2004 JBMO Problems/Problem 3

Let 4x + 3y = a^2 3x + 4y = b^2

Then 7(x + y) = a^2 + b^2

But any perfect square can only be congruent to 0,1,2, or 4 modulo 7

Thus

a = 7p

b = 7q

x + y = 7(p^2 + q^2)

x + y is congruent to 0 mod 7.

4x + 3y = a^2 = 49p^2

3(x+y) + x = 49p^2

x is congruent to 0 mod 7

Similarly, we get y is congruent to 0 mod 7