## Task $4 / 68$

Let $0<a<1, n$ be a natural number. Which numbers $a$ differ from $n$ by their reciprocal?

What value is obtained specifically for $n=1$?
From the condition required in the task, we immediately obtain the equation

$$
n=\frac{1}{a}-a
$$

or, after multiplying by $a \neq 0$ and rearranging, $a^{2}+n a-1=0$ with the (due to $a>0$) unique solution

$$
a=\frac{1}{2}\left(\sqrt{n^{2}+4}-n\right)
$$

Specifically, for $n=1$, we get $a=\frac{1}{2}(\sqrt{5}-1) \approx 0.618 \ldots$, which is the measure of the Golden Ratio. Indeed, $\frac{1}{0.618 \ldots}=1.618 \ldots$

All numbers $a$ that satisfy the equation have the required property. There are, as one can easily verify, countably infinitely many.