Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$. (Mykola Moroz)

The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

The football tournament, held in one round, involved $16$ teams, each two of which scored a different number of points. ($3$ points were given for a victory, $1$ point for a draw, $0$ points for a defeat.) It turned out that the Chisel team lost to all the teams that ultimately scored fewer points. What is the best result that the Chisel team could achieve (insert location)?

It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$ a right-angled triangle can be formed?

Al has three red marbles and four blue marbles. He draws two different marbles at the same time. What is the probability that one is red and the other is blue?

Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.

Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board: If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$. Prove that Pavel can always fulfil his plan.

Find all real number $p$, such that the three roots of the equation $5x^3-5(p+1)x^2+(71p-1)x+1=66p$ are all positive integers.

Alice and Bob play the following game. They write the expressions $x + y$, $x - y$, $x^2+xy+y^2$ and $x^2-xy+y^2$ each on a separate card. The four cards are shuffled and placed face down on a table. One of the cards is turned over, revealing the expression written on it, after which Alice chooses any two of the four cards, and gives the other two to Bob. All cards are then revealed. Now Alice picks one of the variables $x$ and $y$, assigns a real value to it, and tells Bob what value she assigned and to which variable. Then Bob assigns a real value to the other variable. Finally, they both evaluate the product of the expressions on their two cards. Whoever gets the larger result, wins. Which player, if any, has a winning strategy?

Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying $$ a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N. $$

Let $f(x) = x^2 + bx + 1$ for some real number $b$. Across all possible values of $b$, find all possible values for the number of integers $x$ that satisfy $f(f(x) + x) < 0$.

Let $n$ and $k$ be positive integers satisfying $k\leq2n^2$. Lee and Sunny play a game with a $2n\times2n$ grid paper. First, Lee writes a non-negative real number no greater than $1$ in each of the cells, so that the sum of all numbers on the paper is $k$. Then, Sunny divides the paper into few pieces such that each piece is constructed by several complete and connected cells, and the sum of all numbers on each piece is at most $1$. There are no restrictions on the shape of each piece. (Cells are connected if they share a common edge.) Let $M$ be the number of pieces. Lee wants to maximize $M$, while Sunny wants to minimize $M$. Find the value of $M$ when Lee and Sunny both play optimally.

In the triangle $ABC$, it is known that $AC <AB$. Let $\ell$ be tangent to the circumcircle of triangle $ABC$ drawn at point $A$. A circle with center $A$ and radius $AC$ intersects segment $AB$ at point $D$, and line $\ell$ at points $E$ and $F$. Prove that one of the lines $DE$ and $DF$ passes through the center inscribed circle of triangle $ABC$.

Let $ABC$ be a triangle with incenter and $A$-excenter $I, I_a$, whose incircle touches $BC, CA, AB$ at $D, E, F$. The line $EF$ meets $BC$ at $P$ and $X$ is the midpoint of $PD$. Show that $XI \perp DI_a$.

Let $ABC$ be a scalene triangle, $(O)$ and $H$ be the circumcircle and its orthocenter. A line through $A$ is parallel to $OH$ meets $(O)$ at $K$. A line through $K$ is parallel to $AH$, intersecting $(O)$ again at $L$. A line through $L$ parallel to $OA$ meets $OH$ at $E$. Prove that $B,C,O,E$ are on the same circle. Trần Quang Hùng

$n$ is composite. $1<a_1<a_2<...<a_k<n$ - all divisors of $n$. It is known, that $a_1+1,...,a_k+1$ are all divisors for some $m$ (except $1,m$). Find all such $n$.

Jay is given a permutation $\{p_1, p_2,\ldots, p_8\}$ of $\{1, 2,\ldots, 8\}$. He may take two dividers and split the permutation into three non-empty sets, and he concatenates each set into a single integer. In other words, if Jay chooses $a,b$ with $1\le a< b< 8$, he will get the three integers $\overline{p_1p_2\ldots p_a}$, $\overline{p_{a+1}p_{a+2}\ldots p_{b}}$, and $\overline{p_{b+1}p_{b+2}\ldots p_8}$. Jay then sums the three integers into a sum $N=\overline{p_1p_2\ldots p_a}+\overline{p_{a+1}p_{a+2}\ldots p_b}+\overline{p_{b+1}p_{b+2}\ldots p_8}$. Find the smallest positive integer $M$ such that no matter what permutation Jay is given, he may choose two dividers such that $N\le M$. [i]Proposed by James Lin[/i]

Given a triangle $ABC$, show how to construct the point $P$ such that $\angle PAB= \angle PBC= \angle PCA$. Express this angle in terms of $\angle A,\angle B,\angle C$ using trigonometric functions.

Find the locus of the projections of a given point on all planes containing another point $B$.

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

An ordered pair of sets $(A, B)$ is [i]good[/i] if $A$ is not a subset of $B$ and $B$ is not a subset of $A$. How many ordered pairs of subsets of $\{1, 2, \dots, 2017\}$ are good?

On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? $ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$

Find all functions $f : Z \to Z$ satisfying the following two conditions: (i) for all integers $x$ we have $f(f(x)) = x$, (ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.

Let $\Omega$ be a sphere of radius $4$ and $\Gamma$ be a sphere of radius $2.$ Suppose that the center of $\Gamma$ lies on the surface of $\Omega.$ The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumfrence.