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 <math>\triangle ABC</math> 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 | 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> | <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'' |
− | The | + | |
− | < | + | [[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] | ||
+ | |||
+ | {{IMO box|year=2012|before=[[2011 IMO Problems]]|after=[[2013 IMO Problems]]}} |
Latest revision as of 11:00, 24 February 2021
Problems of the 53st IMO 2012 in Mar del Plata, Argentina.
Contents
Day 1
Problem 1.
Given 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
Resources
2012 IMO (Problems) • Resources | ||
Preceded by 2011 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2013 IMO Problems |
All IMO Problems and Solutions |