% This template was created by Stephen on 27th March 2000
%
% Title:  Euclidean Geometry
% Author: Stephen Farrar
% Date:   16/4/2000
%

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{\footnotesize April 2005}

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%--------------------------------------------------------
% Let the document begin!
\begin{document}
\begin{center} {\Large {\bf Mathematical Relay}} \end{center}
\begin{center} Peter McNamara \end{center}
\begin{center} April 2005 \end{center}

\iffalse
\begin{centre}
  \bf \Large
  Relay Questions (Confidential)
\end{centre}
\fi

\begin{enumerate}

\item[J1] Altitude $AD$ of triangle $ABC$ right angled at $A$ cuts $BC$
into lengths $BD=1$, $DC=4$. What is $AD$?

\ritem[J2] How many ways can an IMO team of 6 students be chosen from a
class of 8 boys and 6 girls if it is to contain at least 3 boys
and at least 2 girls?

\ritem[J4] How many integers less than 2005 can be represented in the
form $6m+8n$ where $m$ and $n$ are positive integers.

\ritem[J3] Find the number of ways of arranging two bishops, two rooks,
two knights, one king and one queen on a $1\times 8$ chessboard
coloured in the usual way, such that the two bishops are on
opposite coloured squares. (Pieces of the same type are considered
indistinguishable.)

\ritem[J12] Norm has an automatic card shuffler. This shuffler only
works on a pack of exactly thirteen cards, and performs the same
shuffle each time (so if the first card gets sent to the sixth
card, for example, then the first card will always get sent to the
sixth card). Ross takes a deck of cards $A,2,3,\ldots,J,Q,K$ and
sends them through the card shuffler twice. The cards end up in
the following order:
\[
Q\ K\ 7\ 2\ 6\ 8\ 9\ A\ 10\ 4\ 5\ 3\ J
\]
What was the order of the cards after one shuffle?

\ritem[S10] An emergency service is to be set up in a national park.
Ranger stations are to be connected by phone lines. Each station
must be able to connect to every other station, either by a direct
connection or else through at most one other station. Each station
can be connected to at most three other stations. What is the
largest number of stations which can be connected in this way?

\ritem[S4] Suppose that $PQS$ is a triangle and $R$ is a point on $QS$
such that $PQ\cdot RS=QS\cdot PR$. Furthermore suppose that
$\angle PQR=37^o$ and $\angle PRQ=47^o$. Find $\angle RPS$.

\ritem[S14] $PQRS$ is a quadrilateral with right angles at $P$ and $S$.
$T$ is a point on $PR$ such that $ST\perp PR$. It is also true
that $\angle TQR=\angle PST=30^o$. If $PS=1$, what are all
possible lengths of $PQ$?

\ritem[S7] Let $O$ be a vertex of a twelve dimensional unit cube. Find
the number of vertices $A$ of this cube for which $|OA|<2$.

\ritem[S2] What is the largest number of knights that can be placed on
a standard chessboard such that no two can capture each other (in one
move)?

\ritem[S9] Let $f\map{\Z}{\Z}$ be a function satisfying the following
equation.
\[
f(n+2)=f(n+1)-f(n)
\]
What is the maximum number of different values of $f(n)$ that can
be attained as $n$ ranges over all integers?

\ritem[S12] Let $ABC$ be a triangle for which $\angle B=30^0$ and
$\angle C=45^o$. Let $\Gamma$ be a circle through $A$, tangent to
$BC$ and to the circumcircle of $ABC$. Let $X$ be the point of
tangency of $\Gamma$ and $BC$. What is the value of the ratio
$\frac{BX}{XC}$?

\ritem[S3] Find all pairs of integers $(F,V)$ for which there exists a
polyhedron with $F$ faces, $V$ vertices and eight edges.

\ritem[J10] Ross is counting his marbles. He notices that the number of
marbles he has is a prime that is less than the number of days in
a leap year. Furthermore, he notices the following interesting
facts about this number.
\begin{enumerate}
\item Its first power is congruent to 1 mod 3. \item Its third
power is congruent to 1 mod 5. \item Its fifth power is congruent
to 1 mod 7.
\end{enumerate}
How many marbles does Ross have?

