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Question

Let a, b, c be the sides of a triangle. The least value of expressionE=ab+ca+bc+ab+ca+bc is ___

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Solution

Put x=b+ca, y=c+ab, z=a+bc, then x, y, z>0 and 2a=y+z, 2b=z+x, 2c=x+yUsing A.M.G.M., we getE=12(y+zx+z+xy+x+yz)3(yx.zx.zy.xy.xz.yz)1/6=3


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