2000 IMO Problems/Problem 2


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

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


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 $LHS<0<RHS$.

See Also

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