Problem:
Find the smallest positive integer for which the expansion of , after like terms have been collected, has at least terms.
Solution:
Notice that
The simplified expansions of and have terms each. When these two expansions are multiplied together, terms of the form are produced. No two of these terms are like terms because they differ in at least one exponent. Hence the expansion of has terms. To make , we need . The smallest such integer is .
The problems on this page are the property of the MAA's American Mathematics Competitions