2024 USAJMO Problems
Contents
Day 1
Problem 1
Let be a cyclic quadrilateral with
and
. Points
and
are selected on line segment
so that
. Points
and
are selected on line segment
so that
. Prove that
is a quadrilateral.
Problem 2
Let and
be positive integers. Let
be the set of integer points
with
and
. A configuration of
rectangles is called happy if each point in
is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Problem 3
Let be the sequence defined by
and
n\geq1
p>2
k
a(n)
p^k$.