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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.

Note by Adarsh Kumar
1 year ago

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

Now it can be solved easily. Harsh Shrivastava · 1 year ago

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@Harsh Shrivastava Yup!That is one of the correct ways! Adarsh Kumar · 1 year ago

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@Adarsh Kumar 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. Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar Yup!That is correct! Adarsh Kumar · 1 year ago

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@Adarsh Kumar Any hint? Lakshya Sinha · 1 year ago

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@Lakshya Sinha Substitution. Adarsh Kumar · 1 year ago

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