2000 IMO Problems/Problem 2

Problem

Let $a, b, c$ be positive real numbers with $abc=1$ . Show that

$\left( a-1+\frac{1}{b} \right)\left( b-1+\frac{1}{c} \right)\left( c-1+\frac{1}{a} \right) \le 1$

Solution

There exist positive reals $x$ , $y$ , $z$ such that $a = \frac{x}{y}$ , $b = \frac{y}{z}$ , $c = \frac{z}{x}$ . The inequality then rewrites as $\left(\frac{x-y+z}{y}\right)\left(\frac{y-z+x}{z}\right)\left(\frac{z-x+y}{x}\right)\leq 1$ or $(x-y+z)(y-z+x)(z-x+y)\leq xyz.$ Set $p=x-y+z$ , $q=y-z+x$ , $r=z-x+y$ , we get $8pqr\leq(p+q)(q+r)(r+p).$ Since at most one of $p,q,r$ can be negative (if 2 or more are negative, then one of $a,b,c$ will become negative), for all positive we apply AM-GM, for one negative we have $LHS<0<RHS$ .

2000 IMO (Problems) • Resources
Preceded by Problem 1	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 3
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2000 IMO Problems/Problem 2

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