1987 USAMO Problems
Problem 1
Find all solutions to , where m and n are non-zero integers.
Problem 2
The feet of the angle bisectors of form a right-angled triangle. If the right-angle is at
, where
is the bisector of
, find all possible values for
.
Problem 3
is the smallest set of polynomials
such that:
1. belongs to
.
2. If
belongs to
, then
and
both belong to
.
Show that if and
are distinct elements of
, then
for any
.
Problem 4
M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that and
. For what value of
is
a minimum?
Problem 5
is a sequence of 0's and 1's. T is the number of triples
with
which are not equal to (0, 1, 0) or (1, 0, 1). For
,
is the number of
with
plus the number of
with
. Show that
. If n is odd, what is the smallest value of T?