Difference between revisions of "2001 USAMO Problems"
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* [[2001 USAMO Problems/Problem 6 | Solution]] | * [[2001 USAMO Problems/Problem 6 | Solution]] | ||
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Latest revision as of 09:56, 20 July 2016
Contents
Day 1
Problem 1
Each of eight boxes contains six balls. Each ball has been colored with one of colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer
for which this is possible.
Problem 2
Let be a triangle and let
be its incircle. Denote by
and
the points where
is tangent to sides
and
, respectively. Denote by
and
the points on sides
and
, respectively, such that
and
, and denote by
the point of intersection of segments
and
. Circle
intersects segment
at two points, the closer of which to the vertex
is denoted by
. Prove that
.
Problem 3
Let and satisfy
![$a^2 + b^2 + c^2 + abc = 4.$](http://latex.artofproblemsolving.com/1/6/e/16e70ab813b2e9287a1015d7b890d16f94a7073e.png)
Show that
![$0\leq ab + bc + ca - abc \leq 2.$](http://latex.artofproblemsolving.com/3/8/a/38a4a88e186d6095e8a62f02ad9d732101b33d1a.png)
Day 2
Problem 4
Let be a point in the plane of triangle
such that the segments
,
, and
are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to
. Prove that
is acute.
Problem 5
Let be a set of integers (not necessarily positive) such that
(a) there exist with
;
(b) if and
are elements of
(possibly equal), then
also belongs to
.
Prove that is the set of all integers.
Problem 6
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
See Also
2001 USAMO (Problems • Resources) | ||
Preceded by 2000 USAMO |
Followed by 2002 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.