2011 USAJMO Problems
Contents
Day 1
Problem 1
Find, with proof, all positive integers for which
is a perfect square.
Problem 2
Let ,
,
be positive real numbers such that
. Prove that
Problem 3
For a point in the coordinate plane, let
denote the line passing through
with slope
. Consider the set of triangles with vertices of the form
,
,
, such that the intersections of the lines
,
,
form an equilateral triangle
. Find the locus of the center of
as
ranges over all such triangles.
Day 2
Problem 4
A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words ,
,
,
be defined as follows:
,
, and for
,
is the word formed by writing
follows by
. Prove that for any
, the word formed by writing
,
,
,
in succession is a palindrome.
Problem 5
Points ,
,
,
,
lie on a circle
and point
lies outside the circle. The given points are such that (i) lines
and
are tangent to
, (ii)
,
,
are collinear, and (iii)
. Prove that
bisects
.
Problem 6
Consider the assertion that for each positive integer , the remainder upon dividing
by
is a power of 4. Either prove the assertion or find (with proof) a counterexample.