PROBLEMS FOR NOVEMBER
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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 $\left\{{a}_{1},{a}_{2},\dots ,{a}_{n}\right\}$ of the set $\left\{1,2,\dots ,n\right\}$ 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 $p\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\dots +{a}_{0}$ with integer coefficients ${a}_{i}$ for which $p\left(m\right)$ 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) ${a}_{n}\ne 0$ for each positive integer $n$;
(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

$\sum _{k=1}^{n}\frac{1}{s-{a}_{k}}\le 1$

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 $d$.