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Difference between revisions of "2010 USAJMO Problems"

(→‎Problem 6)
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=Day 1=
+
==Day 1==
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==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]]
  
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==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]]
  
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==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]]
  
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=Day 2=
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==Day 2==
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==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]]
  
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==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]]
  
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==Problem 6==
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===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

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 $1 \le k \le n$. 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. (a). $x_1 < x_2 < \cdots <x_{n-1}$;
  2. (b). $x_i +x_{n-i} = 2n$ for all $i=1,2,\ldots,n-1$;
  3. (c). 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 $1 \le k\le 2010$. 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

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