[ Irrational Inequalities ]

Find the smallest value of the fraction $x / y$, if $\sqrt{x-1}+\sqrt{y-1}=1$.

#
$\sqrt{y-1}=1-\sqrt{x-1} \geq 0$, that is, $\sqrt{x-1} \leq 1 \Leftrightarrow 1 \leq x \leq 2$. Similarly, $1 \leq y \leq 2$. Therefore, $x / y \geq 1 / 2$.

The value $x / y=1 / 2$ is achieved when $x=1$,

$y=2$, satisfying the original equation.

## Answer

$1 / 2$.