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Difference between revisions of "2011 IMO Problems"

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Problems of the 52st [[IMO]] 2011 in Amsterdam, Netherlands.
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== Day 1 ==
 
== Day 1 ==
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=== Problem 1. ===
 
=== Problem 1. ===
 
Given any set <math>A = \{a_1, a_2, a_3, a_4\}</math> of four distinct positive integers, we denote the sum <math>a_1+a_2+a_3+a_4</math> by <math>s(A)</math>. Let <math>n(A)</math> denote the number of pairs <math>(i,j)</math> with <math>1 \le i < j \le 4</math> for which <math>a_i+a_j</math> divides <math>s(A)</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possible value of <math>n(A)</math>.
 
Given any set <math>A = \{a_1, a_2, a_3, a_4\}</math> of four distinct positive integers, we denote the sum <math>a_1+a_2+a_3+a_4</math> by <math>s(A)</math>. Let <math>n(A)</math> denote the number of pairs <math>(i,j)</math> with <math>1 \le i < j \le 4</math> for which <math>a_i+a_j</math> divides <math>s(A)</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possible value of <math>n(A)</math>.
  
 
''author: Fernando Campos, Mexico''
 
''author: Fernando Campos, Mexico''
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[[2011 IMO Problems/Problem 1 | Solution]]
  
 
=== Problem 2. ===  
 
=== Problem 2. ===  
 
Let <math>S</math> be a finite set of at least two points in the plane. Assume that no three points of <math>S</math> are collinear. A windmill is a process that starts with a line <math>l</math> going through a single point <math>P \in S</math>. The line rotates clockwise about the pivot <math>P</math> until the first time that the line meets some other point belonging to <math>S</math>. This point, <math>Q</math>, takes over as the new pivot, and the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>S</math>. This process continues indefinitely.
 
Let <math>S</math> be a finite set of at least two points in the plane. Assume that no three points of <math>S</math> are collinear. A windmill is a process that starts with a line <math>l</math> going through a single point <math>P \in S</math>. The line rotates clockwise about the pivot <math>P</math> until the first time that the line meets some other point belonging to <math>S</math>. This point, <math>Q</math>, takes over as the new pivot, and the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>S</math>. This process continues indefinitely.
 
Show that we can choose a point <math>P</math> in <math>S</math> and a line <math>l</math> going through <math>P</math> such that the resulting windmill uses each point of <math>S</math> as a pivot infinitely many times.
 
Show that we can choose a point <math>P</math> in <math>S</math> and a line <math>l</math> going through <math>P</math> such that the resulting windmill uses each point of <math>S</math> as a pivot infinitely many times.
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[[2011 IMO Problems/Problem 2 | Solution]]
  
 
=== Problem 3. ===
 
=== Problem 3. ===
 
Let <math>f : R \rightarrow R</math> be a real-valued function defined on the set of real numbers that satisfies
 
Let <math>f : R \rightarrow R</math> be a real-valued function defined on the set of real numbers that satisfies
 
<math>f(x + y) \le yf(x) + f(f(x))</math> for all real numbers <math>x</math> and <math>y</math>. Prove that <math>f(x)=0</math> for all <math>x \le 0</math>.
 
<math>f(x + y) \le yf(x) + f(f(x))</math> for all real numbers <math>x</math> and <math>y</math>. Prove that <math>f(x)=0</math> for all <math>x \le 0</math>.
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[[2011 IMO Problems/Problem 3 | Solution]]
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== Day 2 ==
 
== Day 2 ==
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Let <math>n > 0</math> be an integer. We are given a balance and <math>n</math> weights of weight <math>2^0, 2^1,\ldots, 2^{n-1}</math> . We are to place each of the <math>n</math> weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
 
Let <math>n > 0</math> be an integer. We are given a balance and <math>n</math> weights of weight <math>2^0, 2^1,\ldots, 2^{n-1}</math> . We are to place each of the <math>n</math> weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
 
Determine the number of ways in which this can be done.
 
