Let $a_1,...,a_n,b_1,..,b_n$ be integers satisfying $a_1> a_2 \geq a_3 \geq a_4 ... \geq a_n=1$ and $b_1=1$.
Suppose $a_1b_1+....+a_nb_n \geq 0$ then is it true that $a_1^2b_1+...+a_n^2b_n \geq 0$.
My try: I checked this for a few examples and it seems to be true but I don't see how to prove this. Any hints would be appreciated. Thank you.