×

# 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

Note by Avn Bha
3 years, 4 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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 ]

- 3 years, 4 months ago

Nice reducing it to Nesbitt's Inequality.

Staff - 3 years, 4 months ago

:D I am getting into the mathematician grooves.

- 3 years, 3 months ago