National Math Olympiad

Question

Let a and b be real numbers such that |a| |b| and a+ba-b + a-ba+b = 6. Find the value of the expression a^3 + b^3a^3 - b^3 + a^3 - b^3a^3 + b^3.

Accepted Answer

The equality 6 = a+ba-b + a-ba+b = 2a^2 + 2b^2a^2 - b^2 implies 6a^2 - 6b^2 = 2a^2 + 2b^2, or 4a^2 = 8b^2. From here we get a = b2. Now, we can conclude that a^3 + b^3a^3 - b^3 + a^3 - b^3a^3 + b^3 = 2a^6 + 2b^6a^6 - b^6 = (2 2^3 + 2)b^6(2^3 - 1)b^6 = 187.