HMMT-Nov 2022 gen, Problem 1

Emily's broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily's clock also does not tick, but rather updates continuously.

Solution: Solution: When comparing Emily's clock with a normal clock, the difference between the two times decreases by 6 seconds for every 1 second that passes. Since this difference is treated as 0 whenever it is a multiple of 12 hours, the two clocks must agree once every $\frac{12}{6}=2$ hours. Thus, in a 24 hour period it will agree 12 times. $\fbox{12}$.

Answer: 12