2017 Indonesia MO Problems
Contents
Day 1
Problem 1
is a parallelogram. is a line passing . Prove that the distance from to is either the sum or the difference of the distance from to , and the distance from to .
Problem 2
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people is a trio if one of the following is true:
- A shakes hands with B, and B shakes hands with C, or
- A doesn't shake hands with B, and B doesn't shake hands with C.
If we consider and as the same trio, find the minimum possible number of trios.
Problem 3
A positive integer is special if every integer can be represented as for some integers .
- Find the smallest positive integer that is not special.
- Prove 2017 is special.
Problem 4
Determine all pairs of distinct real numbers such that both of the following are true:
Day 2
Problem 5
A polynomial has integral coefficients, and it has at least 9 different integral roots. Let be an integer such that . Prove that .
Problem 6
Find the number of positive integers not greater than 2017 such that divides for some positive integer .
Problem 7
Let be a parallelogram. and are on respectively such that the triangles and have the same area. Let intersect at respectively. Prove there exists a triangle whose side lengths are .
Problem 8
A field is made of unit squares. Luffy has gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of in which this is possible.
See Also
2017 Indonesia MO (Problems) | ||
Preceded by 2016 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2018 Indonesia MO |
All Indonesia MO Problems and Solutions |