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The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that
\[(a+2b+3c+4d)a^ab^bc^cd^d<1\]

Proposed by Stijn Cambie, Belgium

$(a+2b+3c+4d)(a^2+b^2+c^2+d^2) \leq (a+b+c+d)^3$
Proving this is equivalent to the main problem.