Irish Intervarsity Mathematics Competition 2004

1. Ann and Brian play a game as follows. Ann is given a string of length 2004 cm. She must cut it into two pieces of positive integer (centimeter) lenghts, and hand the two pieces to Brian. Brian discards one piece of his choice, cuts the remaining peice into two positive integer lenghts, and hands them back to Ann. This process continues until one of the players discards one piece of string and is left with a piece of lenght 1 cm. This player is the winner. Assuming both players play as well as possible, who will win the game?
2. Suppose H is a regular hexagon in the complex plane with vertices (labelled in order going around H clockwise) h₁, h₂, h₃, h₄, h₅, h₆, where h₁ = 2 - 3i and h₄ = 2 + 3i.
   
   Calculate h₁ + h₂ + h₃ + h₄ + h₅ + h₆ and h₁h₂h₃h₄h₅h₆.
3. Show that there is no 2-by-2 matrix A such that
   
   for some n ≥ 2.
4. Let n be a positive integer, and let a₁,...,a_n be distinct integers.
   Show that the polynomial (x - a₁)...(x - a_n) - 1 cannot be expressed as a product g(x)h(x), g(x) and h(x) are polynomials with integer coefficients of degree ≥ 1.
5. Find all solutions (in real numbers ) to
   
6. Show that
   
   for all real numbers x, y. Find all pairs x,y for which we get equality.
7. Let p be an odd prime. Show that
   
   
   
   
   
   
   
   
8. Define a sequence by b₁ = 1, and
   
9. The sequence of positive integers a₁ = 1, a₂ = 2, a₃ = 3, .... satisfies this identify
   
   Show that no other sequence of positive real numbers satisfies this identify for all n.
10. Define numbers e_k by e₀ = 0, e_k = exp(e_k - 1) for k ≥ 1. Determine all functions f_k = f_k(x) such that f₀ = x, and
    
    on the interval [e_k, infinity), and f_k = 0.
    [Here f_k' denotes the derivative of f_k.]