The Dutch Mathematics Olympiad 1983/1984

Some questions from the first round

1. A unit cube is projected onto a plane so that the projection is a regular hexagon. What is the length of each side?

2. There is a 10-digit number in which each digit 0, 1, 2, ..., 9 occurs exactly once, so that for every k = 1, ..., 10 the first k digits form a number that is divisible by k. For example, 7891234560 contains every digit once, and 7 is divisible by 1 78 is divisible by 2 789 is divisible by 3 but then it goes wrong: 7891 is not divisible by 4
   What is the number?

3. In how many zeroes does the number 100! = 1 . 2 . 3 . ... 100 end?

Some questions from the second round

1. Set a_n = 1 . 4 . 7 . ... . (3n-2)/2 . 5 . 8 . ... . (3n-1).
   Prove that for any n, 1/(3n+1)^1/2 is less than or equal to a_n and a_n is less than 1/(3n+1)^1/3.

2. Prove that for every odd number n,
   2²ⁿ(2²ⁿ⁺¹-1)
   is a number ending in 28 if written as a decimal number.

Back to Home