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Difficult inequality

Source: Bosnia and Herzegovina TST 2014 day 1 problem 2

May 10, 2014
inequalitiesinequalities unsolved

Problem Statement

Let aa ,bb and cc be distinct real numbers. a)a) Determine value of 1+aba−b⋅1+bcb−c+1+bcb−c⋅1+cac−a+1+cac−a⋅1+aba−b \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b}
b)b) Determine value of 1−aba−b⋅1−bcb−c+1−bcb−c⋅1−cac−a+1−cac−a⋅1−aba−b \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b}
c)c) Prove the following ineqaulity 1+a2b2(a−b)2+1+b2c2(b−c)2+1+c2a2(c−a)2≥32 \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2}
When does eqaulity holds?
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