2019 AIME I Problems/Problem 5
Contents
Problem 5
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point , it moves at random to one of the points , , or , each with probability , independently of its previous moves. The probability that it will hit the coordinate axes at is , where and are positive integers. Find .
Solution
A move from to is labeled as down (), from to is labeled as left (), and from to is labeled as slant (). To arrive at without arriving at an axis first, the particle must first go to then do a slant move. The particle can arrive at through any permutation of the following 4 different cases: , , , and .
There is only permutation of . Including the last move, there are possible moves, making the probability of this move .
There are permutations of , as the ordering of the two slants do not matter. There are possible moves, making the probability of this move .
There are permutations of , as the ordering of the two downs and two lefts do not matter. There are possible moves, making the probability of this move .
There are permutations of , as the ordering of the three downs and three lefts do not matter. There are possible moves, making the probability of this move .
Adding these, the total probability is . Therefore, the answer is .
Solution by Zaxter22
Solution 2
Alternatively, one could recursively compute the probabilities of reaching as the first axes point from any point as for and the base cases are for any not equal to one. We then recursively find so the answer is .
If this algebra seems intimidating, you can watch a nice pictorial explanation of this by On The Spot Stem. https://www.youtube.com/watch?v=XBRuy3_TM9w
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.