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Difference between revisions of "2012 EGMO Problems"

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===Problem 3===
 
===Problem 3===
 
Find all functions <math>f:\mathbb{R}\to\mathbb{R}</math> such that<cmath>f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)</cmath>for all <math>x,y\in\mathbb{R}</math>.
 
Find all functions <math>f:\mathbb{R}\to\mathbb{R}</math> such that<cmath>f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)</cmath>for all <math>x,y\in\mathbb{R}</math>.
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[[2012 EGMO Problems/Problem 3|Solution]]
 
[[2012 EGMO Problems/Problem 3|Solution]]
  
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A set <math>A</math> of integers is called sum-full if <math>A \subseteq A + A</math>, i.e. each element <math>a \in A</math> is the sum of some pair of (not necessarily different) elements <math>b,c \in A</math>. A set <math>A</math> of integers is said to be zero-sum-free if <math>0</math> is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of <math>A</math>.
 
A set <math>A</math> of integers is called sum-full if <math>A \subseteq A + A</math>, i.e. each element <math>a \in A</math> is the sum of some pair of (not necessarily different) elements <math>b,c \in A</math>. A set <math>A</math> of integers is said to be zero-sum-free if <math>0</math> is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of <math>A</math>.
 
Does there exist a sum-full zero-sum-free set of integers?
 
Does there exist a sum-full zero-sum-free set of integers?
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[[2012 EGMO Problems/Problem 4|Solution]]
 
[[2012 EGMO Problems/Problem 4|Solution]]
  
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There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if <math>A</math> is a friend of <math>B</math>, then <math>B</math> is a friend of <math>A</math>.)
 
There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if <math>A</math> is a friend of <math>B</math>, then <math>B</math> is a friend of <math>A</math>.)
 
Each person is required to designate one of their friends as their best friend. If <math>A</math> designates <math>B</math> as her best friend, then (unfortunately) it does not follow that <math>B</math> necessarily designates <math>A</math> as her best friend. Someone designated as a best friend is called a <math>1</math>-best friend. More generally, if <math>n> 1</math> is a positive integer, then a user is an <math>n</math>-best friend provided that they have been designated the best friend of someone who is an <math>(n-1)</math>-best friend. Someone who is a <math>k</math>-best friend for every positive integer <math>k</math> is called popular.
 
Each person is required to designate one of their friends as their best friend. If <math>A</math> designates <math>B</math> as her best friend, then (unfortunately) it does not follow that <math>B</math> necessarily designates <math>A</math> as her best friend. Someone designated as a best friend is called a <math>1</math>-best friend. More generally, if <math>n> 1</math> is a positive integer, then a user is an <math>n</math>-best friend provided that they have been designated the best friend of someone who is an <math>(n-1)</math>-best friend. Someone who is a <math>k</math>-best friend for every positive integer <math>k</math> is called popular.
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(a) Prove that every popular person is the best friend of a popular person.
 
(a) Prove that every popular person is the best friend of a popular person.
 +
 
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.
 
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.
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[[2012 EGMO Problems/Problem 6|Solution]]
 
[[2012 EGMO Problems/Problem 6|Solution]]
  
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===Problem 8===
 
===Problem 8===
 
A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, <math>ababab</math> and <math>abcabc</math> are repetitive, but <math>ababa</math> and <math>aabb</math> are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)
 
A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, <math>ababab</math> and <math>abcabc</math> are repetitive, but <math>ababa</math> and <math>aabb</math> are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)
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[[2012 EGMO Problems/Problem 8|Solution]]
 
[[2012 EGMO Problems/Problem 8|Solution]]

Latest revision as of 09:15, 23 December 2022

Day 1

Problem 1

Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.) Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.

Solution

Problem 2

Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds:\[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\] Solution

Problem 3

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that\[f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)\]for all $x,y\in\mathbb{R}$.

Solution

Problem 4

A set $A$ of integers is called sum-full if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be zero-sum-free if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers?

Solution

Day 2

Problem 5

The numbers $p$ and $q$ are prime and satisfy \[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\] for some positive integer $n$. Find all possible values of $q-p$.


Solution

Problem 6

There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.) Each person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a $1$-best friend. More generally, if $n> 1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular.

(a) Prove that every popular person is the best friend of a popular person.

(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.

Solution

Problem 7

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma$ with the circumcircle of triangle $BLM$. Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)

Solution

Problem 8

A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, $ababab$ and $abcabc$ are repetitive, but $ababa$ and $aabb$ are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)

Solution

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