Problem Set Workbook
Access the downloadable workbook for 2006 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2006 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2006 Forum](https://forums.randommath.com/c/tournaments/MAA/2006-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2006 USAMO Day 1 problems, please refer below:
Problem 1: Let be a prime number and let be an integer with . Prove that there exist integers and with and
if and only if is not a divisor of .
(For a real number, let denote the greatest integer less than or equal to , and let denote the fractional part of .)
Solution:
Problem 2: For a given positive integer find, in terms of , the minimum value of for which there is a set of distinct positive integers that has sum greater than but every subset of size has sum at most .
Solution:
Problem 3: For integral , let be the greatest prime divisor of . By convention, we set 1 and . Find all polynomials with integer coefficients such that the sequence is bounded above. (In particular, this requires for .)
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions