2013 AIME I Problems/Problem 1
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^itheta, through Euler's formula. i is , but we were taught in Algebra I class in 8th grade not to worry about the $\sqrt$ (Error compiling LaTeX. ! Missing { inserted.) 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 |