This problem appeared in the pre RMO test in my school How shall we solve this -

PROVE THAT

\(\frac{a^{2} +1}{b+c} +\frac{b^{2} +1}{a+c} +\frac{c^{2} +1}{b+a}\) >3 OR =3

This problem appeared in the pre RMO test in my school How shall we solve this -

PROVE THAT

\(\frac{a^{2} +1}{b+c} +\frac{b^{2} +1}{a+c} +\frac{c^{2} +1}{b+a}\) >3 OR =3

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TopNewestWithout loss of generality, Take a>b>c.

a^{2} + 1 > 2

a - Similarly for all So I substitute 2a, 2b, 2c in the numerators The new sum acquired is lower than the original.Now take 2 common and send it to the other side of the equation

This now reduces to Nesbitt's Inequality. Nesbitt's Inequality.

Proved.

[By the way even I am preparing for RMO, I think these inequalities are basic- -Nesbitt's Inequality -RMS>AM>GM>HM -Chebycheff Inequality -Rearrangement Inequality -Triangle Inequality ] – Dhruv Singh · 2 years, 2 months ago

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– Calvin Lin Staff · 2 years, 2 months ago

Nice reducing it to Nesbitt's Inequality.Log in to reply

– Dhruv Singh · 2 years, 1 month ago

:D I am getting into the mathematician grooves.Log in to reply