HongKong 2022-23 IMO Selection Tests

Question

Mandy needs to wake up early for attending a mathematics contest. She has set an alarm in her smartphone every 15 minutes since 5:30 am. If an alarm is not pressed off by her or her mother (or anybody else), it will ring for a while, stop for a while, then will ring again 9 minutes later as the first ring, and so on (e.g. if the first alarm is not pressed off, it will ring again at 5:39 am). Also each alarm will work independently. Now suppose each ring-tone lasts for x minutes, and the smartphone has eventually rung for 50 minutes before Mandy wakes up at 6:30 am (assuming no one has pressed off any alarm before that). Find the value of x.

Accepted Answer

If each ring-tone lasts for exactly 3 minutes, the alarm will be ringing at 5:30–5:33, 5:39–5:42, 5:45–5:51 and 5:54–6:30. It will ring for 3 + 3 + 6 + 36 = 48 minutes in total. Therefore, we must have x > 3. Let x = 3 + y where 0 < y < 3. In addition to the above periods, the alarm will ring for y more minutes after each of 5:33, 5:42 and 5:51. Thus, we need 48 + 3y = 50. The only solution is y = 23, and hence x = 113.