2004 Indonesia MO Problems
Contents
Day 1
Problem 1
How many odd and even divisors of 
are there?
Problem 2
A trough, if filled with cold water tap, will be full in 14 minutes. To empty the full trough with opening the hole on the base of the trough, the water will be all out in 21 minutes. If the cold water tap and the hot water tap are opened simultaneously with the opening of the hole, the trough will be full in 12.6 minutes. Then, how long does it take to full the trough when only the hot water tap is opened and the hole is closed?
Problem 3
In how many ways can we change the sign 
with 
or 
, such that the following equation is true?
![\[1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29\]](http://latex.artofproblemsolving.com/2/2/d/22db81cccf46c5a20ab9b552f95dac9e0b2e38ef.png)
Problem 4
There exists 4 circles, 
, such that 
is tangent to both 
and 
, is tangent to both and 
, is both tangent to and , and is both tangent to and . Show that all these tangent points are located on a circle.
Day 2
Problem 5
Given a system of equations:
Then determine the value of 
.
Problem 6
A quadratic equation 
with integers and has roots which are positive integers. Prove that 
is not a prime.
Problem 7
Prove that in a triangle 
with 
as the right angle, where denote the side in front of angle 
, denote the side in front of angle 
, denote the side in front of angle , the diameter of the incircle of equals to 
.
Problem 8
A floor with an area of 
will be covered by 
rugs with various shapes, each having an area of 
. Show that there exist 
overlapping rugs with the overlapped area at least 
.
See Also
| 2004 Indonesia MO (Problems) | ||
| Preceded by 2003 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2005 Indonesia MO |
| All Indonesia MO Problems and Solutions | ||
