Difference between revisions of "2010 USAJMO Problems"
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− | =Day 1= | + | == Day 1 == |
− | ==Problem 1== | + | |
− | A permutation of the set of positive integers <math>[n] = {1,2,\ldots,n}</math> | + | === Problem 1 === |
− | is a sequence <math>(a_1,a_2,\ldots,a_n)</math> such that each element of <math>[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> appears precisely one time as a term of the sequence. For example, <math>(3, 5, 1, 2, 4)</math> is a permutation of <math>[5]</math>. Let <math>P(n)</math> be the number of permutations of <math>[n]</math> for which <math>ka_k</math> is a perfect square for all <math>1\leq k\leq n</math>. Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. |
− | appears precisely one time as a term of the sequence. For example, | ||
− | <math>(3, 5, 1, 2, 4)</math> is a permutation of <math>[5]</math>. Let <math>P(n)</math> be the number of | ||
− | permutations of <math>[n]</math> for which <math>ka_k</math> is a perfect square for all | ||
− | <math>1 \ | ||
− | is a multiple of <math>2010</math>. | ||
[[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 three properties: |
− | <math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following | ||
− | three properties: | ||
<ol style="list-style-type:lower-alpha"> | <ol style="list-style-type:lower-alpha"> | ||
− | <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 that <math>x_i + x_j = x_k</math>. |
− | 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>. | ||
</ol> | </ol> | ||
[[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 lines <math>AX, BX, AZ, BZ</math>, respectively. Prove that the acute angle formed by lines <math>PQ</math> and <math>RS</math> is half the size of <math>\angle XOZ</math>, where <math>O</math> is the midpoint of segment <math>AB</math>. |
− | <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto | ||
− | lines <math>AX, BX, AZ, BZ</math>, respectively. Prove that the acute angle | ||
− | formed by lines <math>PQ</math> and <math>RS</math> is half the size of <math>\angle XOZ</math>, where | ||
− | <math>O</math> is the midpoint of segment <math>AB</math>. | ||
[[2010 USAMO Problems/Problem 1|Solution]] | [[2010 USAMO Problems/Problem 1|Solution]] | ||
− | =Day 2= | + | == Day 2 == |
− | ==Problem 4== | + | |
− | A triangle is called a parabolic triangle if its vertices lie on a | + | === Problem 4 === |
− | parabola <math>y = x^2</math>. Prove that for every nonnegative integer <math>n</math>, there | + | 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 is an odd number <math>m</math> and a parabolic triangle with vertices at three distinct points with integer coordinates with area <math>(2^nm)^2</math>. |
− | is an odd number <math>m</math> and a parabolic triangle with vertices at three | ||
− | distinct points with integer coordinates with area <math>(2^nm)^2</math>. | ||
[[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> are said to intersect if <math>a_k = b_k</math> for some value of <math>k</math> in the range <math>1\leq k\leq 2010</math>. Show that there exist <math>1006</math> permutations |
− | <math>b_1, b_2, \ldots, b_{2010}</math> of the numbers <math>1, 2, \ldots, 2010</math> | + | of the numbers <math>1, 2, \ldots, 2010</math> such that any other such permutation is guaranteed to intersect at least one of these <math>1006</math> permutations. |
− | are said to intersect if <math>a_k = b_k</math> for some value of <math>k</math> in the | ||
− | range <math>1 \ | ||
− | of the numbers <math>1, 2, \ldots, 2010</math> such that any other such | ||
− | permutation is guaranteed to intersect at least one of these <math>1006</math> | ||
− | permutations. | ||
[[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 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. |
− | 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 | + | [[2010 USAJMO Problems/Problem 6|Solution]] |
− | <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. | ||
− | [[ | + | == See Also == |
+ | {{USAJMO box|year=2010|before=First USAJMO|after=[[2011 USAJMO Problems|2011 USAJMO]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 13:55, 16 June 2020
Contents
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.