1993 AMC 12 Problem 29

Question

Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a “box”]? (An external diagonal is a diagonal of one of the rectangular faces of the box.)

\{4,5,6\}

\{4,5,7\}

\{4,6,7\}

\{5,6,7\}

\{5,7,8\}

Answer: B

Concepts:rectangular prism face diagonals Pythagorean inequalities

Difficulty rating: 2030

Small hint:

If

p\le q\le r

are the face diagonals, express the edge squares in terms of

p^2,q^2,r^2

Big hint:

A necessary and sufficient positivity condition is

r^2\lt p^2+q^2

Solution:

If the edge lengths are

a,

b,

c,

then the three face-diagonal squares are

a^2+b^2,

a^2+c^2,

b^2+c^2.

For sorted diagonals

p\le q\le r,

a^2=\frac{p^2+q^2-r^2}{2},

so necessarily

r^2\lt p^2+q^2;

the analogous formulas show this is also sufficient. Only

\{4,5,7\}

fails, since

7^2=49\gt16+25=41.

Thus the correct answer is B.