2014 Indonesia MO Problems
Contents
[hide]Day 1
Problem 1
Is it possible to fill a grid with each of the numbers
once each such that the sum of any two numbers sharing a side is prime?
Problem 2
For some positive integers , the system
and
has exactly one integral solution
. Determine all possible values of
.
Problem 3
Let be a trapezoid (quadrilateral with one pair of parallel sides) such that
. Suppose that
and
meet at
and
and
meet at
. Construct the parallelograms
and
. Prove that
passes through the midpoint of the segment
.
Problem 4
Determine all polynomials with integral coefficients such that if
are the sides of a right-angled triangle, then
are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if
is the hypotenuse of the first triangle, it's not necessary that
is the hypotenuse of the second triangle, and similar with the others.)
Day 2
Problem 5
A sequence of positive integers satisfies
for all positive integers
satisfying
. Prove that if
divides
then
.
Problem 6
Let be a triangle. Suppose
is on
such that
bisects
. Suppose
is on
such that
, and
is on
such that
. If
and
intersect on
, prove that
.
Problem 7
Suppose that are positive integers with
. Prove that:
Problem 8
A positive integer is called beautiful if it can be represented in the form for two distinct positive integers
. A positive integer that is not beautiful is ugly.
a) Prove that is a product of a beautiful number and an ugly number.
b) Prove that the product of two ugly numbers is also ugly.
See Also
2014 Indonesia MO (Problems) | ||
Preceded by 2013 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2015 Indonesia MO |
All Indonesia MO Problems and Solutions |