2nd OMCC 2000


A1. Find all three digit numbers abc (with a ≠ 0) such that a² + b² + c² divides 26.
A2. The diagram shows two pentominos made from unit squares. For which n > 1 can we tile a 15 x n rectangle with these pentominos?
A3. ABCDE is a convex pentagon. Show that the centroids of the 4 triangles ABE, BCE, CDE, DAE from a parallelogram with whose area is 2/9 area ABCD.
B1. Write an integer in each small triangles so that every triangle with at least two neighbors has a number equal to the difference between the numbers in two of its neighbors.
B2. ABC is acute-angled. The circle diameter AC meets AB again at F, and the circle diameter AB meets AC again at E. BE meets the circle diameter AC at P, and CF meets the circle diameter AB at Q. Show that AP = AQ.
B3. A nice representation of a positive integer n is a representation of n as sum of powers of 2 with each power appearing at most twice. For example, 5 = 4 + 1 = 2 + 2 + 1. Which positive integers have an even number of nice representations?

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