MathDB

IMO Shortlist 2011, Combinatorics 6

Source: IMO Shortlist 2011, Combinatorics 6

7/12/2012

Let

n

be a positive integer, and let

W = \ldots x_{-1}x_0x_1x_2 \ldots

be an infinite periodic word, consisting of just letters

a

and/or

b

. Suppose that the minimal period

N

of

W

is greater than

2^n

.
A finite nonempty word

U

is said to appear in

W

if there exist indices

k \leq \ell

such that

U=x_k x_{k+1} \ldots x_{\ell}

. A finite word

U

is called ubiquitous if the four words

Ua

,

Ub

,

aU

, and

bU

all appear in

W

. Prove that there are at least

n

ubiquitous finite nonempty words.
Proposed by Grigory Chelnokov, Russia

combinatoricsCombinatorics of wordsIMO Shortlist

IMO Shortlist 2011, G6

Source: IMO Shortlist 2011, G6

7/13/2012

Let

ABC

be a triangle with

AB=AC

and let

D

be the midpoint of

AC

. The angle bisector of

\angle BAC

intersects the circle through

D,B

and

C

at the point

E

inside the triangle

ABC

. The line

BD

intersects the circle through

A,E

and

B

in two points

B

and

F

. The lines

AF

and

BE

meet at a point

I

, and the lines

CI

and

BD

meet at a point

K

. Show that

I

is the incentre of triangle

KAB

.
Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea

geometryincentersymmetryreflectionIMO Shortlist

IMO Shortlist 2011, Number Theory 6

Source: IMO Shortlist 2011, Number Theory 6

7/11/2012

Let

P(x)

and

Q(x)

be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both

P(x)

and

Q(x).

Suppose that for every positive integer

n

the integers

P(n)

and

Q(n)

are positive, and

2^{Q(n)}-1

divides

3^{P(n)}-1.

Prove that

Q(x)

is a constant polynomial.
Proposed by Oleksiy Klurman, Ukraine

algebrapolynomialnumber theoryIMO ShortlistDivisibility

6

Problems(3)