Problem
ALG-B2-M05-P024 Reciprocal squares
#24
★★★★★ Level 5 of 5
Let \(x_1,\ldots,x_n>0\) and \(x_1+\cdots+x_n=n\). Prove \[\frac1{x_1^2}+\frac1{x_2^2}+\cdots+\frac1{x_n^2}\ge n.\]
Hint. The function \(f(x)=1/x^2\) is convex for \(x>0\).
By Jensen, \[\frac1n\sum_{i=1}^n\frac1{x_i^2}\ge \frac{1}{\left((x_1+\cdots+x_n)/n\right)^2}=1.\] Multiplying by \(n\), we get the result.
Final problem on the general form and domain control.