Determine the number of ways in which this can be done.
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[[2011 IMO Problems/Problem 4 | Solution]]
  
 
=== Problem 5. ===
 
=== Problem 5. ===
 
Let <math>f</math> be a function from the set of integers to the set of positive integers. Suppose that, for any two integers <math>m</math> and <math>n</math>, the difference <math>f(m) - f(n)</math> is divisible by <math>f(m - n)</math>. Prove that, for all integers <math>m</math> and <math>n</math> with <math>f(m) \le f(n)</math>, the number <math>f(n)</math> is divisible by <math>f(m)</math>.
 
Let <math>f</math> be a function from the set of integers to the set of positive integers. Suppose that, for any two integers <math>m</math> and <math>n</math>, the difference <math>f(m) - f(n)</math> is divisible by <math>f(m - n)</math>. Prove that, for all integers <math>m</math> and <math>n</math> with <math>f(m) \le f(n)</math>, the number <math>f(n)</math> is divisible by <math>f(m)</math>.
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[[2011 IMO Problems/Problem 5 | Solution]]
  
 
=== Problem 6. ===
 
=== Problem 6. ===
 
Let <math>ABC</math> be an acute triangle with circumcircle <math>\Gamma</math>. Let <math>l</math> be a tangent line to <math>\Gamma</math>, and let <math>l_a</math>, <math>l_b</math> and <math>l_c</math> be the lines obtained by reflecting <math>l</math> in the lines <math>BC</math>, <math>CA</math> and <math>AB</math>, respectively. Show that the circumcircle of the triangle determined by the lines <math>l_a</math>, <math>l_b</math> and <math>l_c</math> is tangent to the circle <math>\Gamma</math>.
 
Let <math>ABC</math> be an acute triangle with circumcircle <math>\Gamma</math>. Let <math>l</math> be a tangent line to <math>\Gamma</math>, and let <math>l_a</math>, <math>l_b</math> and <math>l_c</math> be the lines obtained by reflecting <math>l</math> in the lines <math>BC</math>, <math>CA</math> and <math>AB</math>, respectively. Show that the circumcircle of the triangle determined by the lines <math>l_a</math>, <math>l_b</math> and <math>l_c</math> is tangent to the circle <math>\Gamma</math>.
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[[2011 IMO Problems/Problem 6 | Solution]]
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== Resources ==
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* [[2011 IMO]]
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* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2011&sid=8fa5e6d20f9aad1a4dd4efa73519b417 2011 IMO Problems on the Resources page]

Revision as of 12:59, 21 June 2012

Problems of the 52st IMO 2011 in Amsterdam, Netherlands.

Day 1

Problem 1.

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1+a_2+a_3+a_4$ by $s(A)$. Let $n(A)$ denote the number of pairs $(i,j)$ with $1 \le i < j \le 4$ for which $a_i+a_j$ divides $s(A)$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n(A)$.

author: Fernando Campos, Mexico

Solution

Problem 2.

Let $S$ be a finite set of at least two points in the plane. Assume that no three points of $S$ are collinear. A windmill is a process that starts with a line $l$ going through a single point $P \in S$. The line rotates clockwise about the pivot $P$ until the first time that the line meets some other point belonging to $S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $S$. This process continues indefinitely. Show that we can choose a point $P$ in $S$ and a line $l$ going through $P$ such that the resulting windmill uses each point of $S$ as a pivot infinitely many times.

Solution

Problem 3.

Let $f : R \rightarrow R$ be a real-valued function defined on the set of real numbers that satisfies $f(x + y) \le yf(x) + f(f(x))$ for all real numbers $x$ and $y$. Prove that $f(x)=0$ for all $x \le 0$.

Solution


Day 2

Problem 4.

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1,\ldots, 2^{n-1}$ . We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done.

Solution

Problem 5.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m - n)$. Prove that, for all integers $m$ and $n$ with $f(m) \le f(n)$, the number $f(n)$ is divisible by $f(m)$.

Solution

Problem 6.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $l$ be a tangent line to $\Gamma$, and let $l_a$, $l_b$ and $l_c$ be the lines obtained by reflecting $l$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $l_a$, $l_b$ and $l_c$ is tangent to the circle $\Gamma$.

Solution


Resources

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