Problem

Source: Kyiv City MO 2022 Round 1, Problem 9.2

Tags: algebra, inequalities



For any reals $x, y$, show the following inequality: $$\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20$$ (Proposed by Bogdan Rublov)