Three o'clock

Algebra Level 2

Consider a wall clock with three hands: an hour hand, a minute hand and a second hand. The second hand of this clock moves continuously (not on one-second intervals, as with many other clocks). Suppose that this clock is placed on the wall of the lobby of Palma Hall.

Now, here's the situation. Nikko, Isah and Sylvan agreed with one another to arrive at the lobby on or before 3 in the afternoon to discuss their project in Philo 1.

Ironically, Isah arrived at the lobby $x$ seconds past three, when the second hand of the clock bisects the angle formed by the remaining hands (for the first time),

Nikko arrived $y$ seconds past three, when the hour hand of the clock bisects the angle formed by the remaining hands (for the first time).

And finally, Sylvan reached the lobby $z$ seconds past three, when the minute hand of the clock bisects the angle formed by the remaining hands (for the first time).

If $x$ , $y$ and $z$ are real numbers, find $\lfloor x \rfloor+\lfloor y \rfloor+\lfloor z \rfloor$ .

Image credit: blogs.telegraph.co.uk

79 86 77 82

1 solution

Chew-Seong Cheong
Oct 19, 2018

Assuming the hour-hand and minute-hand move step-by-step instead of continuous. Before 1 minute pass three, the hour-hand is pointing at the 3-hour mark and the minute-hand is pointing the 12-hour mark.

For the second-hand to bisect the right angle between the hour-hand and minute-hand, $x=7.5$ seconds after three.
For the hour-hand to bisect the other two hands, $y=30$ seconds after three.
For the minute-hand to bisects the other two hands, $z=45$ seconds after three.

Therefore, $\lfloor x \rfloor + \lfloor y \rfloor + \lfloor z \rfloor = \lfloor 7.5 \rfloor + \lfloor 30 \rfloor + \lfloor 45 \rfloor = 7+30+45=\boxed{82}$ .

Three o'clock

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