Solutions should be submitted to
Dr. Dragos Hrimiuk
Department of Mathematics
University of Alberta
Edmonton, AB T6G 2G1
no later than December 31, 2000.

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

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

S= a2 - a1 + a3 - a2 ++ an - an-1

has maximum value.

Prove that there is no polynomial p(x)= an xn + an-1 xn-1 ++ a0 with integer coefficients ai for which p(m) is a prime number for every integer m.

Let a1 =2, an+1 = an +2 1-2 an for n=1,2,. Prove that
(a) an 0 for each positive integer n;
(b) there is no integer p1 for which an+p = an for every integer n1 (i.e., the sequence is not periodic).

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

k=1 n 1 s- ak 1

where s=1+ a1 + a2 ++ an .

Let A1 A2 An be a regular n-gon and d an arbitrary line. The parallels through Ai to d intersect its circumcircle respectively at Bi ( i=1,2,,n. Prove that the sum

S= A1 B1 2 ++ An Bn 2

is independent of d.