MathDB

1

Part of 2001 CentroAmerican

Problems(2)

Number theory prob.

Source: Central American Olympiad 2001, problem 4

8/12/2009
Determine the smallest positive integer n n such that there exists positive integers a1,a2,⋯ ,an a_1,a_2,\cdots,a_n, that smaller than or equal to 15 15 and are not necessarily distinct, such that the last four digits of the sum, a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n! Is 2001 2001.
Game

Source: Central American Olympiad 2001, problem 1

8/12/2009
Two players A A, B B and another 2001 people form a circle, such that A A and B B are not in consecutive positions. A A and B B play in alternating turns, starting with A A. A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. The winner is the last man standing. Show that one of the two players has a winning strategy, and give such strategy. Note: A player has a winning strategy if he/she is able to win no matter what the opponent does.