It was a beautiful day in the Winter of 1994. I had slept in because it was Saturday, but my older brother, Nikos, was not so lucky. He had just come back from a regional math competition named after the Greek philosopher Θαλής (Thales of Miletus) and he did not look happy. I asked him how he did, but he did not answer; he just reached into his backpack and took out the paper that contained the problems from the competition. Looking back, what I did next defines much of how I approach life since that day: I took the paper and started solving the problems one after the other until I was done. It took me three days – the competition lasted three hours.

Problem 1: If $f(0,y) = y+1, , f(x+1,0) = f(x,1)$ and $f(x+1,y+1) = f(x, f(x+1,y))$, for all non-negative integers $x,y$, find $f(4,2009)$.