Question 1
In the plane there is an infinite chessboard. For any pair of positive integers $m$ and $n$, consider a right-angled triangle with vertices at lattice points and whose legs, of lengths $m$ and $n$, lie along edges of the squares. Let $S_1$ be the total area of the black part of the triangle and $S_2$ be the total area of the white part. Let $f(m,n) = | S_1 - S_2 |$.
(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.
(b) Prove that $f(m,n) \leq \frac 12 \max \{m,n\}$ for all $m$ and $n$.
(c) Show that there is no constant $C$ such that $f(m,n) \lt C$ for all $m$ and $n$.
(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.
(b) Prove that $f(m,n) \leq \frac 12 \max \{m,n\}$ for all $m$ and $n$.
(c) Show that there is no constant $C$ such that $f(m,n) \lt C$ for all $m$ and $n$.