Time allowed: 4 hours
No calculators are to be used
Each question is worth 7 points
Determine all finite nonempty sets of positive integers satisfying is an element of for all ; where is the greatest common divisor of and .
Let be the circumcentre and the orthocentre of an acute triangle . Prove that the area of one of the triangles and is equal to the sum of the areas of the other two.
Let a set of points in the plane be given, no three of which are collinear. Let denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of with at most two colours, such that for any points of , the number of lines in which separate from is odd if and only if and have the same colour.
Note: A line separates two points and if and lie on opposite sides of with neither point on .
For a real number , let stand for the largest integer that is less than or equal to . Prove that is even for every positive integer .
ចូរបង្ហាញថា ចំពោះគ្រប់ចំនួនពិត .