\ritem[J9] Find the number of positive integers, strictly less than
2005, which are divisible by none of 33, 5 and 6.

\ritem[J5] Nick is rowing a boat upstream. He rows at a speed of 17
metres per second relative to the current, which flows at a rate
of 5 metres per second. At precisely 12 noon, Nick's hat blows
off, and is carried downwind by the current. At 1pm, Nick realises
that he has lost his hat, abruptly turns around, and begins rowing
downstream. At what time will Nick reach his hat?

\ritem[J7] Find the area of a cyclic quadrilateral with sides of length
(in order) 1,7,8,4.

\ritem[S15] Find all composite $r$ which can be written as the sum of
two nonzero squares, and is also the remainder upon dividing some
prime number by 210.

\ritem[J14] Suppose $0\leq m\leq n $ are integers. Find all such pairs
$(m,n)$ that additionally satisfy the following equation:
$$\frac{m^2+mn+n^2}{m+n}=\frac{49}{4}.$$

\ritem[S11] Let $n$ be a positive integer, and $G$ a graph with the
following properties \begin{enumerate} \item $G$ contains $n$
vertices. \item $G$ contains at least one edge. \item There exists
two vertices of $G$, without an edge between them. \item Every
vertex of $G$ has the same degree. \item If there are three
vertices $A,B,C$ of $G$ such that there is an edge between $A$ and
$B$, and an edge between $A$ and $C$, then there is an edge
between $B$ and $C$.
\end{enumerate} Find the number of integers $n\leq 100$ for which
there exists such a graph $G$.

\ritem[J6] Suppose that the incircle of triangle $ABC$ touches the
sides $AB$, $BC$ and $CA$ at points $F$, $E$ and $D$ respectively.
Suppose that $D$, $E$ and $F$ divide the circumference of the
incircle in the ratio $DE:EF:FD=16:11:9$. What is the size of
$\angle BAC$?

\ritem[J8] (General Knowledge - 5 marks) Everybody has heard of the M\" obius strip, named after one
August Ferdinand M\" obius. But the question is, in what century
did M\"obius live the majority of his life in?

\ritem[S13] What is the last nonzero digit of $99!$, when written in
decimal notation?

\ritem[S5] In an $n\times n$ chessboard, aligned with the co-ordinate
axes, how many pairs of ($1\times1$) squares are there with one
lying above and to the right of the other?

\ritem[J11] Two rods of length 1 are divided in to 400 and 501 parts
respectively. They are then placed side by side so that their ends
coincide. Find the minimum distance apart of the division marks.

\ritem[S8] (General Knowledge - 5 marks) How may children did Euler have? [hint: it is prime]

\ritem[S1] David is older than Stewart. Ivan comes along and says: "Hey
guys, check this out, if $p(x)$ is the monic quadratic polynomial,
whose roots are David and Stewart's age (which are assumed to be
non-negative integers), then $p(1)$ is prime!

How old is Stewart?

\ritem[J13] Let $AB$ be a chord of a circle with centre $O$. $C$ is a
point on $AB$, and the radius through $C$ intersects the circle at
$D$. $AC=3$, $BC=5$ and $CD=1$. What is the radius of the circle?

\ritem[S6] What is the maximum number of terms of a geometric
progression with common ratio greater than one, whose entries are
all integers from 25 to 250 inclusive?

\ritem[J15] Alice: I am insane.

Bob: I am pure.

Charlie: I am applied

Dorothy: I am sane.

Alice: Charlie is pure.

Bob: Dorothy is insane.

Charlie: Bob is applied.

Dorothy: Charlie is sane.

Describe Alice's sanity and purity, given that pure mathematicians
tell the truth about their beliefs, applied mathematicians lie
about their beliefs, sane mathematicians' beliefs are correct and
insane mathematicians' beliefs are incorrect.


\end{enumerate}


\end{document}
