Prove the number of possible scoreboards is divisible by $4$ but not by $8$.

Let $X =\{x_0, x_1,\ldots , x_n\}$ be the basis set of a finite metric space, where the points are inductively listed such that $x_k$ maximizes the product of the distances from the points $\{x_0, x_1,\ldots , x_{k-1}\}$ for each $1\leqslant k\leqslant n.$ Prove that if for each $x\in X$ we let $\Pi_x$ be the product of the distances from $x{}$ to every other point, then $\Pi_{x_n}\leqslant 2^{n-1}\Pi_x$ for any $x\in X.$

Prove that there are no pairs of nonnegative integers $(x,y)$ that satisfy the equality $$x^3-y^3=x-y+2^{x-y}.$$

Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle $ABC$.

$3n - 2$ participants took part in the chess festival, some of them played one game of chess with each other. Prove that at least one of the following statements holds: (A) One can find $n$ chess players $A_1 , A_2 , . . . , A_n$ suchthat Ai has played a game with $A_{i+1}$ for all $i = 1, ...,n -1$. (B) Seven chess players can be found in $B_1 , . . . , B_7$, who have not played with each other, except perhaps three pairs $(B_1, B_2)$, $(B_3, B_4)$ and $(B_5, B_6)$, each of whom may or may not have played a game of chess.