2010 Question 2

Solve

2009 Question 6

Find all pairs of positive integers m,n > 3 for which there exist infinitely many positive integers a such that it became the same as the picture shown above.

2009 Question 5

Let n be a positive integer.Let T be the set of points (x,y) in the plane where x and y are non-negative integers and x + y < n.
Each point of T is coloured red or blue.If a point (x,y) is red,then so are all points (x¹,y¹) of T with both x¹ and y¹ < y.Define an X-set to be a set of n blue points having distinct x-coordinates,and a Y-set to be a set of n blue points having distinct y-coordinates.Prove that the number of X-sets is equal to the number of Y-sets.

2009 Question 2

The figure shows a rectangle 9cm long and 6 cm wide was divided into 4 triangles with areas S1,S2,S3 and S4 respectively.Given that S1=S2=S3+S4.Find S4