Follow the links for the solutions in AS3, but if you felt tempted give them first a try, some of these problems can even make a nice programming interview exercise (if the position at hand requires minimal mathematical skills from your candidates).
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.
It's a simple loop with known limits. I guess the average programmer can solve this quickly.
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.
A bit involved, because now your loop doesn't have a limit:
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 two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
Now we are talking.... Raise your hand if you know what a permutation is. Ok. Now every one of you have to code a working version in five minutes. Counting! :D
I couldn't do it. As always the optimized permutation generator slips through the holes in my mind. So I had to resort to my simple but inefficient recursive generator, it has the advantage that if I forget it, I can deduce it again. Here my ruby version:
Once you have a working generator of permutations the rest is just loop through all the possible outcomes and checking if the conditions are satisfied.
Hmm... pretty boring for human minds, even if there are only 5! = 120 possibilities. It's no wonder that the solution for this problem in my book is a one liner without explanation. I guess this problem deserves the nukes!