Math Is Fun Forum

You can check that

(Expand LHS and show it's equal to RHS.) Hence

Adding up …

Rearrange, noting that

Is there an R?

Take your time.

@julianthemath

Q2 (trickier): Let θ be an arbitrary angle greater than 0° and smaller than 180°. How many times in a 12-hour period do the two hands of a clock make an angle of θ between themselves?

And these are the times at which the hands are perpendicular (using 12-hour notation):

I think the problem is for a 12-hour period. The simplest way is to count how many times the hands are at right angles in every hourly.

00:00 to 01:00 – 2
01:00 to 02:00 – 2
02:00 to 03:00 – 2 (including 3 o'clock)
03:00 to 04:00 – 1
04:00 to 05:00 – 2
05:00 to 06:00 – 2
06:00 to 07:00 – 2
07:00 to 08:00 – 2
08:00 to 09:00 – 2 (including 9 o'clock)
09:00 to 10:00 – 1
10:00 to 11:00 – 2
11:00 to 12:00 – 2

The hands are at right angles twice every hourly except for the hours 03:00–04:00 and 09:00–10:00. So it's 22 times every twelve hours.

And even Isaac Newton himself admitted that there were people smarter than he was – giants on whose shoulders he stood.

http://en.wikipedia.org/wiki/Standing_o … _of_giants

I'm not sure you you did there. Maybe part of my first post isn't very clear, so I'll try and explain.

The park has two ends, A and B, where A is the end closer to the shopping mall and B the end closer to the restaurant. The route taken by Mandy is: shopping mall → A → park → B → restaurant. Sandy's route is the reverse: restaurant → B → park → A → shopping mall. Thus when I say that the girls enter the park at the same time, I mean that Mandy reaches point A and Sandy reaches point B simultaneously.

Correct.

Next question (harder): At what time did both girls enter the park?