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

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

(b) there is no integer
$p\ge 1$ for which
${a}_{n+p}={a}_{n}$ for every integer
$n\ge 1$ (*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

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

is independent of $d$.