Let $ ABC$ be an acute triangle, and let $ D$ and $ E$ be the feet of the altitudes drawn from vertexes $ A$ and $ B$, respectively. Show that if, \[ Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]\] then, the triangle is isosceles.

Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.

Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\leq d$

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

$2023$ points are chosen on a circle. Determine the parity of the number of ways to color the chosen points blue and red (each in one color, not necessarily using both), such that among any $31$ consecutive points, there is at least one red point.

A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner. [i]Proposed by Nathan Ramesh

Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.

$AB < AC$ on $\triangle ABC$. The midpoint of arc $BC$ which doesn't include $A$ is $T$ and which includes $A$ is $S$. On segment $AB,AC$, $D,E$ exist so that $DE$ and $BC$ are parallel. The outer angle bisector of $\angle ABE$ and $\angle ACD$ meets $AS$ at $P$ and $Q$. Prove that the circumcircle of $\triangle PBE$ and $\triangle QCD$ meets on $AT$.

A finite collection of digits $0$ and $1$ is written around a circle. An [i]arc[/i] of length $L\ge 0$ consists of $L$ consecutive digits around the circle. For each arc $w,$ let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w,$ respectively. Assume that $|Z(w)-Z(w')|\le 1$ for any two arcs $w,w'$ of the same length. Suppose that some arcs $w_1,\dots,w_k$ have the property that \[Z=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\sum_{j=1}^k N(w_j)\] are both integers. Prove that there exists an arc $w$ with $Z(w)=Z$ and $N(w)=N.$

(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$. (b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.

Find all the odd positive integers $n$ such that there are $n$ odd integers $x_1, x_2,..., x_n$ such that $$x_1^2+x_2^2+...+x_n^2=n^4$$

Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6, DB = 2$, and the ratio $\tfrac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5.$ Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

In a certain city, exactly 3 streets converge at any intersection. The streets are painted in three colors so that they converge at each intersection streets of three different colors. Three roads leave the city. Prove that they have different colors.

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Inside the square ${ABCD}$, the equilateral triangle $\vartriangle ABE$ is constructed. Let ${M}$ be an interior point of the triangle $\vartriangle ABE$ such that $MB=\sqrt{2}$, $MC=\sqrt{6}$, $MD=\sqrt{5}$ and ${ME=\sqrt{3}}$. Find the area of the square ${ABCD}$.

Given is an acute-angled triangle $ABC$ in which holds $|BC|: |AC| = 3:$2. Let $D$ be the midpoint of the side $\overline{AC}$, and P the midpoint of the segment $\overline{BD}$. A point $X$ is given on the line $AC$ so that $|AX| = |BC|$, where $A$ is between $X$ and $C$. The line $XP$ intersects the side $\overline{BC}$ at point $E$. The line $DE$ intersects the line $AP$ at point $Y$. Prove that the points $A$, $X$, $Y$, $E$ lie on one circle if and only if $|AB| = |BC|$.

In the addition problem \[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a different digit. Find the number represented by the answer PICNIC.

$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$, and $C$. What is the magnitude of $\angle B AC$ in degrees?

What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac {1}{1000}$?

Let $f(x), g(x)$ be real polynomials of degrees $2$ and $3$, respectively. Could it happen that $f(g(x))$ has $6$ distinct roots, which are powers of $2$?

Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

Determine the largest possible value of$ M$ for which it holds that: $$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,$$ for all real numbers $x, y, z > 0$ that satisfy the equation $xy + yz + zx = 1$.

Katherine has a piece of string that is $2016$ millimeters long. She cuts the string at a location chosen uniformly at random, and takes the left half. She continues this process until the remaining string is less than one millimeter long. What is the expected number of cuts that she makes?

Real numbers $x$, $y$, and $z$ satisfy the following equality: \[4(x+y+z)=x^2+y^2+z^2\] Let $M$ be the maximum of $xy+yz+zx$, and let $m$ be the minimum of $xy+yz+zx$. Find $M+10m$.