Difference between revisions of "2002 Indonesia MO Problems"
Rockmanex3 (talk | contribs) (Created page with "==Problem 1== Show that <math>n^4 - n^2</math> is divisible by <math>12</math> for any integers <math>n > 1</math>. Solution ==Prob...") |
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|after=[[2003 Indonesia MO]] | |after=[[2003 Indonesia MO]] | ||
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Latest revision as of 00:08, 4 August 2018
Contents
Problem 1
Show that is divisible by for any integers .
Problem 2
Five regular dices are thrown, one at each time, then the product of the numbers shown are calculated. Which probability is bigger; the product is or the product is ?
Problem 3
Find all real solutions from the following system of equations:
Problem 4
Given a triangle with . On the circumcircle of triangle there exists point , which is the midpoint of arc that contains . Let be a point on such that is perpendicular to . Prove that .
Problem 5
Nine of the following ten numbers: are going to be filled into empty spaces in the table shown below. After all spaces are filled, the sum of the numbers on each row will be the same. And so with the sum of the numbers on each column, will also be the same. Determine all possible fillings.
Problem 6
Find all prime number such that and are also prime.
Problem 7
Let be a rhombus with , and is the intersection of diagonals and . Let , , and are three points on the rhombus' perimeter. If is also a rhombus, show that exactly one of , , and is located on the vertices of rhombus .
See Also
2002 Indonesia MO (Problems) | ||
Preceded by First Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by 2003 Indonesia MO |
All Indonesia MO Problems and Solutions |