Problem
ALG-B1-M07-P007 Two Cauchy fractions
#7
★★☆☆☆ Level 2 of 5
Let \(x,y>0\). Prove that \(\frac{x^2}{y}+\frac{y^2}{x}\ge x+y\).
Use Engel form: \(\sum \frac{u_i^2}{v_i}\ge\frac{(\sum u_i)^2}{\sum v_i}\).
By Cauchy, \(\frac{x^2}{y}+\frac{y^2}{x}\ge\frac{(x+y)^2}{x+y}=x+y\). Equality holds when \(x=y\).
This is the first independent problem on Engel form.