MathDB

8

Part of 2004 AIME Problems

Problems(2)

Stars

Source:

11/27/2005
Define a regular nn-pointed star to be the union of nn line segments P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 such that ∙\bullet the points P1,P2,…,PnP_1, P_2,\ldots, P_n are coplanar and no three of them are collinear, ∙\bullet each of the nn line segments intersects at least one of the other line segments at a point other than an endpoint, ∙\bullet all of the angles at P1,P2,…,PnP_1, P_2,\ldots, P_n are congruent, ∙\bullet all of the nn line segments P2P3,…,PnP1P_2P_3,\ldots, P_nP_1 are congruent, and ∙\bullet the path P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?
number theoryleast common multiplefunctionrelatively prime
2004^2004

Source:

7/13/2005
How many positive integer divisors of 200420042004^{2004} are divisible by exactly 20042004 positive integers?
AMCAIMEAIME I