1985 AIME Problem 9

Question

In a circle, parallel chords of lengths

2,

3,

and

4

determine central angles of

\alpha,

\beta,

and

\alpha+\beta

radians, respectively, where

\alpha+\beta\lt\pi.

If

\cos\alpha,

which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Answer

A chord subtending angle

\theta

has length

2R\sin(\frac{\theta}{2})

Let

x=\cos(\frac{\alpha}{2})

and

y=\cos(\frac{\beta}{2}),

then eliminate the common radius

Put

k=\frac{1}{2R},

x=\cos(\frac{\alpha}{2}),

and

y=\cos(\frac{\beta}{2}).

The chord data give

\begin{aligned} \sin\frac\alpha2&=2k,\\ \sin\frac\beta2&=3k,\\ \sin\frac{\alpha+\beta}{2}&=4k. \end{aligned}

The addition formula yields

2y+3x=4.

Also

\frac{1-x^2}{4}=\frac{1-y^2}{9},

so

9x^2-4y^2=5.

Substituting

y=\frac{4-3x}{2}

gives

x=\frac{7}{8}.

Hence

\cos\alpha=2x^2-1=\frac{17}{32},

and the requested sum is

17+32=49.

Problem 9 in other years