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{\centerline{\bf THE 1993 ASIAN PACIFIC MATHEMATICAL OLYMPIAD}}
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{\it Time allowed: 4 hours}

{\it NO calculators are to be used.}

{\it Each question is worth seven points.}
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{\bf Question 1}
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Let $ABCD$ be a quadrilateral such that all sides have equal length and angle
$ABC$ is $60 \deg$.  Let $l$ be a line passing through $D$ and not intersecting
the quadrilateral (except at $D$).  Let $E$ and $F$ be the points of
intersection of $l$ with $AB$ and $BC$ respectively.  Let $M$ be the point of
intersection of $CE$ and $AF$.

Prove that $CA^2 = CM \times CE$.
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{\bf Question 2}
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Find the total number of different integer values the function $$f(x) = [x] +
[2x] + [\frac{5x}{3}] + [3x] + [4x]$$ takes for real numbers $x$ with $0 \leq x
\leq 100$.
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{\bf Question 3}
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Let
\begin{eqnarray*}
f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\
g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0
\end{eqnarray*}
be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for
some real number $r$.  If $a = \max(|a_n|, \ldots, |a_0|)$ and $c =
\max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.
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{\bf Question 4}
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Determine all positive integers $n$ for which the equation $$x^n + (2+x)^n +
(2-x)^n = 0$$ has an integer as a solution.
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{\bf Question 5}
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Let $P_1$, $P_2$, \ldots, $P_{1993} = P_0$ be distinct points in the $xy$-plane
with the following properties:
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(i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$;
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(ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment
joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0,
1, \ldots, 1992$.
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Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with
coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such
that both $2q_x$ and $2q_y$ are odd integers.
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