2009 Indonesia MO Problems
Contents
Day 1
Problem 1
Find all positive integers such that is divisible by .
Problem 2
For any real , let be the largest integer that is not more than . Given a sequence of positive integers such that and Prove that holds for every positive integer .
Problem 3
For every triangle , let be a point located on segment , respectively. Let be the intersection of and . Prove that:
Problem 4
In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: )
Day 2
Problem 5
In a drawer, there are at most balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is . Determine the maximum amount of white balls in the drawer, such that the probability statement is true?
Problem 6
Find the lowest possible values from the function for any real numbers .
Problem 7
A pair of integers is called good if Given 2 positive integers which are relatively prime, prove that there exists a good pair with and , but and .
Problem 8
Given an acute triangle . The incircle of triangle touches respectively at . The angle bisector of cuts and respectively at and . Suppose is one of the altitudes of triangle , and be the midpoint of . (a) Prove that and are perpendicular with the angle bisector of . (b) Show that is a cyclic quadrilateral.
See Also
2009 Indonesia MO (Problems) | ||
Preceded by 2008 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2010 Indonesia MO |
All Indonesia MO Problems and Solutions |