OSN Proposal: Inequality

I proposed this problem to OSN 2009 but it was not selected.

For $a,b,c$ positive numbers such that $a+b+c=3$, prove that:

$2(a^{11} + b^{11} + c^{11}) + 3a^3b^3c^3(ab+bc+ca) \geq 5(a^4b^4 + b^4c^4 + c^4a^4)$

OSN 2009 Problem 4

I proposed this problem to OSN (Indonesian Science Olympiad) 2009 committee and it was selected as problem #4.

There are 7 cities that are connected by a railroad network. Each railroad segment connects two cities, and each city is connected by at least 3 segments. Prove that there is a route that visits exactly 4 cities, visits them exactly once, and goes back to the city of origin.

(Example: A-B-C-D-A)