18th Iberoamerican Mathematical Olympiad Problems 2003

A1.  Let A, B be two sets of N consecutive integers. If N = 2003, can we form N pairs (a, b) with a ∈ A, b ∈ B such that the sums of the pairs are N consecutive integers? What about N = 2004? A2.  C is a point on the semicircle with diameter AB. D is a point on the arc BC. M, P, N are the midpoints of AC, CD and BD. The circumcenters of ACP and BDP are O, O'

17th Iberoamerican Mathematical Olympiad Problems 2002

A1.  The numbers 1, 2, ... , 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ... , 3k+1th, ... numbers in the list are erased. Then the 1st, 4th, 7th, ... 3k+1th numbers in the remaining list are erased (leaving 3, 5, 8, 9, 12, ... ). This process is carried out repeatedly until there are no numbers left. What is the last number to be erased?

16th Iberoamerican Mathematical Olympiad Problems 2001

A1.  Show that there are arbitrarily large numbers n such that: (1) all its digits are 2 or more; and (2) the product of any four of its digits divides n. A2.  ABC is a triangle. The incircle has center I and

15th Iberoamerican Mathematical Olympiad Problems 2000

15th Iberoamerican Mathematical Olympiad Problems 2000A1.  Label the vertices of a regular n-gon from 1 to n > 3. Draw all the diagonals. Show that if n is odd then we can label each side and diagonal with a number from 1 to n different from the labels of its endpoints so that at each vertex the sides and diagonals all have different labels.