Question 1
(BLR) IMO1. An infinite square grid is colored in the chessboard pattern. For any pair of positive integers $m,n$ consider a right-angled triangle whose vertices are grid points and whose legs, of lengths $m$ and $n$, run along the lines of the grid. Let $S_b$ be the total area of the black part of the triangle and $S_w$ the total area of its white part. Define the function $f(m,n)=|S_b-S_w|$.
(a) Calculate $f(m,n)$ for all $m,n$ that have the same parity.
(b) Prove that $f(m,n)\le\frac12\max(m,n)$.
(c) Show that $f(m,n)$ is not bounded from above.
(a) Calculate $f(m,n)$ for all $m,n$ that have the same parity.
(b) Prove that $f(m,n)\le\frac12\max(m,n)$.
(c) Show that $f(m,n)$ is not bounded from above.