Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.
Let be the centroid of triangle and be the midpoint of . Let be on and on such that the points and are collinear and and are parallel.
Suppose that and intersect at and and intersect at . Show that triangle is similar to triangle .
Suppose there are 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 red points in
the plane. Can you find a special case with exactly red points?
Let be positive real numbers such that . Show that
During a break, 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 for which eventually, perhaps
after many rounds, all children will have at least one candy each.
Given are two tangent circles and a point 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 .