2013 Singapore MO Open

2nd Round

1

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 $a_i=a_j$?

2

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 $AP=BQ=CR$.

3

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

4

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.

5

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