# Difference between revisions of "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 $r$ represent the rate Tom swims in miles per minute. Then we have

$\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255$

Solving for $r$, we find $r = 1/50$, so the time Tom spends biking is $\frac{30}{(10)(1/50)} = \boxed{150}$

## Solution Two

Tom trips and dies, therefore, because the decomposition of his body is similar to the fractional decomposition of the formula $\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255$, we can easily express it in terms of e^i$\pi$theta, through Euler's formula. i is $\sqrt-1$, 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 $\boxed{000}$