33rd Hellenic Mathematical Olympiad

Question

The real numbers x, y, z, with x z, are mutually different and nonzero and they satisfy the following equations: (x+y)^2 + (2-xy) = 9, (y+z)^2 - (3+yz) = 4. Determine the value of the expression A = ( xy + y^2x^2 + z^3x^2y ) ( yz + z^2y^2 + x^3y^2z ) ( zx + x^2z^2 + y^3z^2x ).

Accepted Answer

The given equalities can be written: x^2 + y^2 + xy = 7, (1) y^2 + z^2 + yz = 7, (2) By subtraction we get: x^2 - z^2 + xy - yz = 0 (x-z)(x+z) + y(x-z) = 0 (x-z)(x+z+y) = 0. Since x - z 0, we get: x + y + z = 0. (3) Then with simple manipulations for the factors of A we find A = ( x^3 + y^3 + z^3xyz )^3 (4) From (3), using Euler's identity, we find x^3 + y^3 + z^3 = 3xyz (5) Hence from (4) and (5) we get A = 27