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IMOC 2017 G6 ((x + y)^2+(y + z)^2 +(z + x)^2)/(x + y + z)<= a + b + c

Source: https://artofproblemsolving.com/community/c6h1740077p11309077

March 20, 2020
geometrygeometric inequality

Problem Statement

A point PP lies inside △ABC\vartriangle ABC such that the values of areas of △PAB,△PBC,△PCA\vartriangle PAB, \vartriangle PBC, \vartriangle PCA can form a triangle. Let BC=a,CA=b,AB=c,PA=x,PB=y,PC=zBC = a,CA = b,AB = c, PA = x,PB = y, PC = z, prove that (x+y)2+(y+z)2+(z+x)2x+y+z≤a+b+c\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c
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