PROBLEMS FOR NOVEMBER
Solutions should be submitted to
Dr. Dragos Hrimiuk
Department of Mathematics
University of Alberta
Edmonton, AB T6G 2G1
no later than
December 31, 2000.
 43.

Two players pay a game: the first player thinkgs
of n integers x_{1}, x_{2}, ¼, x_{n}, each with one
digit, and the second player selects some numbers
a_{1}, a_{2}, ¼, a_{n} and asks what is the vlaue
of the sum a_{1}x_{1} + a_{2}x_{2} + ¼+ a_{n}x_{n}. What is the
minimum number of questions used by the second player to find
the integers a_{1}, x_{2}, ¼, x_{n}?
 44.

Find the permutation { a_{1}, a_{2}, ¼, a_{n} }
of the set { 1, 2, ¼, n } for which the sum
S = a_{2}  a_{1} + a_{3}  a_{2} + ¼+ a_{n}  a_{n1}  

has maximum value.
 45.

Prove that there is no polynomial
p(x) = a_{n} x^{n} + a_{n1}x^{n1} + ¼+ a_{0}
with integer coefficients a_{i} for which
p(m) is a prime number for every integer m.
 46.

Let a_{1} = 2, a_{n+1} = [(a_{n} + 2)/(1  2a_{n})] for n = 1, 2, ¼.
Prove that
(a) a
_{n} ¹ 0 for each positive integer n;
(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 a_{1}, a_{2}, ¼, a_{n} be positive real
numbers such that a_{1} a_{2} ¼a_{n} = 1. Prove that
where s = 1 + a
_{1} + a
_{2} +
¼+ a
_{n}.
 48.

Let A_{1}A_{2} ¼A_{n} be a regular ngon 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
S = A_{1}B_{1} ^{2} + ¼+ A_{n}B_{n} ^{2} 

is independent of d.