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# 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.

2 years ago

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

Now it can be solved easily.

- 2 years ago

Yup!That is one of the correct ways!

- 2 years 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.

- 2 years ago

Yup!That is correct!

- 2 years ago

- 2 years ago

Any hint?

- 2 years ago

Substitution.

- 2 years ago