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sum a^2/(b + c) =? if sum a/(b + c)=1

Source: Switzerland - 2015 Swiss MO Final Round p7

January 14, 2023

algebraSum

Problem Statement

Let

a, b, c

be real numbers such that:

\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1

Determine all values which the following expression can take :

\frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b}.