MathDB

10

Part of 2006 AIME Problems

Problems(2)

Stacked Circles

Source:

3/8/2006
This is the one with the 8 circles? I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid...
geometryanalytic geometrygraphing linesslopegeometric transformationparallelogramrectangle
Soccer

Source: 2006 AIME II 10

3/28/2006
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a 50%50\% chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team AA beats team BB. The probability that team AA finishes with more points than team BB is m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n.
probabilitysymmetrynumber theoryrelatively primeAMC