IrMO 2018

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$ . She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary's turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$ . Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

The triangle $ABC$ is right-angled at $A$ . Its incentre is $I$ , and $H$ is the foot of the perpendicular from $I$ on $AB$ . The perpendicular from $H$ on $BC$ meets $BC$ at $E$ , and it meets the bisector of $\angle ABC$ at $D$ . The perpendicular from $A$ on $BC$ meets $BC$ at $F$ . Prove that $\angle EFD = 45^o$

Find all functions $f(x) = ax^2 + bx + c$ , with $a \ne 0$ , such that $f(f(1)) = f(f(0)) = f(f(-1))$ .

We say that a rectangle with side lengths $a$ and $b$ fits inside a rectangle with side lengths $c$ and $d$ if either ( $a \le c$ and $b \le d$ ) or ( $a \le d$ and $b \le c$ ). For instance, a rectangle with side lengths $1$ and $5$ fits inside another rectangle with side lengths $1$ and $5$ , and also fits inside a rectangle with side lengths $6$ and $2$ . Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$ , $B$ , and $C$ in $S$ such that $A$ fits inside $B$ , and $B$ fits inside $C$ .

Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$ . A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$ . The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$ .

Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$ .

Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$

Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$ . The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$ . The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$ , and $|MD| = |ME|$ . The point $F$ is the intersection of $MD$ and $AB$ . Prove that $\angle BFM = \angle BME$ .

The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$ . Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.

The game of Greed starts with an initial configuration of one or more piles of stones. Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$ . At each turn, a player has two choices: • take one stone from any one of the piles (a simple move); • take one stone from each of the remaining piles (a greedy move). The player who takes the last stone wins. Consider the following two initial configurations: (a) There are $2018$ piles, with either $20$ or $18$ stones in each pile. (b) There are four piles, with $17, 18, 19$ , and $20$ stones, respectively. In each case, find an appropriate strategy that guarantees victory to one of the players.