ELMO 2013

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$ . Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$ . What is the minimum possible value of $A$ ?

Proposed by Ray Li

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$ . Prove that $a^a b^b c^c \ge 1$ .

Proposed by Evan Chen

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$ . Prove that there is a positive integer $N$ such that $$N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right)$$ and for each $i = 1, 2, ..., 2013$ , there does not exist $a \in A_i$ such that $m_i$ divides $N-a$ .

Proposed by Victor Wang

Triangle $ABC$ is inscribed in circle $\omega$ . A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$ , respectively. Segments $BR$ and $CS$ meet at $L$ , and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$ , respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$ . Prove that if $BE = BR$ , then $\angle ELK = \tfrac{1}{2} \angle BCD$ .

Proposed by Evan Chen

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$ , the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$ ?

Proposed by Andre Arslan

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$ , there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$ . Is it possible that for some integer $n \ge 0$ , there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$ ?

Proposed by David Yang