2013 Indonesia MO Problems
Contents
Day 1
Problem 1
In a grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.
Problem 2
Let be an acute triangle and
be its circumcircle. The bisector of
intersects
at [another point]
. Let
be a point on
and inside
. Lines passing
that are parallel to
and
intersects
on
respectively. Lines
intersects
at points
respectively. Prove that
are concurrent.
Problem 3
Determine all positive real such that for any positive reals
, at least one of
is greater than or equal to
.
Problem 4
Supposeis a prime number and
Prove that
is divisible by
.
Day 2
Problem 5
Let be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics
such that:
have positive leading coefficients and zero discriminants (and hence each has a double root)
- The roots of
are different
Problem 6
A positive integer is called "strong" if there exists a positive integer
such that
is divisible by
.
a. Prove that is strong.
b. If
is strong, determine the smallest
(in terms of
) such that
is divisible by
.
Problem 7
Let be a parallelogram. Construct squares
on the outer side of the parallelogram. Construct a square having
as one of its sides and it is on the outer side of
and call its center
. Similarly do it for
to obtain
. Prove that
.
Problem 8
Let be a set of positive integers.
is called "balanced" if [and only if] the number of 3-element subsets of
whose elements add up to a multiple of
is equal to the number of 3-element subsets of
whose elements add up to not a multiple of
.
a. Find a 9-element balanced set.
b. Prove that no set of elements can be balanced.
See Also
2013 Indonesia MO (Problems) | ||
Preceded by 2012 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2014 Indonesia MO |
All Indonesia MO Problems and Solutions |