Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.
Let be a quadrilateral such that all sides have equal length and angle is .
Let be a line passing through and not intersecting the quadrilateral (except at ). Let and be the points of intersection of with and respectively. Let be the point of intersection of and .
Prove that .
Find the total number of different integer values the function
takes for real numbers with .
be non-zero polynomials with real coeffcients such that for some real number . If and , prove that .
Determine all positive integers n for which the equation
has an integer as a solution.
Let be distinct points in the -plane with the following properties:
(i) both coordinates of are integers, for ;
(ii) there is no point other than and on the line segment joining with whose coordinates are both integers, for .
Prove that for some , there exists a point with coordinates on the line segment joining with such that both and are odd integers.