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\begin{center}
{\Large \bf THE 1999 CANADIAN MATHEMATICAL OLYMPIAD}
\end{center}
\begin{enumerate}
\item
Find all real solutions to the equation $4x^2 - 40[x] + 51 = 0$.
Here, if $x$ is a real number, then $[x]$ denotes the greatest integer that
is less than or equal to $x$.
\item
Let $ABC$ be an equilateral triangle of altitude 1. A circle
with radius 1 and center on the same side of $AB$ as $C$ rolls along
the segment $AB$. Prove that the arc of the circle that is inside the
triangle always has the same length.
\item
Determine all positive integers $n$ with the property that
$n = (d(n))^2$. Here $d(n)$ denotes the number of positive
divisors of $n$.
\item
Suppose $a_1,a_2,\ldots,a_8$ are eight distinct integers from
$\{1,2,\ldots,16,17\}$. Show that there is an integer $k > 0$ such
that the equation $a_i - a_j = k$ has at least three different solutions.
Also, find a specific set of 7 distinct integers
from $\{1,2,\ldots,16,17\}$ such that
the equation $a_i - a_j = k$ does not have three distinct solutions
for any $k > 0$.
\item
Let $x$, $y$, and $z$ be non-negative real numbers satisfying
$x + y + z = 1$. Show that
$$x^2 y + y^2 z + z^2 x \leq \frac{4}{27}\, ,$$
and find when equality occurs.
\end{enumerate}
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