IMO 2020 Q2 #
The real numbers
d are such that
a ≥ b ≥ c ≥ d > 0 and
a + b + c + d = 1.
(a + 2b + 3c + 4d) a^a b^b c^c d^d < 1.
A solution is to eliminate the powers using weighted AM-GM and replace
(a+b+c+d)^3, leaving a homogeneous inequality that can be
proved in many ways by expanding, rearranging and comparing individual
terms. The version here using factors such as
a+3b+3c+3d is from
the official solutions.