Time allowed: 4 hours
No calculators to be used
Each question is worth 7 points
For a positive integer let be the sum of digits in the decimal representation of . Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of is called a stump of . Let be the sum of all stumps of . Prove that
Find the largest positive integer so that the number of integers in the set which are divisible by is equal to the number of integers which are divisible by or (or both).
Let two equal regular -gons and be located in the plane such that their intersection is a -gon (). The sides of the polygon are coloured in red and the sides of in blue.
Prove that the sum of the lengths of the blue sides of the polygon is equal to the sum of the lengths of its red sides.
A point in the plane with a Cartesian coordinate system is called a mixed point if one of its coordinates
is rational and the other one is irrational. Find all polynomials with real coefficients such that their
graphs do not contain any mixed point.
Find the greatest integer , such that there are points in the plane with that satisfy the following condition: for each triangles and are equal.