2003 Indonesia MO Problems
Contents
[hide]Day 1
Problem 1
Prove that is divisible by
for every integers
.
Problem 2
Given a quadrilateral . Let
,
,
, and
are the midpoints of
,
,
, and
, respectively.
and
intersects at
. Prove that
and
.
Problem 3
Find all real solutions of the equation .
[Note: For any real number ,
is the largest integer less than or equal to
, and
denote the smallest integer more than or equal to
.]
Problem 4
Given a matrix, where each element is valued
or
. Let
be the product of all elements at the
row, and
be the product of all elements at the
column. Prove that:
Day 2
Problem 5
For every real number , prove the following inequality
and determine when the equality holds.
Problem 6
A hall of a palace is in a shape of regular hexagon, where the sidelength is . The floor of the hall is covered with an equilateral triangle-shaped tile with sidelength
. Every tile is divided into
congruent triangles (refer to the figure). Every triangle-region is colored with a certain color so that each tile has
different colors. The King wants to ensure that no two tiles have the same color pattern. At least, how many colors are needed?
Problem 7
Let be positive integers such that
and the greatest common divisor of
and
is
. Prove that if
divides
, then
.
Problem 8
Given a triangle with
as the right angle, and the sidelengths of the triangle are integers. Determine the possible side lengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle.
See Also
2003 Indonesia MO (Problems) | ||
Preceded by 2002 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2004 Indonesia MO |
All Indonesia MO Problems and Solutions |