Problem Set Workbook
Access the downloadable workbook for 2003 USAMO problems here.
Discussion Forum
Engage in discussion about the 2003 USAMO math contest by visiting Random Math USAMO 2003 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2003 USAMO problems, please refer below:
Problem 1: Prove that for every positive integer there exists an -digit number divisible by all of whose digits are odd.
Solution:
Problem 2: A convex polygon in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
Solution:
Problem 3: Let . For every sequence of integers
satisfying , for , define another sequence
by setting to be the number of terms in the sequence that precede the term and are different from . Show that, starting from any sequence as above, fewer than applications of the transformation lead to a sequence such that .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions