## THE 1991 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.

Question 1
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 $ABC$.
Question 2
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?
Question 3
Let $a_1, a_2, . . . , a_n, b_1, b_2, . . . , b_n$ be positive real numbers such that $a_1 + a_2 +...+ a_n =b_1 + b_2 + ...+ b_n$. Show that

$\displaystyle \frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+...+\frac{a_n^2}{a_n+b_n} \geq \frac{a_1+a_2+...+a_n}{2}$

Question 4
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.
Question 5
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 $P$.