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 Y B and GC intersect at P. Show that triangle MP Q 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

\frac{a_{1}^{2}}{a_{1} + b_{1}} + \frac{a_{2}^{2}}{a_{2} + b_{2}} + ⋯ + \frac{a_{n}^{2}}{a_{n} + b_{n}} ≥ \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.