Solutions should be submitted to

Dr. Dragos Hrimiuk
Department of Mathematics
University of Alberta
Edmonton, AB T6G 2G1

43.

Two players pay a game: the first player thinkgs of n integers x₁, x₂, ¼, x_n, each with one digit, and the second player selects some numbers a₁, a₂, ¼, a_n and asks what is the vlaue of the sum a₁x₁ + a₂x₂ + ¼+ a_nx_n. What is the minimum number of questions used by the second player to find the integers a₁, x₂, ¼, x_n?

44.

Find the permutation { a₁, a₂, ¼, a_n } of the set { 1, 2, ¼, n } for which the sum

45.

Prove that there is no polynomial p(x) = a_n xⁿ + a_n-1x^n-1 + ¼+ a₀ with integer coefficients a_i for which p(m) is a prime number for every integer m.

46.

Let a₁ = 2, a_n+1 = [(a_n + 2)/(1 - 2a_n)] for n = 1, 2, ¼. Prove that

(b) there is no integer p ³ 1 for which a_n+p = a_n for every integer n ³ 1 (i.e., the sequence is not periodic).

47.

Let a₁, a₂, ¼, a_n be positive real numbers such that a₁ a₂ ¼a_n = 1. Prove that

48.

Let A₁A₂ ¼A_n be a regular n-gon and d an arbitrary line. The parallels through A_i to d intersect its circumcircle respectively at B_i (i = 1, 2, ¼, n. Prove that the sum