Let $ABC$ be an acute triangle with an altitude $AD$. Let $H$ be the orthocenter of triangle $ABC$. The circle $\Omega$ passes through the points $A$ and $B$, and touches the line $AC$. Let $BE$ be the diameter of $\Omega$. The lines~$BH$ and $AH$ intersect $\Omega$ for the second time at points $K$ and $L$, respectively. The lines $EK$ and $AB$ intersect at the point~$T$. Prove that $\angle BDK=\angle BLT$.

Show that for some $c > 0$, we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$.

Given is a square board measuring $1 995 \times 1 995$. These cells are painted with black and white paints in a checkerboard pattern, so that the corner cells are black. A spider sitting on one of the black cells can crawl to the cell on the same side as the one it occupies in one step. Prove that a spider can always reach a fly sitting motionless in another black cell by visiting all the cells of the board once.

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?

Given two fractions $a/b$ and $c/d$ we define their [i]pirate sum[/i] as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$. Given an integer $n \ge 3$, initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$. At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction. Determine, in function of $n$, the maximum and the minimum possible value for the last fraction.

Find all positive integers $N$ such that there exists a triangle which can be dissected into $N$ similar quadrilaterals. [i]Proposed by Nikolai Beluhov (Bulgaria) and Morteza Saghafian[/i]

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$? (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.)

Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$

There are $n{}$ passengers in the queue to board a $n{}$-seat plane. The first one in the queue is an absent-minded old lady who, after boarding the plane, sits down at a randomly selected place. Each subsequent passenger sits in his seat if it is free, and in a random seat otherwise. How many passengers will be out of their seats on average? [i]Proposed by A. Zaslavsky[/i]

Let $x+y=a$ and $xy=b$. Calculate exression $ x^4+y^4$ in terms of $a$ and $b$.

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

Let a $9$-digit number be balanced if it has all numerals $1$ to $9$. Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other.

Let $O$ and $H$ be the circumcenter and the orthocenter, respectively, of an acute triangle $ABC$. Points $D$ and $E$ are chosen from sides $AB$ and $AC$, respectively, such that $A$, $D$, $O$, $E$ are concyclic. Let $P$ be a point on the circumcircle of triangle $ABC$. The line passing $P$ and parallel to $OD$ intersects $AB$ at point $X$, while the line passing $P$ and parallel to $OE$ intersects $AC$ at $Y$. Suppose that the perpendicular bisector of $\overline{HP}$ does not coincide with $XY$, but intersect $XY$ at $Q$, and that points $A$, $Q$ lies on the different sides of $DE$. Prove that $\angle EQD = \angle BAC$. [i]Proposed by Shuang-Yen Lee[/i]

Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.

Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$

a) Points $B$ and $C$ are inside the segment $[AD]$. $|AB|=|CD|$. Prove that for all of the points P on the plane holds inequality $$|PA|+|PD|>|PB|+|PC|$$ b) Given four points $A,B,C,D$ on the plane. For all of the points $P$ on the plane holds inequality $$|PA|+|PD| > |PB|+|PC|.$$ Prove that points $B$ and C are inside the segment $[AD]$ and$ |AB|=|CD|$.

Show that in any triangle $ABC$ with $\angle A = 90^o$ the following inequality holds $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^ 2 < 2BC^6.$$

Let $n$ be an integer satisfying $n\geq2$. Find the greatest integer not exceeding the expression: $E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}$

On the surface of a sphere, a non-intersecting closed curve is drawn. It divides the surface of the sphere in two regions, which are coloured red and blue. Prove that there exist two antipodes of different colours. [i]Note: the curve is colourless.[/i] [i]Proposed by Vlad Spătaru[/i]

Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.

For a number consists of $0$ and $1$, one can perform the following operation: change all $1$ into $100$, all $0$ into $1$. For all nonnegative integer $n$, let $A_n$ be the number obtained by performing the operation $n$ times on $1$(starts with $100,10011,10011100100,\dots$), and $a_n$ be the $n$-th digit(from the left side) of $A_n$. Prove or disprove that there exists a positive integer $m$ satisfies the following: For every positive integer $l$, there exists a positive integer $k\le m$ satisfying$$a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}$$

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.