13

Part of 2000 AIME Problems

Problems(2)

Firetruck and Area Covered

Source:

In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at

50

miles per hour along the highways and at

14

miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is

m/n

square miles, where

m

n

are relatively prime positive integers. Find

m+n.

geometrycalculusAMCAIMEnumber theoryrelatively primeAIME I

Source: AIME 2 2002 #13

2000x^6+100x^5+10x^3+x-2=0

has exactly two real roots, one of which is

\frac{m+\sqrt{n}}r,

m, n

r

m

r

are relatively prime, and

r>0.

m+n+r.

algebrapolynomialquadraticsnumber theoryrelatively primequadratic formula