2008 IMO Problems/Problem 2

Problem 2

(i) If $x$ , $y$ and $z$ are three real numbers, all different from $1$ , such that $xyz = 1$ , then prove that $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$ . (With the $\sum$ sign for cyclic summation, this inequality could be rewritten as $\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1$ .)

(ii) Prove that equality is achieved for infinitely many triples of rational numbers $x$ , $y$ and $z$ .

Solution

Consider the transormation $f:\mathbb{R}/\{1\} \rightarrow \mathbb{R}/\{-1\}$ defined by $f(u) = \frac{u}{1-u}$ and put $\alpha = f(x), \beta = f(y), \gamma = f(z)$ . Since $f$ maps rational numbers to rational, the problem is equivalent to showing that $\alpha^2+\beta^2+\gamma^2 \ge 1 \quad (1)$

given that

\[\frac{\alpha}{\alpha+1)\frac{\beta}{\beta+1) \frac{\gamma}{\gamma+1) = 1 \quad (2)\] (Error compiling LaTeX. Unknown error_msg)

and that the equallity holds for infinitely many triplets of $\alpha,\beta,\gamma$ .

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2008 IMO Problems/Problem 2

Problem 2

Solution