Difference between revisions of "2013 AIME I Problems/Problem 1"
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The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling? | The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling? | ||
| − | == Solution == | + | == Solution One == |
Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have | Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have | ||
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Solving for <math>r</math>, we find <math>r = 1/50</math>, so the time Tom spends biking is <math>\frac{30}{(10)(1/50)} = \boxed{150}</math> | Solving for <math>r</math>, we find <math>r = 1/50</math>, so the time Tom spends biking is <math>\frac{30}{(10)(1/50)} = \boxed{150}</math> | ||
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| + | == Solution Two == | ||
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| + | Tom trips and dies, therefore, because the decomposition of his body is similar to the fractional decomposition of the formula <math>\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255</math>, we can easily express it in terms of e^i<math>\pi</math>theta, through Euler's formula. i is <math>\sqrt-1</math>, but we were taught in Algebra class not to worry about the <math>\sqrt </math> of negative numbers, therefore the answer is <math>\boxed{000}</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2013|n=I|before=First Problem|num-a=2}} | {{AIME box|year=2013|n=I|before=First Problem|num-a=2}} | ||
Revision as of 16:23, 20 March 2013
Problem 1
The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
Solution One
Let 
represent the rate Tom swims in miles per minute. Then we have
Solving for , we find 
, so the time Tom spends biking is 
Solution Two
Tom trips and dies, therefore, because the decomposition of his body is similar to the fractional decomposition of the formula , we can easily express it in terms of e^i
theta, through Euler's formula. i is 
, but we were taught in Algebra class not to worry about the $\sqrt$ (Error compiling LaTeX. Unknown error_msg) of negative numbers, therefore the answer is 
See also
| 2013 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
