2008 Indonesia MO Problems
Contents
Day 1
Problem 1
Given triangle . Points outside triangle are chosen such that triangles , , and are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
Problem 2
Prove that for every positive reals and ,
Problem 3
Find all positive integers which can be expressed as where are positive integers that are pairwise relatively prime.
Problem 4
Let .
(a) Find the number of subset of such that the product of its elements is divisible by 7.
(b) Let denotes the number of subset of in which the sum of its elements, when divided by 7, leaves the remainder . Prove that .
Day 2
Problem 5
Let are integers which satisfy and . Is it a must that ?
Problem 6
In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.
Problem 7
Given triangle with sidelengths . Tangents to the incircle of triangle that are parallel with each side of form three small triangles (each of them has one vertex from ). Prove that the sum of area of incircles of these three small triangles and the area of the incircle of triangle is equal to
Problem 8
Let denote the set of all positive integers. Find all functions that satisfies for all natural number .
See Also
2008 Indonesia MO (Problems) | ||
Preceded by 2007 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2009 Indonesia MO |
All Indonesia MO Problems and Solutions |