# Cork Superbrain 1986

1. 1986 is the year Cork 801. Show how to write the number 801 using each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 once and only once, using any mathematical operation you wish.
2. What, in your opinion is the next term of the sequence, and why?
F, 4, E, S, 9, S, E, 5, E, ?
3. Find the value of
cos20° + cos21° + cos22° + ... + cos289°
(where n° is n degrees) correct to twenty decimal places.
4. Which of the following numbers is bigger and why?
or
5. Find the value of
6. A thin rod is broken into three pieces. What is probability that a triangle can be formed from these three pieces.
7. Prove that
 n S r = 1
 1 = 1 + 1 + 1 + ... + 1 r 1 2 3 n
is not an integer for n > 1.
8. A right angled triangle has all its sides an integer length. If the length of the perimeter equals the area, find all such triangles.
9. Is it possible to have a right circular cone in which the total surface area is equal to the volume? Justify your answer.
10. Find the smallest positive integer that can be expressed as the sum of two squares of positive integers, in two different ways.