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Regional Olympiad - FBH 2017 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

September 19, 2018
identityreal numbersalgebra

Problem Statement

Let aa, bb and cc be real numbers such that abc(a+b)(b+c)(c+a)≠0abc(a+b)(b+c)(c+a)\neq0 and (a+b+c)(1a+1b+1c)=10071008(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008} Prove that ab(a+c)(b+c)+bc(b+a)(c+a)+ca(c+b)(a+b)=2017\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017
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