RMO

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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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Without loss of generality, Take a>b>c.

a^{2} + 1 > 2a - 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 - 6 years, 7 months ago

Nice reducing it to Nesbitt's Inequality.

Calvin Lin Staff - 6 years, 7 months ago

:D I am getting into the mathematician grooves.

Dhruv Singh - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$