4
Part of 2019 AIME Problems
Problems(2)
Soccer substitutions
Source: 2019 AIME I #4
3/14/2019
A soccer team has 22 available players. A fixed set of 11 players starts the game, while the other 11 are available as substitutes. During the game, the coach may make as many as 3 substitutions, where any one of the 11 players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when is divided by 1000.
AMCAIMEAIME I2019 AIME I2019 AMC
Perfect Square Dice
Source: 2019 AIME II #4
3/22/2019
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is , where and are relatively prime positive integers. Find .
2019 AMCAMCAIMEAIME II2019 AIME IIprobabilitysimilar triangles