RMO practice 3

If $a,b,c$ are real numbers such that, $0<a,b,c<1,a+b+c=2$ .Then prove that, $\prod_{cyc}\dfrac{a}{1-a} \geq 8.$ .I have two solutions,one of which has been provided by Surya Prakash.

Easy Math Editor

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`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`	$\sum_{i=1}^3$
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Comments

@Harsh Shrivastava @Nihar Mahajan @Dev Sharma @Lakshya Sinha

Adarsh Kumar - 5 years, 7 months ago

Any hint?

Department 8 - 5 years, 7 months ago

Substitution.

Adarsh Kumar - 5 years, 7 months ago

Substitute a= 1-p ; b=1-q and c=1-r.

Now it can be solved easily.

Harsh Shrivastava - 5 years, 7 months ago

Yup!That is one of the correct ways!

Adarsh Kumar - 5 years, 6 months ago

After the substitution, multipling the terms and simplyfy using $p+q+r=1$ , we get $pq+qr+rp \ge 9pqr$ . Then divide the both sides of theb equation by $pqr$ . The inequality will hold by AM-HM inequality.

A Former Brilliant Member - 5 years, 6 months ago

@A Former Brilliant Member – Yup!That is correct!

Adarsh Kumar - 5 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}`	$\boxed{123}$