Difference between revisions of "2012 USAMO Problems"

Latest revision as of 02:37, 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

@@ Line 8: / Line 8: @@
 ===Problem 2===
-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.
+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]]
@@ Line 43: / Line 43: @@
 == See Also ==
 {{USAMO newbox|year= 2012|before=[[2011 USAMO]]|after=[[2013 USAMO]]}}
+{{MAA Notice}}

2012 USAMO (Problems • Resources)
Preceded by 2011 USAMO	Followed by 2013 USAMO
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All USAMO Problems and Solutions

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Difference between revisions of "2012 USAMO Problems"

Latest revision as of 02:37, 7 June 2020

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

See Also