Problem
ALG-B2-M07-P002 Can we normalize
#2
★★☆☆☆ Level 2 of 5
For each inequality, decide whether one may set \(a+b+c=1\) without an extra condition: \(a^2+b^2+c^2\ge ab+bc+ca\); \(a^2+b^2+c^2\ge1\); \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac32\).
Hint. Compare the degrees of the left and right sides.
The first is homogeneous of degree \(2\), so normalization is allowed. The second is not homogeneous: the degrees are \(2\) and \(0\), so normalization is not allowed. The third has degree \(0\), so normalization is allowed for positive variables.
This problem catches the most common mistake in the module.