IMO 2020 #2

Discussion on International Mathematical Olympiad (IMO)
User avatar
FuadAlAlam
Posts:30
Joined:Wed Sep 16, 2020 11:10 am
Location:Dhaka, Bangladesh
Contact:
IMO 2020 #2

Unread post by FuadAlAlam » Fri Dec 04, 2020 12:40 pm

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

Dustan
Posts:71
Joined:Mon Aug 17, 2020 10:02 pm

Re: IMO 2020 #2

Unread post by Dustan » Thu Dec 10, 2020 10:54 pm

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

Post Reply