1. Tom and Jerry are chasing each other on a track shaped like an eight (see figure).

They run in the same direction at constant speeds. Initially, Jerry was directly above Tom. After 20 minutes, Tom was directly above Jerry, and neither of them had completed a full lap. At the moment when Jerry had completed exactly one lap from the start, Tom finally caught up with him. How long did Tom chase

![](https://cdn.mathpix.com/cropped/2024_05_06_0420eb5419af6e3d6971g-1.jpg?height=260&width=520&top_left_y=442&top_left_x=1407)
Jerry
Solution. While Jerry runs the small loop, Tom runs the large one; while Jerry runs the large loop, Tom runs both the large and small loops together. Let's denote the lengths of the large and small loops by $L$ and $l$ respectively. Then we get that $L: l = (L + l): L$. If we denote $L: l$ by $x$, we get: $x = 1 + \frac{1}{x}$, from which $x^2 = x + 1$. Solving the quadratic equation (and considering that $x > 0$), we find that the ratio of the lengths of the loops is the golden ratio, that is, the number $\tau = \frac{1 + \sqrt{5}}{2}$.

We know that Jerry ran the small loop in 20 minutes. Therefore, he ran the large loop in $20 \tau = 10 + 10 \sqrt{5}$ minutes, and the entire circuit in $20 + (10 + 10 \sqrt{5})$ minutes. That's when Tom caught him.

Answer: $30 + 10 \sqrt{5}$ minutes.