×

# 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.

1 year, 2 months ago

Sort by:

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

Now it can be solved easily. · 1 year, 2 months ago

Yup!That is one of the correct ways! · 1 year, 2 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. · 1 year, 2 months ago

Yup!That is correct! · 1 year, 2 months ago

Any hint? · 1 year, 2 months ago

Substitution. · 1 year, 2 months ago

×