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 
.
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Solution
One could recursively compute the probabilities of reaching as the first axes point from any point 
as ![\[P(x,y) = \frac{1}{3} P(x-1,y) + \frac{1}{3} P(x,y-1) + \frac{1}{3} P(x-1,y-1)\]](http://latex.artofproblemsolving.com/c/0/c/c0c68668fdf9c80a5d85ca829509a731584b1d24.png)
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
Solution 2
Obviously, the only way to reach (0,0) is to get to (1,1) and then have a chance to get to (0,0). Let x denote a move left 1 unit, y denote a move down 1 unit, and z denote a move left and down one unit each. The possible cases for these moves are 
and 
. This gives a probability of 
to get to 
. The probability of reaching is 
. This gives 
.
The MAA should have specified that is not divisible by 
or the greatest common divisor of and 
was 1.
See Also
| 2019 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
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