▪ Third Benelux Mathematical Olympiad 2011

Problem 1

An ordered pair of integers with is said to be a Benelux couple if the following two conditions hold: has the same prime divisors as , and has the same prime divisors as .

(a) Find three Benelux couples with .

(b) Prove that there are infinitely many Benelux couples.

Problem 2

Let be a triangle with incentre . The angle bisectors , and meet , and at , , , respectively. The perpendicular bisector of intersects the lines and at and , respectively. Show that , , and lie on a circle.

Problem 3

If is an integer, let denote the largest cube that is less than or equal to . Find all positive integers for which the following sequence is bounded:

and for .

Problem 4

Abby and Brian play the following game: They first choose a positive integer . Then they write numbers on a blackboard in turn. Abby starts by writing a . Thereafter, when one of them has written the number , the other writes down either or , provided that the number is not greater than . The player who writes on the blackboard wins.

(a) Determine which player has a winning strategy if .

(b) Find the number of positive integers for which Brian has a winning strategy.

Λύση του 2ου προβλήματος: Φραγκάκης Νικόλαος