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## Equation of X and Y | AIME I, 1993 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1993 based on Equation of X and Y.

## Equation of X and Y – AIME I, 1993

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centred mid way between the paths . At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let t be amount of time, in seconds, Before Jenny and Kenny, can see each other again. If t is written as a fraction in lowest terms, find the sum of numerator and denominator.

• is 107
• is 163
• is 840
• cannot be determined from the given information

### Key Concepts

Variables

Equations

Algebra

AIME I, 1993, Question 13

Elementary Algebra by Hall and Knight

## Try with Hints

Let circle be of radius 50

Let start points be (-50,100),(-50,-100) then at time t, end points (-50+t,100),(-50+3t,-100)

or, equation and equation of circle is

y=$\frac{-100}{t}x+200 -\frac{5000}{t}$ is first equation

$50^2=x^2+y^2$ is second equation

when they see again then

$\frac{-x}{y}=\frac{-100}{t}$

or, $y=\frac{xt}{100}$

solving in second equation gives $x=\frac{5000}{\sqrt{100^2+t^2}}$

or, $y=\frac{xt}{100}$

solving in first equation for t gives $t=\frac{160}{3}$

or, 160+3=163.