Problem:
For each even positive integer , let denote the greatest power of that divides . For example, and . For each positive integer , let . Find the greatest integer less than such that is a perfect square.
Solution:
Note that is defined as the sum of the greatest powers of that divide the consecutive even numbers . Of these, are divisible by but not are divisible by but not are divisible by but not , and the only number not accounted for is . Thus
In order for to be a perfect square, must be odd, because if were even, then the prime factorization of would have an odd number of factors of . Because is odd, must be a square, and because is even, must be the square of an even integer. The greatest that is less than the square of an even integer is .
The problems on this page are the property of the MAA's American Mathematics Competitions