If ( a+b+c )( 1/a + 1/b + 1/c ) = 1 , prove that either one of a+b or b+c or c+a is equal to 0

Answer 1

We know that if X*Y*Z = 0, then X or Y or Z must be zero.

So, if (a+b)*(b+c)*(a+c) = 0, then (a+b) or (b+c) or (a+c) must be zero.

If we can show that

(a+b+c)(1/a + 1/b + 1/c ) = 1

and

(a+b)*(b+c)*(a+c) = 0

turn into the same expression when we apply some algebraic manipulations, we are done.

And, indeed, it turns out that both equations can be 'simplified' into:

2abc + a²b + a²c + ab² + ac² + b²c + bc² = 0