# Difference between revisions of "2012 IMO Problems"

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+ | Problems of the 53st [[IMO]] 2012 in Mar del Plata, Argentina. | ||

+ | == Day 1 == | ||

+ | === Problem 1. === | ||

+ | Given triangle ABC the point <math>J</math> is the centre of the excircle opposite the vertex <math>A</math>. | ||

+ | This excircle is tangent to the side <math>BC</math> at <math>M</math>, and to the lines <math>AB</math> and <math>AC</math> at <math>K</math> and <math>L</math>, respectively. | ||

+ | The lines <math>LM</math> and <math>BJ</math> meet at <math>F</math>, and the lines <math>KM</math> and <math>CJ</math> meet at <math>G</math>. Let <math>S</math> be the point of | ||

+ | intersection of the lines <math>AF</math> and <math>BC</math>, and let <math>T</math> be the point of intersection of the lines <math>AG</math> and <math>BC</math>. | ||

+ | Prove that <math>M</math> is the midpoint of <math>ST</math>. | ||

+ | (The excircle of <math>ABC</math> opposite the vertex <math>A</math> is the circle that is tangent to the line segment <math>BC</math>, | ||

+ | to the ray <math>AB</math> beyond <math>B</math>, and to the ray <math>AC</math> beyond <math>C</math>.) | ||

+ | |||

+ | ''Author: Evangelos Psychas, Greece'' | ||

+ | |||

+ | [[2012 IMO Problems/Problem 1 | Solution]] | ||

+ | |||

+ | === Problem 2. === | ||

+ | Let <math>{{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}</math> be positive real numbers that satisfy <math>{{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1</math> . Prove that | ||

+ | <cmath> \left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n\gneq n^n</cmath> | ||

+ | |||

+ | ''Author: Angelo di Pasquale, Australia'' | ||

+ | |||

+ | [[2012 IMO Problems/Problem 2 | Solution]] | ||

+ | |||

+ | === Problem 3. === | ||

+ | The ''liar’s guessing game'' is a game played between two players <math>A</math> and <math>B</math>. The rules of the game depend on two positive integers <math>k</math> and <math>n</math> which are known to both players. At the start of the game A chooses integers <math>x</math> and <math>N</math> with <math>1\le x\le N</math>. Player <math>A</math> keeps <math>x</math> secret, and truthfully tells <math>N</math> to player <math>B</math>. Player <math>B</math> now tries to obtain information about <math>x</math> by asking player <math>A</math> questions as follows: each question consists of <math>B</math> specifying an arbitrary set <math>S</math> of positive integers (possibly one specified in some previous question), and asking <math>A</math> whether <math>x</math> belongs to <math>S</math>. Player <math>B</math> may ask as many such questions as he wishes. After each question, player <math>A</math> must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is | ||

+ | that, among any <math>k + 1</math> consecutive answers, at least one answer must be truthful. After <math>B</math> has asked as many questions as he wants, he must specify a set <math>X</math> of at most <math>n</math> positive integers. If <math>x</math> belongs to <math>X</math>, then <math>B</math> wins; otherwise, he loses. Prove that: | ||

+ | # If <math>n\ge {{2}^{k}}</math>, then <math>B</math> can guarantee a win. | ||

+ | # For all sufficiently large <math>k</math>, there exists an integer <math>n\ge {1.99^k}</math> such that <math>B</math> cannot guarantee a win. | ||

+ | |||

+ | ''Author: David Arthur, Canada '' | ||

+ | |||

+ | [[2012 IMO Problems/Problem 3 | Solution]] | ||

+ | |||

+ | |||

+ | == Day 2 == | ||

+ | === Problem 4. === | ||

+ | Find all functions <math>f:\mathbb{Z}\to \mathbb{Z}</math> such that, for all integers <math>a</math>, <math>b</math>, <math>c</math> that satisfy <math>a+b+c = 0</math>, the following equality holds: | ||

