Difference between revisions of "2010 USAJMO Problems"
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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 ABD = \angle DBC</math> and <math>\angle ACE = \angle ECB</math>. Segments <math>BD</math> and <math>CE</math> meet at <math>I</math>. Determine whether or not it is possible for segments <math>AB, AC, BI, ID, CI, IE</math> to all have integer lengths. | 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 ABD = \angle DBC</math> and <math>\angle ACE = \angle ECB</math>. Segments <math>BD</math> and <math>CE</math> meet at <math>I</math>. Determine whether or not it is possible for segments <math>AB, AC, BI, ID, CI, IE</math> to all have integer lengths. | ||
− | [[2010 | + | [[2010 USAJMO Problems/Problem 6|Solution]] |
== See Also == | == See Also == | ||
{{USAJMO box|year=2010|before=First USAJMO|after=[[2011 USAJMO Problems|2011 USAJMO]]}} | {{USAJMO box|year=2010|before=First USAJMO|after=[[2011 USAJMO Problems|2011 USAJMO]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:55, 16 June 2020
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.
See Also
2010 USAJMO (Problems • Resources) | ||
Preceded by First USAJMO |
Followed by 2011 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.