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\title{The Irish Intervarsity Competition in Mathematics}
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\author{Timothy Murphy\\
School of Mathematics, TCD}
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{\bf What, in your opinion, is the next term in the sequence
\[
3, 5, 1, 15, 11, 10?
\]}
{\em This will be a cinch.
Easy one to start with.
Thanks, Des.
Knew he was a good sort.
Let's see.
Probably squares minus 1 \dots
that will explain the 15, anyway.
No, doesn't seem to be that.
Maybe it's not quite as simple as I thought \dots}
The Irish Intervarsity in Mathematics
came out of the fertile brain of Des MacHale.
It was a natural extension of the Superbrain Competition
that Des has been running in UCC since 1984.
(The question above was the opening problem
in the 1987 Superbrain paper.)
{\em I've got it!
Should have thought of that earlier.
It's obvious.
Just a common-or-garden code,
1 for A, 2 for B, and so on.
Let's see\dots
CEAOKJ\dots.
What language is this?
Maybe the Viking name for Cork?
Well, it was worth a try, anyway.}
The Intervarsity was first held in Cork, in 1990.
It moved to TCD for the next 2 years;
and UCG hosted the event this year.
The competition was won by TCD in 1990 and 1991,
and by UCC in 1992 and 1993.
(UCC and UCD tied in 1993;
but the prize went to UCC for the best individual result.)
Although the competition is mainly a team event,
there is also an individual winner each year.
Paul Harrington of TCD won in 1990;
Aiden O'Reilly of Maynooth in 1991;
Cian Dorr of UCC in 1992;
and Peter Hegarty of UCC in 1993.
{\em Maybe it's
the number of steps on the stairs in the UCC Maths Dept \dots
Cork bus numbers \dots
The ages of Des MacHale's children, in alphabetical order \dots
The number of moons of the planets \dots
The winners of the Eurovision Song Contest \dots
I've always suspected that man MacHale had a sadistic streak.
Now calm down.
I've only taken 20 minutes on this question so far.
If he's a sadist,
we must all be mathematical masochists
to just sit here and take this sort of thing from him.
Everyone else seems to be scribbling away.
Even that awful ass from UCD.
What was it Tartakower said,
``Why am I always being beaten by fools?''
Come on, pull yourself together.
Only 25 minutes gone.
Let's abandon this question.
But now I've invested so much time in it.
I know it must really be simple.}
The universities in Northern Ireland are invited each year,
and were expected in 1991, but didn't materialise.
It would be nice to make a special effort to persuade them
to take part next year.
Perhaps with a promise of a meeting in Queens' the following year?
Women are also conspicuous by their absence.
I think there were 2 in Maynooth's team last year,
and that was about it.
I wonder why?
In TCD the student's organise a selection test
(which Richard Timoney and I usually set)
but very few women will take part,
even though they constitute some 30\% of our student numbers (in maths).
In 1991 Helen Joyce---who was our best student for some years,
as far as exam results are concerned
(she went on to get a distinction in Cambridge Part III)---absolutely refused
even under extreme pressure.
{\em I've got it!
Not the letters themselves,
but the numbers of letters in the words.
A well-known saying,
with 5 letters in the first word,
3 in the second, and so on.
Well, what has 15 letters?
Intervarsity? Not quite.
Differentiation? Maybe.
I wonder if Kraft-Ebbing had a category for people like MacHale.
Does he belong to a recognisable criminal type?
Don't you see the similarity between his features
and those of Hannibal Lector?}
The questions in the Intervarsity were set by Des MacHale
in the inaugural year 1990,
and again this year.
Richard Timoney and I set the paper
in the 2 intervening years.
What sort of questions do we set?
Well, without saying this on oath,
they shouldn't require much if anything beyond Leaving Cert.
And though they shouldn't be too predictable,
there are certain recognisable families of problems,
or perhaps one should say,
families of solutions.
First, there are the ``moduli'' problem.
There was a nice one of these in the 1987 Superbrain:
Is 314154314155314156314157314158314159 a prime?
{\em So that must be it!
At last.
Fancy taking all that time to hit on it.
It's just a calculation modulo $n$,
for some $n$.
Probably just multiplication by $a$,
for some $a$.
So that's it \dots
just have to work out $n$ and $a$.
I suppose $n$ must be 16,
since the largest number is 15.
And $a$ is \dots
Damn, I was sure I had it.}
Then there are the infinite series to be summed.
Here there seem to be 2 common themes:
firstly, differentiating power-series and substituting (usually $x = 1$);
secondly, expressing the $n$th term $a(n)$ in the form
\[
a(n) = b(n+1) - b(n),
\]
where $b(n) \to 0$,
so that
\[
\sum_1^\infty a(n) = b(1).
\]
I rather liked a variant of this I hadn't met before,
using the relation
\[
\tan^{-1}a - \tan^{-1}b = tan^{-1} \frac{a - b}{1 + ab},
\]
which is just another way of saying
\[
\tan(\theta - \phi) = \frac{\tan\theta - \tan\phi}{1 + \tan\theta\tan\phi}.
\]
A pretty example of this is the sum
\[
\sum_1^\infty \tan^{-1} \frac{1}{2n^2}.
\]
Another is
\[
\sum_1^\infty \tan^{-1} \frac{1}{n^2 + n + 1}.
\]
A series surprisingly susceptible to the same difference technique
is:
\[
\sum_1^\infty \frac{2^n}{2^{2^n} + 1}.
\]
{\em What do you mean, ten minutes more?
Oh my God, my watch has stopped.
Wait a moment.
Inspiration, where are you?
I never liked that man MacHale.
Did you notice his eyebrows?
The crime rate has rocketed since he started his Superbrain.
Surely parents don't expect their children to be subjected
to this kind of thing when they send them to college.}
Then there are the `sporadic' questions---once-off,
never seen before and never to be seen again.
There was one like that in the 1990 Intervarsity:
Find any solution in positive integers of
\[
x^x y^y = z^z.
\]
The School of Maths in TCD
ground to a halt for a week,
as we all looked for solutions;
and I noticed the mathematicians from UCD
looking very tired and emotional at that time.
{\em What do you mean, is that all I've written?
Bloody supercilious little creep.
I've got better things to do than sit around all day
answering silly questions.
I'm going for a drink.
You see what that man has done to me,
I never drink at this time of day.
I wonder which is his car.
How would he like to find all 4 tyres flat.
I hear sugar in the petrol has a devastating effect.
OK, it must have been easy.
Shall I ask that nasty type sitting in front of me,
who spent the entire time scribbling.
No. It would be too shaming.
There are some things it is better not to know.}
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