+ | <cmath>f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).</cmath> | ||

+ | (Here <math>\mathbb{Z}</math> denotes the set of integers.) | ||

+ | |||

+ | ''Author: Liam Baker, South Africa '' | ||

+ | |||

+ | [[2012 IMO Problems/Problem 4 | Solution]] | ||

+ | |||

+ | === Problem 5. === | ||

+ | Let <math>ABC</math> be a triangle with <math>\angle BCA=90{}^\circ </math>, and let <math>D</math> be the foot of the altitude from <math>C</math>. Let <math>X</math> be a point in the interior of the segment <math>CD</math>. Let K be the point on the segment <math>AX</math> such that <math>BK = BC</math>. Similarly, let <math>L</math> be the point on the segment <math>BX</math> such that <math>AL = AC</math>. Let <math>M</math> be the point of intersection of <math>AL</math> and <math>BK</math>. | ||

+ | Show that <math>MK = ML</math>. | ||

+ | |||

+ | ''Author: Josef Tkadlec, Czech Republic'' | ||

+ | |||

+ | [[2012 IMO Problems/Problem 5 | Solution]] | ||

+ | |||

+ | === Problem 6. === | ||

+ | Find all positive integers n for which there exist non-negative integers <math>a_1</math>, <math>a_2</math>, <math>\ldots</math> , <math>a_n</math> such that | ||

+ | <math>\frac{1}{{{2}^{{{a}_{1}}}}}+\frac{1}{{{2}^{{{a}_{2}}}}}+\cdots +\frac{1}{{{2}^{{{a}_{n}}}}}=\frac{1}{{{3}^{{{a}_{1}}}}}+\frac{2}{{{3}^{{{a}_{2}}}}}+\cdots +\frac{n}{{{3}^{{{a}_{n}}}}}=1</math> | ||

+ | |||

+ | ''Author: Dušan Djukić, Serbia'' | ||

+ | |||

+ | [[2012 IMO Problems/Problem 6 | Solution]] | ||

+ | |||

+ | |||

+ | == Resources == | ||

+ | * [[2012 IMO]] | ||

+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2012&sid=0b159fdf0134b865758167cc5cf255dd 2012 IMO Problems on the Resources page] |

## Revision as of 21:02, 6 January 2013

Problems of the 53st IMO 2012 in Mar del Plata, Argentina.

## Contents

## Day 1

### Problem 1.

Given triangle ABC the point is the centre of the excircle opposite the vertex . This excircle is tangent to the side at , and to the lines and at and , respectively. The lines and meet at , and the lines and meet at . Let be the point of intersection of the lines and , and let be the point of intersection of the lines and . Prove that is the midpoint of . (The excircle of opposite the vertex is the circle that is tangent to the line segment , to the ray beyond , and to the ray beyond .)

*Author: Evangelos Psychas, Greece*

### Problem 2.

Let be positive real numbers that satisfy . Prove that

*Author: Angelo di Pasquale, Australia*

### Problem 3.

The *liar’s guessing game* is a game played between two players and . The rules of the game depend on two positive integers and which are known to both players. At the start of the game A chooses integers and with . Player keeps secret, and truthfully tells to player . Player now tries to obtain information about by asking player questions as follows: each question consists of specifying an arbitrary set of positive integers (possibly one specified in some previous question), and asking whether belongs to . Player may ask as many such questions as he wishes. After each question, player must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is
that, among any consecutive answers, at least one answer must be truthful. After has asked as many questions as he wants, he must specify a set of at most positive integers. If belongs to , then wins; otherwise, he loses. Prove that:

- If , then can guarantee a win.
- For all sufficiently large , there exists an integer such that cannot guarantee a win.

*Author: David Arthur, Canada *

## Day 2

### Problem 4.

Find all functions such that, for all integers , , that satisfy , the following equality holds: (Here denotes the set of integers.)

*Author: Liam Baker, South Africa *

### Problem 5.

Let be a triangle with , and let be the foot of the altitude from . Let be a point in the interior of the segment . Let K be the point on the segment such that . Similarly, let be the point on the segment such that . Let be the point of intersection of and . Show that .

*Author: Josef Tkadlec, Czech Republic*

### Problem 6.

Find all positive integers n for which there exist non-negative integers , , , such that

*Author: Dušan Djukić, Serbia*