MathDB

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Part of 2019 Brazil Undergrad MO

Problems(1)

System of equations

Source: Brazilian University Olympics 2019

11/18/2019
Let a,b,ca,b,c be constants and a,b,ca,b,c are positive real numbers. Prove that the equations 2x+y+z=c2+z2+c2+y22x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2} x+2y+z=b2+x2+b2+z2x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2} x+y+2z=a2+x2+a2+y2x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2} have exactly one real solution (x,y,z)(x,y,z) with x,y,z≥0x,y,z \geq 0.
calculussystem of equationscollege contestsalgebraBrazilian Undergrad MOBrazilian Undergrad MO 2019