2006 Indonesia MO Problems
Contents
Day 1
Problem 1
Find all pairs 
of real numbers which satisfy 
and 
.
Problem 2
Let 
be positive integers. If 
, prove that 
.
Problem 3
Let 
be the set of all triangles 
which have property: 
are positive integers. Prove that all triangles in are similar.
Problem 4
A black pawn and a white pawn are placed on the first square and the last square of a 
chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.
Day 2
Problem 5
In triangle , 
is the midpoint of side 
and 
is the centroid of triangle . A line 
passes through , intersecting line 
at 
and line 
at 
, where 
and 
. If ![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
denotes the area of triangle 
, show that ![$\frac{[BGM]}{[PAG]}+\frac{[CMG]}{[QGA]}=\frac32$](http://latex.artofproblemsolving.com/c/a/e/cae40449c33ad07f639387734db7f85773cb09e8.png)
.
Problem 6
Every phone number in an area consists of eight digits and starts with digit 
. Mr. Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr. Edy gets a phone number which consists of at most five different digits?
Problem 7
Let be real numbers such that 
are rational numbers. Prove that there are integers 
, not all of them are 
, such that 
.
Problem 8
Find the largest 
-digit integer which has property: the sum of its digits equals to the product of its digits.
See Also
| 2006 Indonesia MO (Problems) | ||
| Preceded by 2005 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2007 Indonesia MO |
| All Indonesia MO Problems and Solutions | ||
