If a,b,c are unequal and positive ,show that bc/(b+c ) +ac/(c+a) +ab/(a+b) is less than1/2(a+b+c)

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Solution

(a + b)/2 > 2/(1/a + 1/b)
(b + c)/2 > 2/(1/b + 1/c) and
(c + a)/2 > 2/(1/c + 1/a)

Adding,
a + b + c > 2 * [ab/(a+b) + bc/(b+c) + ca/(c+a)]
=> bc/(b+c) + ca/(c+a) + ab/(a+b) < (a+b+c)/2

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