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.

Mom brought Andriy and Olesya $4$ balls with the numbers $1, 2, 3$ and $4$ written on them (one on each ball). She held $2$ balls in each hand and did not know which numbers were written on the balls in each hand. The mother asked Andriy to take a ball with a higher number from each hand, and then to keep the ball with the lower number from the two balls he took. After that, she asked Olesya to take two other balls, and out of these two, keep the ball with the higher number. Does the mother know with certainty, which child has the ball with the higher number? [i]Proposed by Bogdan Rublov[/i]

For which rational $ x$ is the number $ 1 \plus{} 105 \cdot 2^x$ the square of a rational number?

The set of all points $(x,y)$ in the plane satisfying $x<y$ and $x^3-y^3>x^2-y^2$ has area $A$. What is the value of $A$? [i]Proposed by Adam Bertelli[/i]

A rectangle is divided into several similar isosceles triangles. Determine the possible values of the angles of the triangles.

Let there be an acute triangle $\triangle ABC$ with incenter $I$. $E$ is the foot of the perpendicular from $I$ to $AC$. The line which passes through $A$ and is perpendicular to $BI$ hits line $CI$ at $K$. The line which passes through $A$ and is perpendicular to $CI$ hits the line which passes through $C$ and is perpendicular to $BI$ at $L$. Prove that $E, K, L$ are colinear.

Adam is playing Minesweeper on a $9\times9$ grid of squares, where exactly $\frac13$ (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines. First, he clicks the square directly above the center square, which shows the number $4$. Next, he clicks the square directly below the center square, which shows the number $1$. What is the probability that the center square is a mine? [i]Proposed by Adam Bertelli[/i]

7. What is the minimum value of the product $$\prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}}$$ given that $(a_1, a_2, a_3, a_4, a_5, a_6)$ is a permutation of $(1, 2, 3, 4, 5, 6)$? (note $a_7 = a_1, a_8 = a_2 \ldots$) 8. Danielle picks a positive integer $1 \le n \le 2016$ uniformly at random. What is the probability that $\text{gcd}(n, 2015) = 1$? 9. How many $3$-element subsets of the set $\{1, 2, 3, . . . , 19\}$ have sum of elements divisible by $4$?

Let $a>0$ and $12a+5b+2c>0$. Prove that it is impossible for the equation $ax^2+bx+c=0$ to have two real roots in the interval $(2,3)$.

Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?

Let $ABCD$ be a square with center $O$ , and $P$ be a point on the minor arc $CD$ of its circumcircle. The tangents from $P$ to the incircle of the square meet $CD$ at points $M$ and $N$. The lines $PM$ and $PN$ meet segments $BC$ and $AD$ respectively at points $Q$ and $R$. Prove that the median of triangle $OMN$ from $O$ is perpendicular to the segment $QR$ and equals to its half.

In triangle $ABC$, the incircle touches sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Median of triangle $ABC$ from vertex $B$ meets segment $P Q$ at point $R$. Prove that angle $ARC$ is obtuse.

If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair?

A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.

The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.