Difference between revisions of "2002 USAMO Problems"
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Revision as of 02:29, 21 December 2008
Contents
Day 1
Problem 1
Let be a set with 2002 elements, and let
be an integer with
. Prove that it is possible to color every subset of
either blue or red so that the following conditions hold:
(a) the union of any two red subsets is red;
(b) the union of any two blue subsets is blue;
(c) there are exactly red subsets.
Problem 2
Let be a triangle such that
,
where and
denote its semiperimeter and inradius, respectively. Prove that triangle
is similar to a triangle
whose side lengths are all positive integers with no common divisor and determine those integers.
Problem 3
Prove that any monic polymomial (a polynomial with leading coefficient 1) of degree with real coefficients is the average of two monic polynomials of degree
with
real roots.
Day 2
Problem 4
Let be the set of real numbers. Determine all functions
such that
for all pairs of real numbers and
.
Problem 5
Let be integers greater than 2. Prove that there exists a positive integer
and a finite sequence
of positive integers such that
,
, and
is divisible by
for each
(
).
Problem 6
I have an sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let
be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants
and
such that
![$\dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn$](http://latex.artofproblemsolving.com/f/8/2/f8227e8e6f5e87f8ff9762baf0e1f00748cd32eb.png)
for all .