Difference between revisions of "2017 AMC 12A Problems/Problem 22"
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A square is drawn in the Cartesian coordinate plane with vertices at <math>(2, 2)</math>, <math>(-2, 2)</math>, <math>(-2, -2)</math>, <math>(2, -2)</math>. A particle starts at <math>(0,0)</math>. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is <math>1/8</math> that the particle will move from <math>(x, y)</math> to each of <math>(x, y + 1)</math>, <math>(x + 1, y + 1)</math>, <math>(x + 1, y)</math>, <math>(x + 1, y - 1)</math>, <math>(x, y - 1)</math>, <math>(x - 1, y - 1)</math>, <math>(x - 1, y)</math>, or <math>(x - 1, y + 1)</math>. The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>? | A square is drawn in the Cartesian coordinate plane with vertices at <math>(2, 2)</math>, <math>(-2, 2)</math>, <math>(-2, -2)</math>, <math>(2, -2)</math>. A particle starts at <math>(0,0)</math>. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is <math>1/8</math> that the particle will move from <math>(x, y)</math> to each of <math>(x, y + 1)</math>, <math>(x + 1, y + 1)</math>, <math>(x + 1, y)</math>, <math>(x + 1, y - 1)</math>, <math>(x, y - 1)</math>, <math>(x - 1, y - 1)</math>, <math>(x - 1, y)</math>, or <math>(x - 1, y + 1)</math>. The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>? | ||
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| + | ==Solution== | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2017|ab=A|num-b=21|num-a=23}} | {{AMC12 box|year=2017|ab=A|num-b=21|num-a=23}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 17:59, 8 February 2017
Problem
A square is drawn in the Cartesian coordinate plane with vertices at 
, 
, 
, 
. A particle starts at 
. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is 
that the particle will move from 
to each of 
, 
, 
, 
, 
, 
, 
, or 
. The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is 
, where 
and 
are relatively prime positive integers. What is 
?
Solution
See Also
| 2017 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
