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{\centerline{\bf THE 1994 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 $f:\Bbb{R} \rightarrow \Bbb{R}$ be a function such that
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(i) For all $x,y \in \Bbb{R}$, $$f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y),$$
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(ii) For all $x \in [0,1)$, $f(0) \geq f(x)$,
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(iii) $-f(-1) = f(1) = 1$.
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Find all such functions $f$.
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{\bf Question 2}
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Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$,
and circumradius $R$, prove that $|OH| < 3R$.
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{\bf Question 3}
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Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively
prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$
divides $ab$.  Determine all such $n$.
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{\bf Question 4}
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Is there an infinite set of points in the plane such that no three points are
collinear, and the distance between any two points is rational?
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{\bf Question 5}
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You are given three lists A, B, and C.  List A contains the numbers of the form
$10^k$ in base 10, with $k$ any integer greater than or equal to 1.  Lists B
and C contain the same numbers translated into base 2 and 5 respectively:
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\centerline{
\begin{tabular}{lll}
A & B & C \\
10 & 1010 & 20 \\
100 & 1100100 & 400 \\
1000 & 1111101000 & 13000 \\
\vdots & \vdots & \vdots
\end{tabular}
}
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Prove that for every integer $n > 1$, there is exactly one number in exactly
one of the lists B or C that has exactly $n$ digits.
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