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 1 of 3: Compute equal-parity cases

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In plain words

Compute equal-parity cases

f(m,n)=0\text{ or }1/2

Detailed analysis

Complete the triangle to a rectangle and rotate it by $180^\circ$ about the midpoint of its hypotenuse. If $m,n$ are even, the midpoint is a lattice point and the two halves have equal black and white areas, so $f(m,n)=0$ . If both are odd, the midpoint is the center of a unit square and the rectangle has color excess $1$ , so each half has excess $1/2$ .