10th Eötvös Competition Problems 1903

1.  Prove that 2p-1(2p - 1) is perfect when 2p - 1 is prime. [A perfect number equals the sum of its (positive) divisors, excluding the number itself.] 2.  α and β are real and a = sin α, b = sin β, c = sin(α+β). Find a polynomial p(x, y, z) with integer coefficients, such that p(a, b, c) = 0. Find all values of (a, b) for which there are less than four