Problem 3

Let two equal regular

n

-gons

S

and

T

be located in the plane such that their intersection is a

2n

-gon (

n \ge 3

). The sides of

S

are colored red and the sides of

T

blue. Prove that the sum of the lengths of the blue sides of

S\cap T

equals the sum of the lengths of its red sides.

Step 6 of 6: Use the strict perimeter inequality

x+y<na\implies x-y=0\implies x=y.

Detailed analysis

Since x+y is strictly less than na, the second factor cannot vanish. Hence x=y, exactly the required equality of the blue and red sums.