4th OMCC 2002


A1. For which n > 2 can the numbers 1, 2, ... , n be arranged in a circle so that each number divides the sum of the next two numbers (in a clockwise direction)?
A2. ABC is acute-angled. AD and BE are altitudes. area BDE ≤ area DEA ≤ area EAB ≤ ABD. Show that the triangle is isosceles.
A3. Define the sequence a₁, a₂, a₃, ... by a₁ = A, a_n+1 = a_n + d(an), where d(m) is the largest factor of m which is < m. For which integers A > 1 is 2002 a member of the sequence?
B1. ABC is a triangle. D is the midpoint of BC. E is a point on the side AC such that BE = 2AD. BE and AD meet at F and ∠FAE = 60^o. Find ∠FEA.
B2. Find an infinite set of positive integers such that the sum of any finite number of distinct elements of the set is not a square.
B3. A path from (0,0) to (n,n) on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its 2n+1 vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices (0,0), (n,0), (n,n), (0,n) into two parts with equal area.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.