Problem:
The lengths of the sides of a triangle with positive area are log1012,log1075, and log10n, where n is a positive integer. Find the number of possible values for n.
Solution:
The Triangle Inequality yields
logn<log75+log12=log900, and logn>log75−log12=log(25/4).
Therefore 25/4<n<900, and so 7≤n≤899. Hence there are 899−7+1=893 possible values of n.
The problems on this page are the property of the MAA's American Mathematics Competitions