# IMO 2020 Q2 #

The real numbers `a`

, `b`

, `c`

, `d`

are such that `a ≥ b ≥ c ≥ d > 0`

and `a + b + c + d = 1`

.
Prove that `(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
`1`

by `(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.