SMO 2013

Let $a_1$ , $a_2$ , ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$ , $a_{n+1}$ is the sum of the $2013$ -th powers of the digits of $a_n$ . Do there exist distinct positive integers $i$ , $j$ such that $a_i=a_j$ ?

Let $ABC$ be an acute-angled triangle and let $D$ , $E$ , and $F$ be the midpoints of $BC$ , $CA$ , and $AB$ respectively. Construct a circle, centered at the orthocenter of triangle $ABC$ , such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to intersect the circle at $P$ , $FD$ to intersect the circle at $Q$ and $DE$ to intersect the circle at $R$ . Show that $AP=BQ=CR$ .

Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that $\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$ , $z \in F$ ; $\bullet$ If $1 \leq k \leq n$ and $x_1$ , ..., $x_k \in F$ , then $x_1+\cdots+x_k \neq 0$ .

Show that $F$ has at least $2n+2$ elements.