Difference between revisions of "2010 USAJMO Problems"
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− | <li> | + | <li> <math>x_1 < x_2 < \cdots <x_{n-1}</math>; |
− | <li> | + | <li> <math>x_i +x_{n-i} = 2n</math> for all <math>i=1,2,\ldots,n-1</math>; |
− | <li> | + | <li> given any two indices <math>i</math> and <math>j</math> (not necessarily distinct) |
for which <math>x_i + x_j < 2n</math>, there is an index <math>k</math> such | for which <math>x_i + x_j < 2n</math>, there is an index <math>k</math> such | ||
that <math>x_i+x_j = x_k</math>. | that <math>x_i+x_j = x_k</math>. |
Revision as of 08:18, 12 October 2012
Contents
[hide]Day 1
Problem 1
A permutation of the set of positive integers
is a sequence
such that each element of
appears precisely one time as a term of the sequence. For example,
is a permutation of
. Let
be the number of
permutations of
for which
is a perfect square for all
. Find with proof the smallest
such that
is a multiple of
.
Problem 2
Let be an integer. Find, with proof, all sequences
of positive integers with the following
three properties:
-
;
-
for all
;
- given any two indices
and
(not necessarily distinct) for which
, there is an index
such that
.
Problem 3
Let be a convex pentagon inscribed in a semicircle of diameter
. Denote by
the feet of the perpendiculars from
onto
lines
, respectively. Prove that the acute angle
formed by lines
and
is half the size of
, where
is the midpoint of segment
.
Day 2
Problem 4
A triangle is called a parabolic triangle if its vertices lie on a
parabola . Prove that for every nonnegative integer
, there
is an odd number
and a parabolic triangle with vertices at three
distinct points with integer coordinates with area
.
Problem 5
Two permutations and
of the numbers
are said to intersect if
for some value of
in the
range
. Show that there exist
permutations
of the numbers
such that any other such
permutation is guaranteed to intersect at least one of these
permutations.
Problem 6
Let be a triangle with
. Points
and
lie on sides
and
, respectively, such that
and
. Segments
and
meet at
. Determine whether or not it is possible for
segments
to all have integer lengths.