Difference between revisions of "2010 USAJMO Problems"
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− | =Day 1= | + | ==Day 1== |
− | ==Problem 1== | + | ===Problem 1=== |
A permutation of the set of positive integers <math>[n] = {1,2,\ldots,n}</math> | 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 <math>[n]</math> | is a sequence <math>(a_1,a_2,\ldots,a_n)</math> such that each element of <math>[n]</math> | ||
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[[2010 USAJMO Problems/Problem 1|Solution]] | [[2010 USAJMO Problems/Problem 1|Solution]] | ||
− | ==Problem 2== | + | ===Problem 2=== |
Let <math>n > 1</math> be an integer. Find, with proof, all sequences | Let <math>n > 1</math> be an integer. Find, with proof, all sequences | ||
<math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following | <math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following | ||
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[[2010 USAJMO Problems/Problem 2|Solution]] | [[2010 USAJMO Problems/Problem 2|Solution]] | ||
− | ==Problem 3== | + | ===Problem 3=== |
Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter | Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter | ||
<math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto | <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto | ||
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[[2010 USAMO Problems/Problem 1|Solution]] | [[2010 USAMO Problems/Problem 1|Solution]] | ||
− | =Day 2= | + | ==Day 2== |
− | ==Problem 4== | + | ===Problem 4=== |
A triangle is called a parabolic triangle if its vertices lie on a | A triangle is called a parabolic triangle if its vertices lie on a | ||
parabola <math>y = x^2</math>. Prove that for every nonnegative integer <math>n</math>, there | parabola <math>y = x^2</math>. Prove that for every nonnegative integer <math>n</math>, there | ||
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[[2010 USAJMO Problems/Problem 4|Solution]] | [[2010 USAJMO Problems/Problem 4|Solution]] | ||
− | ==Problem 5== | + | ===Problem 5=== |
Two permutations <math>a_1, a_2, \ldots, a_{2010}</math> and | Two permutations <math>a_1, a_2, \ldots, a_{2010}</math> and | ||
<math>b_1, b_2, \ldots, b_{2010}</math> of the numbers <math>1, 2, \ldots, 2010</math> | <math>b_1, b_2, \ldots, b_{2010}</math> of the numbers <math>1, 2, \ldots, 2010</math> | ||
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[[2010 USAJMO Problems/Problem 5|Solution]] | [[2010 USAJMO Problems/Problem 5|Solution]] | ||
− | ==Problem 6== | + | ===Problem 6=== |
Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\circ}</math>. Points <math>D</math> | 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 |
Revision as of 15:15, 30 August 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:
- (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.