This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .

Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.

Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold

- bulleted - list

bulleted

list

1. numbered 2. list

numbered

list

Note: you must add a full line of space before and after lists for them to show up correctly

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

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestCan you explain what is sym,pls.

Log in to reply

It means symmetry

You can write this thing as

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

Log in to reply

Would it not be $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a}$ only!

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Proof:

NOTE: All sums are symmetric in a,b,c.$\sum \frac{a^2}{b} \geq 3\frac{\sum a^3}{\sum a^2}$

First we can see that $3\sum a^3 \geq 9abc$, by

AM-GMThus we need to show $\sum \frac{a^2}{b} \times \sum a^2 \geq 9abc$.

Also, by

AM-GM$\sum a^2 \geq \sum ab$ Hence $\sum \frac{a^2}{b} \times \sum a^2 \geq \sum \frac{a^2}{b} \times \sum ab$

Thus by Cauchy-Schwarz Inequality :-

$\sum \frac{a^2}{b} \times \sum a^2 \geq \sum \frac{a^2}{b} \times \sum ab \geq (a^\frac{3}{2} + b^\frac{3}{2} + c^\frac{3}{2})^2$

and $(a^\frac{3}{2} + b^\frac{3}{2} + c^\frac{3}{2})^2 \geq 9abc$ by

AM-GMThus $\sum \frac{a^2}{b} \times \sum a^2 \geq \sum \frac{a^2}{b} \times \sum ab \geq 9abc$ ...Q.E.D

If you face any problem in understanding my language or concept, feel free to ask.Log in to reply

It's sum over all permutation of variables

Log in to reply