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

Answer: 450

Concepts:altitude Pythagorean Triple divisibility

Difficulty rating: 2270

Small Hint:

Use similarity to write $BD=\frac{BC^2}{AB}$

Big Hint:

Express $\frac{BC}{AB}=\frac{m}{n}$ in lowest terms and use the prime factorization of $29^3$

Solution:

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 $m+n=29+421=450.$

1994 AIME Problem 10

Answer: 450

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 10 in Other Years