Problem
ALG-B2-M12-P015 Set 4. Three reciprocal squares
#15
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\frac1{1+a^2}+\frac1{1+b^2}+\frac1{1+c^2}\ge\frac32.\]
Hint. Use \(\frac1{1+x^2}\ge\frac{2-x}{2}\).
The difference \(\frac1{1+x^2}-\frac{2-x}{2}\) equals \(\frac{x(x-1)^2}{2(1+x^2)}\ge0\). Sum for \(a,b,c\) and use \(a+b+c=3\).
Mock set 4: the problem is intended for independent method selection without an explicit cue in the statement.