A collection of Math Olympiad problems

(1) Find the smallest natural number which is a multiple of 2009 and whose sum of
(decimal) digits equals 2009.

(2) Find all integer solutions of the equation 3^x - 5^y = z^2

(3) Solve in integers the equation 12^x + y^4 = 2008^z

(4) Determine all pairs of natural numbers (x, n) that satisfy the equation

x^3 + 2x + 1 = 2^n

(5) Determine all pairs (x, y) of integers such that

1 + 2^x + 2^(2x+1) = y^2

(6) Find all primes p such that p^2 – p + 1 is a perfect cube.

(7) Let a, b, c be positive real numbers. Prove the inequality

(a^2 / b) + (b^2 / c) + (c^2 / a) >= a + b + c + 4(a – b)^2 / (a + b + c)

When does equality occur ?