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Difference between revisions of "2010 USAJMO Problems"

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=Day 1=
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== Day 1 ==
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==Problem 1==
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A permutation of the set of positive integers <math>[n] = {1,2,\ldots,n}</math>
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=== Problem 1 ===
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is a sequence <math>(a_1,a_2,\ldots,a_n)</math> such that each element of <math>[n]</math>
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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>.
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appears precisely one time as a term of the sequence. For example,
 
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<math>(3, 5, 1, 2, 4)</math> is a permutation of <math>[5]</math>. Let <math>P(n)</math> be the number of
 
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permutations of <math>[n]</math> for which <math>ka_k</math> is a perfect square for all
 
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<math>1 \le k \le n</math>. Find with proof the smallest <math>n</math> such that <math>P(n)</math>
 
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is a multiple of <math>2010</math>.
 
  
 
[[2010 USAJMO Problems/Problem 1|Solution]]
 
[[2010 USAJMO Problems/Problem 1|Solution]]
  
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==Problem 2==
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=== Problem 2 ===
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Let <math>n > 1</math> be an integer. Find, with proof, all sequences
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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:
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<math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following
 
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three properties:
 
 
<ol style="list-style-type:lower-alpha">
 
<ol style="list-style-type:lower-alpha">
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<li> (a). <math>x_1 < x_2 < \cdots <x_{n-1}</math>;
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<li> <math>x_1 < x_2 < \cdots < x_{n-1}</math>;
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<li> (b). <math>x_i +x_{n-i} = 2n</math> for all <math>i=1,2,\ldots,n-1</math>;
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<li> <math>x_i + x_{n-i} = 2n</math> for all <math>i = 1, 2, \ldots, n - 1</math>;
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<li> (c). given any two indices <math>i</math> and <math>j</math> (not necessarily distinct)
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<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>.
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for which <math>x_i + x_j < 2n</math>, there is an index <math>k</math> such
 
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that <math>x_i+x_j = x_k</math>.
 
 
</ol>
 
</ol>
  
 
[[2010 USAJMO Problems/Problem 2|Solution]]
 
[[2010 USAJMO Problems/Problem 2|Solution]]
  
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==Problem 3==
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=== Problem 3 ===
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Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter
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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>.
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<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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lines <math>AX, BX, AZ, BZ</math>, respectively. Prove that the acute angle
 
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formed by lines <math>PQ</math> and <math>RS</math> is half the size of <math>\angle XOZ</math>, where
 
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<math>O</math> is the midpoint of segment <math>AB</math>.
 
  
 
[[2010 USAMO Problems/Problem 1|Solution]]
 
[[2010 USAMO Problems/Problem 1|Solution]]
  
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=Day 2=
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== Day 2 ==
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==Problem 4==
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A triangle is called a parabolic triangle if its vertices lie on a
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=== Problem 4 ===
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parabola <math>y = x^2</math>. Prove that for every nonnegative integer <math>n</math>, there
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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>.
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is an odd number <math>m</math> and a parabolic triangle with vertices at three
 
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distinct points with integer coordinates with area <math>(2^nm)^2</math>.
 
  
 
[[2010 USAJMO Problems/Problem 4|Solution]]
 
[[2010 USAJMO Problems/Problem 4|Solution]]
  
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==Problem 5==
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=== Problem 5 ===
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Two permutations <math>a_1, a_2, \ldots, a_{2010}</math> and
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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
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<math>b_1, b_2, \ldots, b_{2010}</math> of the numbers <math>1, 2, \ldots, 2010</math>
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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.
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are said to intersect if <math>a_k = b_k</math> for some value of <math>k</math> in the
 
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range <math>1 \le k\le 2010</math>. Show that there exist <math>1006</math> permutations
 
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of the numbers <math>1, 2, \ldots, 2010</math> such that any other such
 
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permutation is guaranteed to intersect at least one of these <math>1006</math>
 
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permutations.
 
  
 
[[2010 USAJMO Problems/Problem 5|Solution]]
 
[[2010 USAJMO Problems/Problem 5|Solution]]
  
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==Problem 6==
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=== Problem 6 ===
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Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\circ}</math>. Points <math>D</math>
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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.
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and <math>E</math> lie on sides <math>AC</math> and <math>AB</math>, respectively, such that <math>\angle
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ABD = \angle DBC</math> and <math>\angle ACE = \angle ECB</math>. Segments <math>BD</math> and
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[[2010 USAJMO Problems/Problem 6|Solution]]
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<math>CE</math> meet at <math>I</math>.  Determine whether or not it is possible for
 
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segments <math>AB, AC, BI, ID, CI, IE</math> to all have integer lengths.
 
  
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[[2010 USAMO Problems/Problem 4|Solution]]
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== See Also ==
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{{USAJMO box|year=2010|before=First USAJMO|after=[[2011 USAJMO Problems|2011 USAJMO]]}}
 +
{{MAA Notice}}

Latest revision as of 12:55, 16 June 2020

Day 1

Problem 1

A permutation of the set of positive integers $[n] = \{1, 2, \ldots, n\}$ is a sequence $(a_1, a_2, \ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\leq k\leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$.

Solution

Problem 2

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1, x_2, \ldots, x_{n-1}$ of positive integers with the following three properties:

  1. $x_1 < x_2 < \cdots < x_{n-1}$;
  2. $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots, n - 1$;
  3. given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

Solution

Problem 3

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto lines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

Solution

Day 2

Problem 4

A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.

Solution

Problem 5

Two permutations $a_1, a_2, \ldots, a_{2010}$ and $b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$ are said to intersect if $a_k = b_k$ for some value of $k$ in the range $1\leq k\leq 2010$. Show that there exist $1006$ permutations of the numbers $1, 2, \ldots, 2010$ such that any other such permutation is guaranteed to intersect at least one of these $1006$ permutations.

Solution

Problem 6

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB, AC, BI, ID, CI, IE$ to all have integer lengths.

Solution

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

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