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

  3. What is the number?

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

Some questions from the second round

• 1. Set an = 1 . 4 . 7 . ... . (3n-2)/2 . 5 . 8 . ... . (3n-1).

  2. Prove that for any n, 1/(3n+1)1/2 is less than or equal to an and an is less than 1/(3n+1)1/3.

  3. Prove that for every odd number n,

  4. 22n(22n+1-1)

  5. is a number ending in 28 if written as a decimal number.