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 19th time. If necessary, round the answer to the hundredths place 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 $19 * 720 / 11=13680 / 11$ minutes.

Answer: $1243.64$.