ELMO 2015

Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$ . Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$ .

Proposed by Sam Korsky

Let $m$ , $n$ , and $x$ be positive integers. Prove that $$\sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right).$$ Proposed by Yang Liu

Let $\omega$ be a circle and $C$ a point outside it; distinct points $A$ and $B$ are selected on $\omega$ so that $\overline{CA}$ and $\overline{CB}$ are tangent to $\omega$ . Let $X$ be the reflection of $A$ across the point $B$ , and denote by $\gamma$ the circumcircle of triangle $BXC$ . Suppose $\gamma$ and $\omega$ meet at $D \neq B$ and line $CD$ intersects $\omega$ at $E \neq D$ . Prove that line $EX$ is tangent to the circle $\gamma$ .

Proposed by David Stoner

Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$ , the number $2^{2^n}+a$ is not prime.

Proposed by Jack Gurev

Let $m, n, k > 1$ be positive integers. For a set $S$ of positive integers, define $S(i,j)$ for $i<j$ to be the number of elements in $S$ strictly between $i$ and $j$ . We say two sets $(X,Y)$ are a fat pair if $$X(i,j)\equiv Y(i,j) \pmod{n}$$ for every $i,j \in X \cap Y$ . (In particular, if $\left\lvert X \cap Y \right\rvert < 2$ then $(X,Y)$ is fat.)

If there are $m$ distinct sets of $k$ positive integers such that no two form a fat pair, show that $m<n^{k-1}$ .

Proposed by Allen Liu