Problem
ALG-B2-M05-P007 Weighted square
#7
★★★☆☆ Level 3 of 5
Let \(0\le p\le1\). Prove \[p x^2+(1-p)y^2\ge(px+(1-p)y)^2.\]
Hint. This is Jensen for \(x^2\) with weights \(p\) and \(1-p\).
Since \(f(t)=t^2\) is convex, \[p f(x)+(1-p)f(y)\ge f(px+(1-p)y).\] This is exactly the required inequality.
Introduces weights without extra machinery.