Difference between revisions of "2016 USAJMO Problems"
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Revision as of 13:56, 21 April 2016
Contents
[hide]Day 1
Problem 1
The isosceles triangle , with
, is inscribed in the circle
. Let
be a variable point on the arc
that does not contain
, and let
and
denote the incenters of triangles
and
, respectively.
Prove that as varies, the circumcircle of triangle
passes through a fixed point.
Problem 2
Prove that there exists a positive integer such that
has six consecutive zeros in its decimal representation.
Problem 3
Let be a sequence of mutually distinct nonempty subsets of a set
. Any two sets
and
are disjoint and their union is not the whole set
, that is,
and
, for all
. Find the smallest possible number of elements in
.
Day 2
Problem 4
Find, with proof, the least integer such that if any
elements are removed from the set
, one can still find
distinct numbers among the remaining elements with sum
.
Problem 5
Let be an acute triangle, with
as its circumcenter. Point
is the foot of the perpendicular from
to line
, and points
and
are the feet of the perpendiculars from
to the lines
and
, respectively.
Given that prove that the points
and
are collinear.
Problem 6
Find all functions such that for all real numbers
and
,
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.