International Mathematical Olympiad July 17, 2009 These three
problems from the International Mathematical Olympiad recently
popped up on the web. IMO 1960 Problem 01 Determine all three-
digit numbers N having the property that N is divisible by 11,
and N/11 is equal to the sum of the squares of the digits of N.
IMO 1962 Problem 01 Find the smallest natural number n which has
the following properties: (a) Its decimal representation has 6
as the last digit. (b) If the last digit 6 is erased and placed
in front of the remaining digits, the resulting number is four
times as large as the original number n. IMO 1963 Problem 06
Five students, A,B,C,D,E, took part in a contest. One prediction
was that the contestants would finish in the order ABCDE. This
prediction was very poor. In fact no contestant finished in the
position predicted, and no two contestants predicted to finish
consecutively actually did so. A second prediction had the
contestants finishing in the order DAECB. This prediction was
better. Exactly tw