RS-SMO 2016

Let $n$ be a positive integer. Let $f$ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$ , $f (1, 1)=n$ , and $f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}]$ for positive integers $i, j$ such that $i*j>1$ . Find the number of pairs $(i,j)$ such that $f (i, j)$ is an odd number.( $[x]$ is the floor function).

Let $ABC$ be a triangle, and $I$ the incenter, $M$ midpoint of $BC$ , $D$ the touch point of incircle and $BC$ . Prove that perpendiculars from $M, D, A$ to $AI, IM, BC$ respectively are concurrent

There are $2n-1$ twoelement subsets of set $1,2,...,n$ . Prove that one can choose $n$ out of these such that their union contains no more than $\frac{2}{3}n+1$ elements.