How many times per day, and at what time(s) do the hour and minute hands on a clock form EXACTLY a 60 degree angle?

Answer: 44 times PER 24 hrs

SoluTioN : Per 24 hrs the minute hand performs 24 revolutions-

the hour hand only 2. THEREFORE THEY MEET 22 TIMES.

Prior to each meeting and after each meeting there is a situation of 60 degrrees between them, ergo... 44 times.

If you mean "per day" as opposed to " per night" - at this time of year i.e. around the spring equinoctium- the division is 50-50;

22 per day and 22 per night.