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{\centerline {\bf THE 1989 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 $x_1$, $x_2$, \ldots, $x_n$ be positive real numbers, and let $$S = x_1 +
x_2 + \cdots + x_n.$$  Prove that $$(1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1
+ S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \ .$$
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{\bf Question 2}
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Prove that the equation $$6(6a^2 + 3b^2 + c^2) = 5n^2$$ has no solutions in
integers except $a = b = c = n = 0$.
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{\bf Question 3}
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Let $A_1$, $A_2$, $A_3$ be three points in the plane, and for convenience, let
$A_4= A_1$, $A_5 = A_2$.  For $n = 1$, 2, and 3, suppose that $B_n$ is the
midpoint of $A_n A_{n+1}$, and suppose that $C_n$ is the midpoint of $A_n B_n$.
Suppose that $A_n C_{n+1}$ and $B_n A_{n+2}$ meet at $D_n$, and that $A_n
B_{n+1}$ and $C_n A_{n+2}$ meet at $E_n$.  Calculate the ratio of the area of
triangle $D_1 D_2 D_3$ to the area of triangle $E_1 E_2 E_3$.
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{\bf Question 4}
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Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the
property that \linebreak $1 \leq a < b \leq n$.  Show that there are at least
$$4m \cdot \frac{(m - \frac{n^2}{4})}{3n}$$ triples $(a,b,c)$ such that
$(a,b)$, $(a,c)$, and $(b,c)$ belong to S.
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{\bf Question 5}
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Determine all functions $f$ from the reals to the reals for which

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(1) $f(x)$ is strictly increasing,

(2) $f(x) + g(x) = 2x$ for all real $x$,

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where $g(x)$ is the composition inverse function to $f(x)$.  (Note: $f$ and $g$
are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all
real $x$.)
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