2. We started the Mathematical Olympiad punctually at 9:00, as I checked on my watch, which was working correctly at that time. When I finished, at 13:00, I looked at the watch again and saw that the hands had come off their axis but maintained the position they were in when the watch was working. Curiously, the hour and minute hands were exactly superimposed, one on top of the other, forming a non-zero angle less than $120^{\circ}$ with the second hand. At what time did my watch break? (Give the answer in hours, minutes, and seconds with a maximum error of one second; assume that, when it was working, the hands of the watch moved continuously.)
Solution. If we measure time $t$ in seconds from 00:00 and angles in degrees, in a clockwise direction and starting from the position of the hands at 00:00, the angle swept by the hour hand at time $t$ is $\alpha_{hor}(t)=t / 120$ and the angle swept by the minute hand is $\alpha_{\min}(t)=t / 10$. Since both hands have overlapped, the two angles must coincide at the moment $t$ when the clock broke. The minute hand may have completed some full turns, so we must have

$$
\frac{t}{10}=\frac{t}{120}+360 k
$$

with $k \geq 0$ an integer, that is, $t=\frac{360 \times 120}{11} k$. Since the malfunction occurred between 09:00 and 13:00, it must be $9 \leq k \leq 12$. The angle for the second hand is $\alpha_{\text{seg}}(t)=6 t$, so the difference

$$
6 t-\frac{t}{120}=\frac{360 \times 719}{11} k=\left(360 \times 65+\frac{360 \times 4}{11}\right) k
$$

must be, except for multiples of 360, a number $\beta$ between -120 and 120. If $k=9$, $\beta=(360 \times 3) / 11$, which is indeed within this range. However, if $k=10$ or $12, \beta= \pm(360 \times 4) / 11$, which is outside this interval. The case $k=11$ is also excluded since $\beta=0$ would result, and the three hands are not superimposed. Therefore, the only possible case is $k=9$, which corresponds to the moment

$$
t=\frac{360 \times 120 \times 9}{11}=3600 \times 9+60 \times 49+5+\frac{5}{11}
$$

which means the clock broke at 09:49:05.