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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 x1, x2, ¼, xn, each with one digit, and the second player selects some numbers a1, a2, ¼, an and asks what is the vlaue of the sum a1x1 + a2x2 + ¼+ anxn. What is the minimum number of questions used by the second player to find the integers a1, x2, ¼, xn?

44.
Find the permutation { a1, a2, ¼, an } of the set { 1, 2, ¼, n } for which the sum
 S = |a2 - a1 |+ |a3 - a2 |+ ¼+ |an - an-1 |
has maximum value.

45.
Prove that there is no polynomial p(x) = an xn + an-1xn-1 + ¼+ a0 with integer coefficients ai for which p(m) is a prime number for every integer m.

46.
Let a1 = 2, an+1 = [(an + 2)/(1 - 2an)] for n = 1, 2, ¼. Prove that
(a) an ¹ 0 for each positive integer n;

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

47.
Let a1, a2, ¼, an be positive real numbers such that a1 a2 ¼an = 1. Prove that
 n å k = 1 1s - ak £ 1
where s = 1 + a1 + a2 + ¼+ an.

48.
Let A1A2 ¼An 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, ¼, n. Prove that the sum
 S = |A1B1 |2 + ¼+ |AnBn |2
is independent of d.
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