The hands of a clock overlap how many times, form a right angle, and a straight angle with each other in one hour, and how many times in 12 hours?
Suppose the two hands overlap and will overlap again after $x$ minutes. During this time, the minute hand will make one full revolution and advance another $x-60$ minutes, so

$$
x-60=\frac{x}{12}
$$

from which

$$
x=65 \frac{5}{11}
$$

that is, the hands overlap every $65 \frac{5}{11}$ minutes, so 11 times in 12 hours. Since the hands form a right angle twice and a straight angle once as they move from one overlapping position to the next, in 12 hours they form a right angle 22 times and a straight angle 11 times.

(Kovács Gyula, Budapest.)

The problem was also solved by: Auer Gy., Ehrenfeld N., Erdős V., Fried E., Füstös E., Kirchknopf E., Paunz A., Petrik S., Sárközy P., Tóth B, Viola R., Wáhl V., Weisz S.