2006 USAMO Problems
Let be a prime number and let be an integer with . Prove that there exists integers and with and
if and only if is not a divisor of .
Note: For a real number, let denote the greatest integer less than or equal to , and let denote the fractional part of x.
For a given positive integer k find, in terms of k, the minimum value of for which there is a set of distinct positive integers that has sum greater than but every subset of size k has sum at most .
For integral , let be the greatest prime divisor of . By convention, we set and . Find all polynomial with integer coefficients such that the sequence
is bounded above. (In particular, this requires for