Difference between revisions of "2001 AIME I Problems/Problem 10"
m |
(solution (written by JesusFreak197)) |
||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | Let <math>S</math> be the set of points whose | + | Let <math>S</math> be the [[set]] of points whose [[coordinate]]s <math>x,</math> <math>y,</math> and <math>z</math> are integers that satisfy <math>0\le x\le2,</math> <math>0\le y\le3,</math> and <math>0\le z\le4.</math> Two distinct points are randomly chosen from <math>S.</math> The [[probability]] that the [[midpoint]] of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> |
== Solution == | == Solution == | ||
− | {{ | + | The distance between the <math>x</math>, <math>y</math>, and <math>z</math> coordinates must be even so that the midpoint can have integer coordinates. Therefore, |
+ | *For <math>x</math>, we have the possibilities <math>(0,0)</math>, <math>(1,1)</math>, <math>(2,2)</math>, <math>(0,2)</math>, and <math>(2,0)</math>, <math>5</math> possibilities. | ||
+ | *For <math>y</math>, we have the possibilities <math>(0,0)</math>, <math>(1,1)</math>, <math>(2,2)</math>, <math>(3,3)</math>, <math>(0,2)</math>, <math>(2,0)</math>, <math>(1,3)</math>, and <math>(3,1)</math>, <math>8</math> possibilities. | ||
+ | *For <math>z</math>, we have the possibilities <math>(0,0)</math>, <math>(1,1)</math>, <math>(2,2)</math>, <math>(3,3)</math>, <math>(4,4)</math>, <math>(0,2)</math>, <math>(0,4)</math>, <math>(2,0)</math>, <math>(4,0)</math>, <math>(2,4)</math>, <math>(4,2)</math>, <math>(1,3)</math>, and <math>(3,1)</math>, <math>13</math> possibilities. | ||
+ | However, we have <math>3\cdot 4\cdot 5 = 60</math> cases where we have simply taken the same point twice, so we subtract those. Therefore, our answer is <math>\frac {5\cdot 8\cdot 13 - 60}{60\cdot 59} = \frac {23}{177}\Longrightarrow m+n = \boxed{200}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2001|n=I|num-b=9|num-a=11}} | {{AIME box|year=2001|n=I|num-b=9|num-a=11}} | ||
+ | |||
+ | [[Category:Intermediate Combinatorics Problems]] |
Revision as of 18:04, 12 June 2008
Problem
Let be the set of points whose coordinates
and
are integers that satisfy
and
Two distinct points are randomly chosen from
The probability that the midpoint of the segment they determine also belongs to
is
where
and
are relatively prime positive integers. Find
Solution
The distance between the ,
, and
coordinates must be even so that the midpoint can have integer coordinates. Therefore,
- For
, we have the possibilities
,
,
,
, and
,
possibilities.
- For
, we have the possibilities
,
,
,
,
,
,
, and
,
possibilities.
- For
, we have the possibilities
,
,
,
,
,
,
,
,
,
,
,
, and
,
possibilities.
However, we have cases where we have simply taken the same point twice, so we subtract those. Therefore, our answer is
.
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |