IMO 1968 SL 11

Origin: ROM

Problem

Find all solutions $(x_1, x_2, \ldots, x_n)$ of the equation

$$1 + \frac{1}{x_1} + \frac{x_1 + 1}{x_1x_2} + \frac{(x_1 + 1)(x_2 + 1)}{x_1x_2x_3} + \cdots + \frac{(x_1 + 1) \cdots (x_{n-1} + 1)}{x_1x_2 \cdots x_n} = 0.$$

Solution

Introducing $y_i = \frac{1}{x_i}$, we transform our equation to

$$\begin{aligned} 0 &= 1 + y_1 + (1 + y_1)y_2 + \cdots + (1 + y_1) \cdots (1 + y_{n-1})y_n \ &= (1 + y_1)(1 + y_2) \cdots (1 + y_n). \end{aligned}$$

The solutions are $n$-tuples $(y_1, \ldots, y_n)$ with $y_i \neq 0$ for all $i$ and $y_j = -1$ for at least one index $j$. Returning to $x_i$, we conclude that the solutions are all the $n$-tuples $(x_1, \ldots, x_n)$ with $x_i \neq 0$ for all $i$, and $x_j = -1$ for at least one index $j$.