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{\centerline {\bf THE 1991 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 $G$ be the centroid of triangle $ABC$ and $M$ be the midpoint of $BC$. Let
$X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are
collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect
at $Q$ and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar
to triangle $ABC$.
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{\bf Question 2}
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Suppose there are 997 points given in a plane. If every two points are joined
by a line segment with its midpoint coloured in red, show that there are at
least 1991 red points in the plane. Can you find a special case with exactly
1991 red points?
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{\bf Question 3}
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Let $a_1$, $a_2$, \ldots, $a_n$, $b_1$, $b_2$, \ldots, $b_n$ be positive real
numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$. Show
that $$\frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots +
\frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2} \ .$$
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{\bf Question 4}
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During a break, $n$ children at school sit in a circle around their teacher to
play a game. The teacher walks clockwise close to the children and hands out
candies to some of them according to the following rule. He selects one child
and gives him a candy, then he skips the next child and gives a candy to the
next one, then he skips 2 and gives a candy to the next one, then he skips 3,
and so on. Determine the values of $n$ for which eventually, perhaps after
many rounds, all children will have at least one candy each.
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{\bf Question 5}
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Given are two tangent circles and a point $P$ on their common tangent
perpendicular to the lines joining their centres. Construct with ruler and
compass all the circles that are tangent to these two circles and pass through
the point $P$.
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