SMO 2013 Open Round 2

Let $a_1,a_2,\ldots$ be a sequence of integers defined recursively by $a_1=2013$ and, for $n\ge1$, let $a_{n+1}$ be 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 $D,E,F$ be the midpoints of $BC,CA,AB$, respectively. Construct a circle centred at the orthocentre of triangle $ABC$ such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to meet the circle at $P$, $FD$ to meet it at $Q$, and $DE$ to meet it at $R$. Show that $AP=BQ=CR$.

Let $N$ be a positive integer. Prove that there exists a positive integer $n$ such that
$$n^{2013}-n^{20}+n^{13}-2013$$
has at least $N$ distinct prime factors.

Let $F$ be a finite nonempty set of integers and let $n$ be a positive integer. Suppose that:

- Every $x\in F$ can be written as $x=y+z$ for some $y,z\in F$.
- If $1\le k\le n$ and $x_1,\ldots,x_k\in F$, then $x_1+\cdots+x_k\ne0$.

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

Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.