Problem 1

In the plane, points with integer coordinates are vertices of unit squares, colored alternately black and white as on a chessboard. For positive integers

m,n

, consider a right triangle with integer-coordinate vertices and legs of lengths

m,n

along square edges. Let

S_1

be the black area and

S_2

the white area. Define

f(m,n)=|S_1-S_2|

. (a) Calculate

f(m,n)

when

m,n

are both even or both odd. (b) Prove

f(m,n)\le\max(m,n)/2

. (c) Show that no constant

C

satisfies

f(m,n)<C

for all

m,n

Step 2 of 3: Handle opposite parity

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In plain words

Handle opposite parity

f(m,n)\le\dfrac{\max(m,n)}2

Detailed analysis

Suppose $m$ is odd and $n$ even. Increase the odd leg to $m+1$ ; the enlarged triangle has equal legs even and hence zero excess. The added triangle has base $1$ and height $n$ , area $n/2$ , so the original excess is at most $n/2\le\max(m,n)/2$ .