Problem

Source: Romanian TST 1978, Day 3, P6

Tags: algebra



a) Prove that $ 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \} $ b) Prove that there exist three positive rational numbers $ a,b,c $ such that the expression $ E(x,y,z):=xa+yb+zc $ vanishes for infinitely many integer triples $ (x,y,z), $ but it doesn´t get arbitrarily close to $ 0. $