IR-MO-R2 2013

Find all pairs $(a,b)$ of positive integers for which $\gcd(a,b)=1$ , and $\frac{a}{b}=\overline{b.a}$ . (For example, if $a=92$ and $b=13$ , then $b/a=13.92$ )

Let $n$ be a natural number and suppose that $w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a Perfect Setif we can achieve all of the $1,2, \ldots, W$ weights with sums of $w_1, w_2, \ldots , w_n$ , where $W=\sum_{i=1}^n w_i$ . Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$ . Suppose that the altitude drawn from $A$ intersects the circle at $N$ . Draw two lines through circumcenter $O$ of $ABC$ paralell to $MB$ and $MC$ , which intersect $AB$ and $AC$ at $K$ and $L$ , respectively. Prove that $NK=NL$ .