55rd Ukrainian National Mathematical Olympiad - Fourth Round

Question

It is known that the arithmetic average of the numbers *a*, *b* is equal to the number *c*, so c = 12(a+b), and that the geometric average number of *a*, *c* is equal to the number *b*, so b = ac. Is it necessary that numbers *a*, *b*, *c* are equal?

Accepted Answer

Let's rewrite the equality b^2 = ac using c = a+b2: b^2 = a a+b2 2b^2 = ba + a^2 (b-a)(2b+a) = 0. Now let's denote, for example, b=2, which means a=-4 and c=-1, hence we receive three different numbers satisfying the conditions.