1994 AIME Problem 10

Question

In triangle

ABC,

angle

C

is a right angle and the altitude from

C

meets

\overline{AB}

at

D.

The lengths of the sides of

\triangle ABC

are integers,

BD=29^3,

and

\cos B=\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m+n.

Answer

Use similarity to write

BD=\frac{BC^2}{AB}

Express

\frac{BC}{AB}=\frac{m}{n}

in lowest terms and use the prime factorization of

29^3

Write

\frac{BC}{AB}=\frac{m}{n}

in lowest terms, with

BC=km

and

AB=kn.

Similarity gives

29^3=BD=\frac{BC^2}{AB}=\frac{km^2}{n}.

Hence

k

is divisible by

n;

writing

k=nt

gives

tm^2=29^3.

The possibility

m=1

cannot be a leg of a nondegenerate integer right triangle, so

m=29.

Writing the other leg of the primitive triple as

u,

we have

(n-u)(n+u)=29^2,

giving

n=421

and

u=420.

Therefore

m+n=29+421=450.

Problem 10 in other years