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Inequalities

Source: 2011 IrMO Paper 2 Problem 4

February 8, 2018

inequalities

Problem Statement

Suppose that

x,y

and

z

are positive numbers such that

1=2xyz+xy+yz+zx

Prove that (i)

\frac{3}{4}\le xy+yz+zx<1

(ii)

xyz\le \frac{1}{8}

Using (i) or otherwise, deduce that

x+y+z\ge \frac{3}{2}

and derive the case of equality.

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