TargetIIT

[im]http://codecogs.izyba.com/gif.latex?if%20%5C%20a+b+c%20%3D1%20%2C%3B%20a%2Cb%2C%20c%20%5Crightarrow%20R%5E+%20%5C%5C%20prove%20%5C%20that%20%5C%20%5C%5C%20%5C%5C%20%5Csum_%7Bcyclic%7D%7B%7D%20%7B%5Cfrac%7Ba%7D%7B1+bc%7D%7D%5Cgeq%20%5Cfrac%7B9%7D%7B10%7D[/im]

I assume u r looking for a solution:

Using Cauchy Schwarz

[im]http://codecogs.izyba.com/gif.latex?%5Csum%20%5Cfrac%7Ba%7D%7B1+bc%7D%20%3D%20%5Csum%20%5Cfrac%7Ba%5E2%7D%7Ba+abc%7D%20%5Cge%20%5Cfrac%7B%28a+b+c%29%5E2%7D%7B%5Csum%20a%20+%203abc%7D%20%3D%20%5Cfrac%7B1%7D%7B1+3abc%7D[/im]

From AM-GM [im]http://codecogs.izyba.com/gif.latex?a+b+c%20%3D1%20%5CRightarrow%203abc%20%5Cle%20%5Cfrac%7B1%7D%7B9%7D%20%5CRightarrow%20%5Cfrac%7B1%7D%7B1+3abc%7D%20%5Cge%20%5Cfrac%7B9%7D%7B10%7D[/im]