35th CanMO 2003 Problem 1 The angle between the hour and minute hands of a standard 12-hour clock is exactly 1 o . The time is an integral number n of minutes after noon (where 0 < n < 720). Find the possible values of n.
The angles of the minute, hour hands are 6n, n/2 degrees, so we want 11n/2 = 360k +- 1 for some k. Hence n = 65k + (5k +- 2)/11. Since 0 < n < 720, we have 0 < k < 11. Also 5k = 2 or -2 mod 11, so k = 4 or 7, giving n = 262 or 458. 35th CanMO 2003 © John Scholes jscholes@kalva.demon.co.uk 14 Aug 2003 Last corrected/updated 14 Aug 03