An Infinite Number of Angles
Show there exists an infinite number of angles between 0 and
p/2 whose sine and cosine are both rational.Hint: Consider integer solutions to x2+y2=z2 (known as Pythagorean triples).
November 1999
Find the Smallest Natural Number
Part 1. Find the smallest natural number that can be expressed as the sum of the squares of two distinct natural numbers in two different ways. (That would entail altogether, four distinct natural numbers that are squared.)
Part 2. Find the smallest natural number that can be expressed as the sum of the squares of two, not necessarily distinct, natural numbers in two different ways.