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3/2 <= sum (b + c) /( b + c + 2a) <= 5/3 for a,b,c triangle sidelenghts

Source: 2016 Romanian NMO grade VIII P3

September 3, 2024
geometrygeometric inequalityGeometric Inequalities

Problem Statement

If a,ba, b and cc are the length of the sides of a triangle, show that 32≤b+cb+c+2a+a+ca+c+2b+a+ba+b+2c≤53.\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.
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