jmc

Question

The lengths of the sides of a triangle with positive area are _1012, _1075, and _10n, where n is a positive integer. Find the number of possible values for n.

Accepted Answer

By the triangle inequality, a nondegenerate triangle with these side lengths exists if and only if

⎩ ⎨ ⎧ lo g_{10} 75 + lo g_{10} n lo g_{10} 12 + lo g_{10} 75 lo g_{10} 12 + lo g_{10} n > lo g_{10} 12, > lo g_{10} n, > lo g_{10} 75.

The first inequality is always true, because

lo g_{10} 75 > lo g_{10} 12

and

lo g_{10} n > 0.

The second inequality gives

lo g_{10} (12 \cdot 75) > lo g_{10} n,

so

12 \cdot 75 = 900 > n .

The third inequality gives

lo g_{10} (12 n) > lo g_{10} 75,

so

12 n > 75,

or

n > \frac{75}{12} = 6.25.

Thus, the possible values for

n

are

n = 7, 8, 9, \dots, 899,

which makes

899 - 7 + 1 = 893

values of

n .