5. At a certain moment, Karel noticed on his precisely running watch that the end of the minute hand, the end of the hour hand, and a suitable point on the circumference of the dial form the vertices of an equilateral triangle. Before this phenomenon occurred again, a time $t$ had elapsed. Find the greatest possible $t$ in terms of the ratio $k$ of the lengths of the two hands $(k>1)$, given that the radius of the dial's circumference is equal to the length of the minute hand.

(Jaromír Šimša)
Solution. We will show that the sought maximum $t$ is equal to $4 / 11$ hours, independent of the ratio $k$ of the lengths of the hands.

Let $c$ be the circle of the clock face, $S$ its center, and $M$ the end of the hour hand (Fig. 2). First, we explain why, for a fixed position of point $M$, there are exactly two equilateral triangles $M X Y$ with vertices $X, Y$ on the circle $c$. Since the line $S M$ must be the axis of the chord $X Y$, and thus also the axis of the angle $X M Y$, both lines $M X, M Y$ form an angle of $30^{\circ}$ with the line $S M$. Therefore, the triangle $M X Y$ is identical to one of the equilateral triangles $M V_{1} V_{2}, M V_{3} V_{4}$ constructed in Fig. 2.

The points $V_{i}$ divide the circle $c$ into four arcs. The arcs $V_{2} V_{3}$ and $V_{4} V_{1}$ correspond to the inscribed angles $V_{2} V_{4} V_{3}, V_{1} V_{3} V_{4}$, each of size $60^{\circ}$. Therefore, according to the theorem of inscribed and central angles, the first two of the following equalities hold:

$$
\left|\angle V_{2} S V_{3}\right|=\left|\angle V_{4} S V_{1}\right|=120^{\circ} \text { and } \quad\left|\angle V_{1} S V_{2}\right|+\left|\angle V_{3} S V_{4}\right|=120^{\circ} \text {, }
$$

the third equality is a consequence of these (by completing the full angle at vertex $S$). It follows that both central angles $V_{1} S V_{2}, V_{3} S V_{4}$ are less than $120^{\circ}$.

We can imagine that the hour hand of the clock is stationary and the minute hand rotates around the center $S$ at an angular speed of $(360-30)^{\circ}=330^{\circ}$ per hour. As we have found, the phenomenon in question occurs precisely when the end $V$ of the minute hand coincides with one of the four points $V_{i}$. Therefore, between two consecutive events, the minute hand rotates through an angle that, in two cases, has a size of $120^{\circ}$ and in the remaining two cases, sizes of $\left|\angle V_{1} S V_{2}\right|$ and $\left|\angle V_{3} S V_{4}\right|$, which are less than $120^{\circ}$ (and depend on the ratio $k$). The longest time $t$ is thus independent of the ratio $k$ and is equal to 120/330 hours.

![](https://cdn.mathpix.com/cropped/2024_04_17_d6cc3cffaf5852457704g-7.jpg?height=733&width=808&top_left_y=156&top_left_x=615)

Fig. 2