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{\centerline {\bf THE 1990 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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Given triagnle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$
respectively and let $G$ be the centroid of the triangle.
For each value of $\angle BAC$, how many non-similar triangles are there in
which $AEGF$ is a cyclic quadrilateral?
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{\bf Question 2}
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Let $a_1$, $a_2$, \ldots, $a_n$ be positive real numbers, and let $S_k$ be the
sum of the products of $a_1$, $a_2$, \ldots, $a_n$ taken $k$ at a time. Show
that
$$S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n$$
for $k = 1$, 2, \ldots, $n - 1$.
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{\bf Question 3}
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Consider all the triangles $ABC$ which have a fixed base $AB$ and whose
altitude from $C$ is a constant $h$. For which of these triangles is the
product of its altitudes a maximum?
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{\bf Question 4}
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A set of 1990 persons is divided into non-intersecting subsets in such a way
that
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1. No one in a subset knows all the others in the subset,
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2. Among any three persons in a subset, there are always at least two who do
not know each other, and
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3. For any two persons in a subset who do not know each other, there is exactly
one person in the same subset knowing both of them.
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(a) Prove that within each subset, every person has the same number of
acquaintances.
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(b) Determine the maximum possible number of subsets.
Note: It is understood that if a person $A$ knows person $B$, then person $B$
will know person $A$; an acquaintance is someone who is known. Every person is
assumed to know one's self.
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{\bf Question 5}
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Show that for every integer $n \geq 6$, there exists a convex hexagon which can
be dissected into exactly $n$ congruent triangles.
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