2014 Indonesia MO Problems
Contents
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 |