1. The clock shows $00:00$, at which the hour and minute hands of the clock coincide. Considering this coincidence as number 0, determine after what interval of time (in minutes) they will coincide for the 21st time. If the answer is not an integer, round the result to the hundredths according to rounding rules.
# Solution:

The minute hand passes 1 circle in an hour, while the hour hand passes $1 / 12$ of a circle, so their closing speed is $11 / 12$ of a circle per hour, and one closing takes $1 /(11 / 12)=12 / 11$ hours or 720/11 minutes. 19 closings is $21 * 720 / 11=15120 / 11=1374.55$ minutes.

Answer: 1374.55.