11.3. The fractional part of a positive number, its integer part, and the number itself form an increasing geometric progression. Find all such numbers.
Answer. The number is unique $\frac{\sqrt{5}+1}{2}$.

Let $a$ be the desired number, $b$ its integer part, $c$ its fractional part, and $q$ the common ratio of the progression. Then $a=b+c$ and $c q^{2}=c q+c$. Given $c>0$, dividing by $c$, we get the quadratic equation $q^{2}-q-1=0$, which has two roots, one of which is $q=\frac{\sqrt{5}+1}{2}$ and is greater than 1. According to the condition, the integer part of the number is greater than its fractional part, so $b \geq 1$. On the other hand, from the inequality $b=c q<1 \cdot \frac{\sqrt{5}+1}{2}<2$ it follows that $b=1$ and the desired number $a=q=\frac{\sqrt{5}+1}{2}$.