A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$. Anna and Bob play a game on this grid. Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first. 1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line. 2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted. As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins. Does Anna have a winning strategy? [i]Proposed by Demetres Christofides, Cyprus[/i]

A company of $n$ soldiers is such that (i) $n$ is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively. Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.

Points $ A$ and $ B$ lie on a circle centered at $ O$, and $ \angle AOB\equal{}60^\circ$. A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle? $ \textbf{(A)}\ \frac{1}{16} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{4}$

One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fifty segments intersect one another?

Let $\mathcal C$ be a circle of radius $1$. (a) Determine the triangles $ABC$ inscribed in $\mathcal C$ for which $AB^2+BC^2+CA^2$ is maximal. (b) Determine the quadrilaterals $ABCD$ inscribed in $\mathcal C$ for which $AB^2+AC^2+AD^2+BC^2+BD^2+CD^2$ is maximal.

Suppose $x, y$, and $ 1$ are side lengths of a triangle$ T$ such that $x < 1$ and $y < 1$. Given $x$ and $y$ are chosen uniformly at random from all possible pairs $(x, y)$, determine the probability that $T$ is obtuse.

Suppose that $ a$, $ b$, and $ c$ are positive real numbers such that $ a^{\log_3 7} \equal{} 27$, $ b^{\log_7 11} \equal{} 49$, and $ c^{\log_{11} 25} \equal{} \sqrt {11}$. Find \[ a^{(\log_3 7)^2} \plus{} b^{(\log_7 11)^2} \plus{} c^{(\log_{11} 25)^2}. \]

Let $\triangle ABC$ have $M$ as the midpoint of $BC$ and let $P$ and $Q$ be the feet of the altitudes from $M$ to $AB$ and $AC$ respectively. Find $\angle BAC$ if $[MPQ]=\frac{1}{4}[ABC]$ and $P$ and $Q$ lie on the segments $AB$ and $AC$.

A triangle $ABC$ with orthocenter $H$ is given. $P$ is a variable point on line $BC$. The perpendicular to $BC$ through $P$ meets $BH$, $CH$ at $X$, $Y$ respectively. The line through $H$ parallel to $BC$ meets $AP$ at $Q$. Lines $QX$ and $QY$ meet $BC$ at $U$, $V$ respectively. Find the shape of the locus of the incenters of the triangles $QUV$.

Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear. [img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]

Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?

Let $ABC$ be an acute triangle with $AB\neq AC$. Let $D, E, F$ be the midpoints of the sides $BC$, $CA$, and $AB$ respectively, and $M, N$ be the midpoints of minor arc $BC$ not containing $A$ and major arc $BAC$ respectively. Suppose $W, X, Y, Z$ are the incenter, $D$-excenter, $E$-excenter, and $F$-excenter of triangle $DEF$ respectively. Prove that the circumcircles of the triangles $ABC$, $WNX$, $YMZ$ meet at a common point. [i]Proposed by Ivan Chan Kai Chin[/i]

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

There are $n$ boxes $A_1, ..., A_n$ with non-negative number of pebbles inside it(so it can be empty). Let $a_n$ be the number of pebbles in the box $A_n$. There are total $3n$ pebbles in the boxes. From now on, Alice plays the following operation. In each operation, Alice choose one of these boxes which is non-empty. Then she divide this pebbles into $n$ group such that difference of number of pebbles in any two group is at most 1, and put these $n$ group of pebbles into $n$ boxes one by one. This continues until only one box has all the pebbles, and the rest of them are empty. And when it's over, define $Length$ as the total number of operations done by Alice. Let $f(a_1, ..., a_n)$ be the smallest value of $Length$ among all the possible operations on $(a_1, ..., a_n)$. Find the maximum possible value of $f(a_1, ..., a_n)$ among all the ordered pair $(a_1, ..., a_n)$, and find all the ordered pair $(a_1, ..., a_n)$ that equality holds.

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of $ABC$ paralell to $MB$ and $MC$, which intersect $AB$ and $AC$ at $K$ and $L$, respectively. Prove that $NK=NL$.

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

Points $B_1$ and $C_1$ are marked respectively on the sides $AB$ and $AC$ of an acute isosceles triangle $ABC$( $AB=AC$) such that $BB_1=AC_1$. The points $B,C$ and $S$ lie in the same half-plane with respect to the line $B_1C_1$ so that $\angle SB_1C_1=\angle SC_1B_1 = \angle BAC$ Prove that $B,C,S$ are colinear if and only if the triangle $ABC$ is equilateral.

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that: 1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences; 2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences, Find the smallest possible value of $k$ (as a function of $n$.) Note: The common differences might not be positive. Proposed by Chu-Lan Kao

Find the number of ordered pairs $(a, b, c, d)$ of positive integers satisfying the equation: \[a + 2b + 3c + 1000d = 2024.\] [i]Proposed by Irakli Khutsishvili, Georgia [/i]

Given $n$ positive real numbers $a_1, a_2, \ldots , a_n$ such that $a_1a_2 \cdots a_n = 1$, prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.\]

Let $m,n$ be positive integers, greater than 2.Find the number of polynomials of degree $2n-1$ with distinct coefficients from the set $\left\{ 1,2,\ldots,m\right\}$ which are divisible by $x^{n-1}+x^{n-2}+\ldots+1.$

Given is a cyclic quadrilateral $ABCD$ and a point $E$ lies on the segment $DA$ such that $2\angle EBD = \angle ABC$. Prove that $DE= \frac {AC.BD}{AB+BC}$.

Let $A\subset \{1,2,\ldots,4014\}$, $|A|=2007$, such that $a$ does not divide $b$ for all distinct elements $a,b\in A$. For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$. Find $\min m_{A}$ (for all $A$ with the above properties).

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)