SELECTION TESTS FOR THE 2019 BMO AND IMO

Question

Determine the largest value the expression _1 i < j 4 (x_i + x_j) x_i x_j may achieve, as x_1, x_2, x_3, x_4 run through the non-negative real numbers that add up to 1. Determine also the x_i at which the maximum is achieved.

Accepted Answer

The required maximum is 3/4 and is achieved if and only if the x_i are all equal to 1/4. To prove this, use the binomial expansion of (x_i - x_j)^4 to write 4(x_i + x_j)x_i x_j = x_i^2 + 6x_i x_j + x_j^2 - (x_i - x_j)^4, aligned 4 _1 i < j 4 (x_i + x_j) x_i x_j &= 3(x_1 + x_2 + x_3 + x_4)^2 - _1 i < j 4 (x_i - x_j)^4 & 3(x_1 + x_2 + x_3 + x_4)^2 = 3; aligned $ clearly, equality holds if and only if the x_i are all equal, and the constraint forces them all equal to 1/4$.