Problem

Source: ELMO Shortlist 2012, C9

Tags: analytic geometry, geometry, geometric transformation, combinatorics, number theory



For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? David Yang.