MathDB

13

Part of 2016 AIME Problems

Problems(2)

Kelvin the Frog

Source: 2016 AIME I #13

3/4/2016
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line y=24y = 24. A fence is located at the horizontal line y=0y = 0. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where y=0y=0, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where y<0y < 0. Freddy starts his search at the point (0,21)(0, 21) and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.
2016 AIME IAIME
How about a nice game of chess?

Source: 2016 AIME II #13

3/17/2016
Beatrix is going to place six rooks on a 6×66\times6 chessboard where both the rows and columns are labelled 11 to 66; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
AMCAIMEAIME II