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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 $x_{1}$ , $x_{2}$ , $\dots$ , $x_{n}$ , each with one digit, and the second player selects some numbers $a_{1}$ , $a_{2}$ , $\dots$ , $a_{n}$ and asks what is the vlaue of the sum $a_{1} x_{1} + a_{2} x_{2} + \dots + a_{n} x_{n}$ . What is the minimum number of questions used by the second player to find the integers $a_{1}$ , $x_{2}$ , $\dots$ , $x_{n}$ ?

44.: Find the permutation ${a_{1}, a_{2}, \dots, a_{n}}$ of the set ${1, 2, \dots, n}$ for which the sum

S = ‖ a_{2} - a_{1} ‖ + ‖ a_{3} - a_{2} ‖ + \dots + ‖ a_{n} - a_{n - 1} ‖

has maximum value.

45.: Prove that there is no polynomial $p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{0}$ with integer coefficients $a_{i}$ for which $p (m)$ is a prime number for every integer $m$ .

46.: Let $a_{1} = 2$ , $a_{n + 1} = \frac{a_{n} + 2}{1 - 2 a_{n}}$ for $n = 1, 2, \dots$ . Prove that

(a)

a_{n} \neq 0

for each positive integer

n

;

(b) there is no integer

p \geq 1

for which

a_{n + p} = a_{n}

for every integer

n \geq 1

(i.e., the sequence is not periodic).

47.: Let $a_{1}, a_{2}, \dots, a_{n}$ be positive real numbers such that $a_{1} a_{2} \dots a_{n} = 1$ . Prove that

\sum_{k = 1}^{n} \frac{1}{s - a_{k}} \leq 1

where

s = 1 + a_{1} + a_{2} + \dots + a_{n}

48.: Let $A_{1} A_{2} \dots A_{n}$ be a regular $n -$ gon and $d$ an arbitrary line. The parallels through $A_{i}$ to $d$ intersect its circumcircle respectively at $B_{i}$ ( $i = 1, 2, \dots, n$ . Prove that the sum

S = ‖ A_{1} B_{1} ‖^{2} + \dots + ‖ A_{n} B_{n} ‖^{2}

is independent of

d