Maths Olympiad Prep

Library / /1 of 3

Algebra Difficulty 4.6 AIME Prove it North Macedonia

Let aa, bb and cc be positive real numbers such that abc=1abc = 1. Prove the inequality
(a5+a4+a3+a2+a+1)(b5+b4+b3+b2+b+1)(c5+c4+c3+c2+c+1)≥8(a2+a+1)(b2+b+1)(c2+c+1) (a^5 + a^4 + a^3 + a^2 + a + 1)(b^5 + b^4 + b^3 + b^2 + b + 1)(c^5 + c^4 + c^3 + c^2 + c + 1) \geq 8(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)

Solution

By factorizing we get
(a5+a4+a3+a2+a+1)=(a3+1)(a2+a+1). (a^5 + a^4 + a^3 + a^2 + a + 1) = (a^3 + 1)(a^2 + a + 1).
We apply same thing to the other terms and simply to get (a3+1)(b3+1)(c3+1)≥8(a^3 + 1)(b^3 + 1)(c^3 + 1) \geq 8.
By AM≥GMAM \geq GM we have
a3+1≥2a3 a^3 + 1 \geq 2\sqrt{a^3}
b3+1≥2b3 b^3 + 1 \geq 2\sqrt{b^3}
c3+1≥2c3 c^3 + 1 \geq 2\sqrt{c^3}
(a3+1)(b3+1)(c3+1)≥8a3b3c3 (a^3 + 1)(b^3 + 1)(c^3 + 1) \geq 8\sqrt{a^3 b^3 c^3}
Equality holds if and only if a=b=c=1a = b = c = 1.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.