MathDB

System of equations

Source: Brazilian University Olympics 2019

November 18, 2019

calculussystem of equationscollege contestsalgebraBrazilian Undergrad MOBrazilian Undergrad MO 2019

Problem Statement

Let

a,b,c

be constants and

a,b,c

are positive real numbers. Prove that the equations

2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}

x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}

x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}

have exactly one real solution

(x,y,z)

with

x,y,z \geq 0