Difference between revisions of "2012 AIME II Problems/Problem 4"

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== Problem 4 ==
 
== Problem 4 ==
 
Ana, Bob, and Cao bike at constant rates of <math>8.6</math> meters per second, <math>6.2</math> meters per second, and <math>5</math> meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point <math>D</math> on the south edge of the field. Cao arrives at point <math>D</math> at the same time that Ana and Bob arrive at <math>D</math> for the first time. The ratio of the field's length to the field's width to the distance from point <math>D</math> to the southeast corner of the field can be represented as <math>p : q : r</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are positive integers with <math>p</math> and <math>q</math> relatively prime. Find <math>p+q+r</math>.
 
Ana, Bob, and Cao bike at constant rates of <math>8.6</math> meters per second, <math>6.2</math> meters per second, and <math>5</math> meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point <math>D</math> on the south edge of the field. Cao arrives at point <math>D</math> at the same time that Ana and Bob arrive at <math>D</math> for the first time. The ratio of the field's length to the field's width to the distance from point <math>D</math> to the southeast corner of the field can be represented as <math>p : q : r</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are positive integers with <math>p</math> and <math>q</math> relatively prime. Find <math>p+q+r</math>.
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== Solution ==
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== See also ==
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{{AIME box|year=2012|n=II|num-b=3|num-a=5}}

Revision as of 17:20, 31 March 2012

Problem 4

Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.


Solution

See also

2012 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions