PRMO 2019


Q. One day I went for a walk in the morning at xx minutes past 5 O’ clock, where xx is a two-digit number. When I returned, it was yy minutes past 6 O’ clock, and I noticed that
(i) I walked exactly for xx minutes and
(ii) yy was a 2-digit number obtained by reversing the digits of xx. How many minutes did I walk?
[PRE-RMO–2019]

This is quite simple actually.
Let x = ab and y = ba
=> x = 10a + b, y = 10b + a
Now x minutes past 5 = 300 + x since 5 hours have 300 minutes.
So 360 + y - (300 + x) = x
=> 60 + y = 2x
=> 30 + y/2 = x
=> 30 + (10b+a)/2 = 10a + b
=> 30 + 5b + a/2 = 10a + b
=> a = 2/19(30 + 4b)
=> a = (60 + 8b)/19

Now a,b are between 0,9
So just keep putting values of 'b' to find an answer.
On b = 2, a = 4
So x = 42 and y = 24

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