MathDB

1

Make the word repetitive

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.)
Romania (Dan Schwarz)

7

1

$KH$, $EM$ and $BC$ are concurrent

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.)
Luxembourg (Pierre Haas)

6

1

Every popular person is the best friend of a popular person?

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

5

1

Find all possible values of q-p

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

.
Luxembourg (Pierre Haas)

1

Prove DK and BC are perpendicular.

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.
Netherlands (Merlijn Staps)

4

1

Does there exist a sum-full zero-sum-free set?

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?
Romania (Dan Schwarz)

2

1

Find the greatest possible m of the m x n table

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

Poland (Tomasz Kobos)

3

1

Find all real functions

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}

.
Netherlands (Birgit van Dalen)

2012 EGMO

Subcontests

Make the word repetitive

$KH$, $EM$ and $BC$ are concurrent

Every popular person is the best friend of a popular person?

Find all possible values of q-p

Prove DK and BC are perpendicular.

Does there exist a sum-full zero-sum-free set?

Find the greatest possible m of the m x n table

Find all real functions