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\title{%
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Royal Life Ireland\\[3mm]
Irish Intervarsity Mathematics Competition}
\author{Trinity College Dublin 1991}
\date{9.30--12.30 February 9}
\maketitle
\thispagestyle{empty}
\begin{quote}
\sl
Answer all questions.\\
Calculators permitted.
\end{quote}
\begin{enumerate}
\item The prime factorisations of $r+1$ positive integers
($r \geq 1$) together involve only $r$ primes. Prove that there is a
subset of these integers whose product is a perfect square.
\item Consider a polynomial $p(x) = x^n + nx^{n-1} + a_2 x^{n-2}
+ \cdots + a_n$ which has real roots $r_1, r_2, \ldots, r_n$. If
\[
r_1^{16} + r_2^{16} + \cdots + r_n^{16} = n,
\]
find all the roots.
\item An $n$-inch cube ($n$ a positive integer) is painted on all
sides and then cut into 1-inch cubes. If the number of small cubes
with one painted side is the same as the number with two painted
sides, what could $n$ have been?
\item How many ways are there of painting the 6 faces of a cube in 6
different colours, if two colourings are considered the same when one
can be obtained from the other by rotating the cube?
\item How many positive integers $x \leq 1991$ are such that 7
divides $2^x - x^2$?
\item How many ways can 1,000,000 be expressed as a product of 3 positive
integers? Factorisations different only in order are considered to be
the same.
\item Prove that $2^n$ can begin with any sequence of digits.
\item Imagine a point $P$ inside a square $ABCD$. If $|PA|=5$, $|PB|=3$
and $|PC|=7$, what is the side of the square?
\item
Let $f(x)$ be a function such that $f(1) = 1$ and, for $x\geq 1$
\[
f'(x) = \frac{1}{x^2 + f^2(x)}.
\]
Prove that
\[
\lim_{x \to \infty} f(x)
\]
exists and is less than $1+\pi/4$.
\item Prove that the number of odd binomial coefficients in each row of
Pascal's triangle is a power of 2. [In Pascal's triangle
\[
\begin{array}{ccccccc}
& & & 1 \\
& & 1 & & 1 \\
& 1 & & 2 & & 1 \\
1 & & 3 & & 3 & & 1\\
& & & \vdots
\end{array}
\]
each entry is the sum of the entries directly above it.]
\end{enumerate}
\end{document}
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