Problem
ALG-B2-M12-P002 Set 1. Three identical traps
#2
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[\frac{a}{a^2+a+1}+\frac{b}{b^2+b+1}+\frac{c}{c^2+c+1}\le1.\]
Hint. Bound each term from above by \(\frac13\).
For \(x>0\), \(\frac{x}{x^2+x+1}\le\frac13\), because \(x^2-2x+1=(x-1)^2\ge0\). The sum of three such fractions is at most \(1\).
Mock set 1: the problem is intended for independent method selection without an explicit cue in the statement.