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ax^2+by^2+cz^2>=0 if x+y+z=0 , a+b+c>= 0, ab+bc+ca >=0

Source: 2017 Swedish Mathematical Competition p6

May 1, 2021
algebrainequalities

Problem Statement

Let a,b,c,x,y,za,b,c,x,y,z be real numbers such that x+y+z=0x+y+z=0, a+b+c≥0a+b+c\geq 0, ab+bc+ca≥0ab+bc+ca \ge 0. Prove that ax2+by2+cz2≥0 ax^2+by^2+cz^2\ge 0
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