AT-OEMO-FED1 2018

Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur? (Proposed by Walther Janous)

Let $ABC$ be a triangle with incenter $I$ . The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$ , respectively. Let $P$ denote the common point of lines $AI$ and $DE$ , and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$ , respectively. Prove that points $M, N$ and $P$ are collinear.

(Proposed by Karl Czakler)

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$ . Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$ . Can Alice avoid that?

(Proposed by Richard Henner)

Let $M$ be a set containing positive integers with the following three properties: (1) $2018 \in M$ . (2) If $m \in M$ , then all positive divisors of m are also elements of $M$ . (3) For all elements $k, m \in M$ with $1 < k < m$ , the number $km + 1$ is also an element of $M$ . Prove that $M = Z_{\ge 1}$ .

(Proposed by Walther Janous)