1. Triangle BCF has a right angle at B. Let A be the point on line CF
such that FA=FB and F lies between A and C.
Point D is chosen so that DA=DC and AC is the bisector of ∠DAB.
Point E is chosen so that EA=ED and AD is the bisector of ∠EAC.
Let M be the midpoint of CF. Let X be the point such that AMXE
is a parallelogram. Prove that BD,FX and ME are concurrent.
2. Find all integers n for which each cell n*n table can be filled with
one of the letters I,M and O in such a way that:
‧in each row and each colume, one third of the entries are I,
one third are M and one third are O; and
‧in any diagonal, if the number of entries on the diagonal is a
multiple of three, then one third of the entries are I,
one third are M and one third are O
Note: the roes and columns of n*n table are each labelled 1 to n in a
natural order. Thus each cell corresponds to a pair of positive integer
(i,j) with 1≦i,j≦n For n>1, the table has 4n-2 diagonals of two types.
A diagonal of 1st type consists all cells (i,j) for which i+j is constant.
A diagonal of 2nd type consists all cells (i,j) for which i-j is constant.
3. Let P = A1A2...Ak be a convex polygon in the plane. The vertices
A1,A2,...,Ak have integral coordinates and lie on a circle. Let S be the
area of P. An odd positive integer n is given such that the squares of
the side lengths of P are integers divisible by n. Prove that 2S ia an
integer divisible by n.