2006 AIME II Problem 2

Question

The lengths of the sides of a triangle with positive area are

\log_{10} 12,

\log_{10} 75,

and

\log_{10} n,

where

n

is a positive integer. Find the number of possible values for

n.

Answer

The three lengths must satisfy the triangle inequality; two of the three conditions bound

\log_{10} n

from both sides.

\log 75 - \log 12

\lt \log n \lt

\log 75 + \log 12,

so

\frac{25}{4} \lt n \lt 900.

The triangle inequality requires

\log n \lt \log 12 + \log 75 = \log 900

and

\log 12 + \log n \gt \log 75,

that is

\log n \gt \log 75 - \log 12 = \log \frac{25}{4}.

The remaining inequality,

\log 75 + \log n \gt \log 12,

is automatic because

n \ge 1

and

75 \gt 12.

So

\frac{25}{4} \lt n \lt 900,

which for integers means

7 \le n \le 899.

That gives

899 - 7 + 1 = 893

possible values of

n.

Problem 2 in other years