2003 USAMO Problems
Contents
Day 1
Problem 1
Prove that for every positive integer there exists an
-digit number divisible by
all of whose digits are odd.
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.
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
.