MathDB

IMO Shortlist 2009 - Problem A5

Source:

7/5/2010

Let

f

be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers

x

and

y

such that

f\left(x-f(y)\right)>yf(x)+x

Proposed by Igor Voronovich, Belarus

functioninequalitiesalgebraFunctional inequalityIMO Shortlist

IMO Shortlist 2009 - Problem C5

Source:

7/5/2010

Five identical empty buckets of

2

-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
Proposed by Gerhard Woeginger, Netherlands

combinatoricsgameIMO ShortlistinvariantGerhard Woeginger

IMO Shortlist 2009 - Problem G5

Source:

7/5/2010

Let

P

be a polygon that is convex and symmetric to some point

O

. Prove that for some parallelogram

R

satisfying

P\subset R

we have

\frac{|R|}{|P|}\leq \sqrt 2

where

|R|

and

|P|

denote the area of the sets

R

and

P

, respectively.
Proposed by Witold Szczechla, Poland

geometryparallelogramgeometric inequalityareapolygonIMO Shortlist

IMO Shortlist 2009 - Problem N5

Source:

7/5/2010

Let

P(x)

be a non-constant polynomial with integer coefficients. Prove that there is no function

T

from the set of integers into the set of integers such that the number of integers

x

with

T^n(x)=x

is equal to

P(n)

for every

n\geq 1

, where

T^n

denotes the

n

-fold application of

T

.
Proposed by Jozsef Pelikan, Hungary

algebrapolynomialnumber theorypermutationIMO Shortlist

5

Problems(4)