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por biank »
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Multiplicando por $a + b + c$ obtenemos $$
\frac{a(a + b + c)}{b + c} + \frac{b(a + b + c)}{c + a} + \frac{c(a + b + c)}{a + b} = a + b + c \\
\Leftrightarrow \frac{a^2 + a(b + c)}{b + c} + \frac{b^2 + b(c + a)}{c + a} + \frac{c^2 + c(a + b)}{a + b} = a + b + c \\
\Leftrightarrow \frac{a^2}{b + c} + a + \frac{b^2}{c + a} + b + \frac{c^2}{a + b} + c = a + b + c \\
\Leftrightarrow \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} = 0 \quad \blacksquare
$$
