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Difference between revisions of "2012 USAMO Problems"

(Created page with "=Day 1= ==Problem 1== Find all integers <math>n \ge 3</math> such that among any <math>n</math> positive real numbers <math>a_1</math>, <math>a_2</math>, <math>\dots</math>, <mat...")
 
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=Day 1=
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==Day 1==
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==Problem 1==
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===Problem 1===
 
Find all integers <math>n \ge 3</math> such that among any <math>n</math> positive real numbers <math>a_1</math>, <math>a_2</math>, <math>\dots</math>, <math>a_n</math> with
 
Find all integers <math>n \ge 3</math> such that among any <math>n</math> positive real numbers <math>a_1</math>, <math>a_2</math>, <math>\dots</math>, <math>a_n</math> with
 
<cmath>\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),</cmath>
 
<cmath>\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),</cmath>
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[[2012 USAMO Problems/Problem 1|Solution]]
 
[[2012 USAMO Problems/Problem 1|Solution]]
  
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==Problem 2==
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===Problem 2===
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A circle is divided into 432 congruent arcs by 432 points.  The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow.  Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
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A circle is divided into <math>432</math> congruent arcs by <math>432</math> points.  The points are colored in four colors such that <math>108</math> points are colored red, <math>108</math> points are colored green, <math>108</math> points are colored blue  and the remaining <math>108</math> points are colored yellow.  Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
  
 
[[2012 USAMO Problems/Problem 2|Solution]]
 
[[2012 USAMO Problems/Problem 2|Solution]]
  
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==Problem 3==
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===Problem 3===
 
Determine which integers <math>n > 1</math> have the property that there exists an infinite sequence <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, <math>\dots</math> of nonzero integers such that the equality
 
Determine which integers <math>n > 1</math> have the property that there exists an infinite sequence <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, <math>\dots</math> of nonzero integers such that the equality
 
<cmath>a_k + 2a_{2k} + \dots + na_{nk} = 0</cmath>
 
<cmath>a_k + 2a_{2k} + \dots + na_{nk} = 0</cmath>
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[[2012 USAMO Problems/Problem 3|Solution]]
 
[[2012 USAMO Problems/Problem 3|Solution]]
  
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=Day 2=
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==Day 2==
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==Problem 4==
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===Problem 4===
 
Find all functions <math>f : \mathbb{Z}^+ \to \mathbb{Z}^+</math> (where <math>\mathbb{Z}^+</math> is the set of positive integers) such that <math>f(n!) = f(n)!</math> for all positive integers <math>n</math> and such that <math>m - n</math> divides <math>f(m) - f(n)</math> for all distinct positive integers <math>m</math>, <math>n</math>.
 
Find all functions <math>f : \mathbb{Z}^+ \to \mathbb{Z}^+</math> (where <math>\mathbb{Z}^+</math> is the set of positive integers) such that <math>f(n!) = f(n)!</math> for all positive integers <math>n</math> and such that <math>m - n</math> divides <math>f(m) - f(n)</math> for all distinct positive integers <math>m</math>, <math>n</math>.
  
 
[[2012 USAMO Problems/Problem 4|Solution]]
 
[[2012 USAMO Problems/Problem 4|Solution]]
  
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==Problem 5==
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===Problem 5===
 
Let <math>P</math> be a point in the plane of triangle <math>ABC</math>, and <math>\gamma</math> a line passing through <math>P</math>.  Let <math>A'</math>, <math>B'</math>, <math>C'</math> be the points where the reflections of lines <math>PA</math>, <math>PB</math>, <math>PC</math> with respect to <math>\gamma</math> intersect lines <math>BC</math>, <math>AC</math>, <math>AB</math>, respectively.  Prove that <math>A'</math>, <math>B'</math>, <math>C'</math> are collinear.
 
Let <math>P</math> be a point in the plane of triangle <math>ABC</math>, and <math>\gamma</math> a line passing through <math>P</math>.  Let <math>A'</math>, <math>B'</math>, <math>C'</math> be the points where the reflections of lines <math>PA</math>, <math>PB</math>, <math>PC</math> with respect to <math>\gamma</math> intersect lines <math>BC</math>, <math>AC</math>, <math>AB</math>, respectively.  Prove that <math>A'</math>, <math>B'</math>, <math>C'</math> are collinear.
  
 
[[2012 USAMO Problems/Problem 5|Solution]]
 
[[2012 USAMO Problems/Problem 5|Solution]]
  
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==Problem 6==
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===Problem 6===
 
For integer <math>n \ge 2</math>, let <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_n</math> be real numbers satisfying
 
For integer <math>n \ge 2</math>, let <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_n</math> be real numbers satisfying
 
<cmath>x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.</cmath>
 
<cmath>x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.</cmath>
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[[2012 USAMO Problems/Problem 6|Solution]]
 
[[2012 USAMO Problems/Problem 6|Solution]]
  
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= See also =
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== See Also ==
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*[[USAMO Problems and Solutions]]
 
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{{USAMO newbox|year= 2012|before=[[2011 USAMO]]|after=[[2013 USAMO]]}}
 
{{USAMO newbox|year= 2012|before=[[2011 USAMO]]|after=[[2013 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 03:37, 7 June 2020

Day 1

Problem 1

Find all integers $n \ge 3$ such that among any $n$ positive real numbers $a_1$, $a_2$, $\dots$, $a_n$ with \[\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),\] there exist three that are the side lengths of an acute triangle.

Solution

Problem 2

A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that $108$ points are colored red, $108$ points are colored green, $108$ points are colored blue and the remaining $108$ points are colored yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

Solution

Problem 3

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\dots$ of nonzero integers such that the equality \[a_k + 2a_{2k} + \dots + na_{nk} = 0\] holds for every positive integer $k$.

Solution

Day 2

Problem 4

Find all functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m - n$ divides $f(m) - f(n)$ for all distinct positive integers $m$, $n$.

Solution

Problem 5

Let $P$ be a point in the plane of triangle $ABC$, and $\gamma$ a line passing through $P$. Let $A'$, $B'$, $C'$ be the points where the reflections of lines $PA$, $PB$, $PC$ with respect to $\gamma$ intersect lines $BC$, $AC$, $AB$, respectively. Prove that $A'$, $B'$, $C'$ are collinear.

Solution

Problem 6

For integer $n \ge 2$, let $x_1$, $x_2$, $\dots$, $x_n$ be real numbers satisfying \[x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.\] For each subset $A \subseteq \{1, 2, \dots, n\}$, define \[S_A = \sum_{i \in A} x_i.\] (If $A$ is the empty set, then $S_A = 0$.)

Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A \ge \lambda$ is at most $2^{n - 3}/\lambda^2$. For what choices of $x_1$, $x_2$, $\dots$, $x_n$, $\lambda$ does equality hold?

Solution

See Also

2012 USAMO (Problems • Resources)
Preceded by
2011 USAMO 		Followed by
2013 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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