\documentstyle{amsppt} \magnification=\magstephalf \nologo 
\def\Z{\Bbb Z} \def\R{\Bbb R} \def\F{\Bbb F_p} \def\a{\alpha} \def\v{\varepsilon}
\null\vskip.8cm\heading Problems Given At The St\. Andrews Colloquium, 1996\endheading\vskip.8cm


\bigskip\noindent{\bf First Day: Number Theory}
\medskip
1. Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1}\equiv a\pmod n$
holds for all $a$ if $n$ is prime.  Describe the set of integers $n$ for which this property is in fact true.
\medskip
2. Prove or disprove: for every odd number $k$ there is a prime of the form $2^nk+1$.
\medskip
3. Show that if $(a^2+b^2)/(ab+1)$ ($a,\,b\in\Bbb N$) is integral, then it is a perfect square.
\medskip


\bigskip\noindent{\bf Second Day: Algebra}
\medskip
1. Show that there is no ring (=commutative, with 1) with exactly five units.
\medskip
2.  Show that if a group has only finitely many elements of finite order, then
these form a subgroup.
\medskip
3. Let $P$ and $S$ denote the direct product and direct sum of countably
many copies of $\Z$, respectively, i.e\. $P$ is the set of all sequences
of integers $(a_1,a_2,a_3,\ldots)$, with the abelian group structure given by
componentwise addition, and $S$ is the subgroup consisting of all sequences 
with $a_i=0$ for $i\gg0$.  Show that $\,\text{Hom}_\Z(P,\Z)\approx S$.

\bigskip\noindent{\bf Third Day: Polynomials}
\medskip
1. Associate to a prime the polynomial whose coefficients are the decimal digits of 
the prime (e.g\. $9x^3+4x^2+3$ for the prime 9403).  Show that this polynomial is always irreducible.
\medskip
2. Suppose that $\sum_{n\ge0}b_nx^n\in\R[x]$ has only real roots.  Show that the polynomial 
$\sum_{n\ge0} b_nx^n/n!\,$ has the same property.
\medskip
3. Let $p=2m-1$  be an odd prime.  Show that the polynomial  $(1-x)^m+1+x^m$  is twice a square in  $\F[x]$.  
\medskip

\bigskip\noindent{\bf Fourth Day: Geometry}
\medskip
1. Let $A$ and $B$ be two {\it adjacent} vertices of an equilateral polygon.  If the angles at all other 
vertices are known to be rational (when measured in degrees), show that the angles at $A$ and $B$ are 
also rational.  Give a counterexample to this statement when $A$ and $B$ are not assumed to be adjacent.
\medskip
2. A rectangle $R$ is the union of finitely many smaller rectangles (non-overlapping except on their
boundaries), each one of which has at least one rational side.  Show that $R$ has the same property.
\medskip
3. Mark an angle $\a$ ($0<\a<2\pi$) on a pie-plate, and pick another angle $\beta$ ($0<\beta<2\pi$).
Define an operation on pie as follows: cut out the slice of pie over the marked angle, lift it up,
turn it over, replace it, and rotate the whole pie on the plate by the angle $\beta$.  Show that,
whatever the values of $\alpha$ and $\beta$, this operation has finite order (i.e., after a finite
number of iterations every piece of the pie is in its original position).
\medskip



\bigskip\noindent{\bf Fifth Day: Sequences and recursions}
\medskip
1. Define a sequence $\{t_n\}_{n\ge1}$ by the recursion $t_{n+5}=\bigl(t_{n+4}t_{n+1}+t_{n+3}t_{n+2}\bigr)/t_n\,$,
with initial values $(1,1,1,1,1)$.    Prove the statement that all of the $t_n$ are integral.
\medskip
2. Define a sequence $\{u_n\}_{n\ge1}$ by the formula $u_n=\sum\limits_{k=1}^n\dfrac{(2n+k)!}{(2n-2k)!\,(3k+1)!}\,$. 
Show that the statement ``none of the $u_n$ are integral" is false, but that 
the first counterexample is approximately $10^{2019025}\,$.
\medskip
3. Define a sequence $\{v_n\}_{n\ge1}$ by the recursion $v_n=(2+v_1^2+\cdots+v_{n-1}^2)/n$.  
Show that the statement ``all of the $v_n$ are integral" is false, but that 
the first counterexample is  approximately $10^{178485291567}\,$. 

\end
