MathDB

3

Part of 2000 CentroAmerican

Problems(2)

Area of a parallelogram

Source: Central American Olympiad 2000, problem 3

8/8/2007
Let ABCDE ABCDE be a convex pentagon. If P P, Q Q, R R and S S are the respective centroids of the triangles ABE ABE, BCE BCE, CDE CDE and DAE DAE, show that PQRS PQRS is a parallelogram and its area is 2/9 2/9 of that of ABCD ABCD.
geometryparallelogramtrigonometryanalytic geometrygraphing linesslopecomplex numbers
"Nice" representations

Source: Central American Olympiad 2000, problem 6

8/9/2007
Let's say we have a nice representation of the positive integer n n if we write it as a sum of powers of 2 in such a way that there are at most two equal powers in the sum (representations differing only in the order of their summands are considered to be the same). a) Write down the 5 nice representations of 10. b) Find all positive integers with an even number of nice representations.
inductionstrong induction