Difference between revisions of "2010 USAJMO Problems"
(Created page with '=Day 1= ==Problem 1== A permutation of the set of positive integers <math>[n] = {1,2,\ldots,n}</math> is a sequence <math>(a_1,a_2,\ldots,a_n)</math> such that each element of <m…') |
(→Problem 6) |
||
Line 55: | Line 55: | ||
==Problem 6== | ==Problem 6== | ||
− | Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\ | + | Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\circ}</math>. Points <math>D</math> |
and <math>E</math> lie on sides <math>AC</math> and <math>AB</math>, respectively, such that <math>\angle | and <math>E</math> lie on sides <math>AC</math> and <math>AB</math>, respectively, such that <math>\angle | ||
ABD = \angle DBC</math> and <math>\angle ACE = \angle ECB</math>. Segments <math>BD</math> and | ABD = \angle DBC</math> and <math>\angle ACE = \angle ECB</math>. Segments <math>BD</math> and |
Revision as of 00:24, 7 May 2010
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:
- (a).
;
- (b).
for all
;
- (c). 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.