MathDB

IMO Shortlist 2009 - Problem A4

Source:

7/5/2010

Let

a

,

b

,

c

be positive real numbers such that

ab+bc+ca\leq 3abc

. Prove that

\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)

Proposed by Dzianis Pirshtuk, Belarus

inequalitiesIMO Shortlist

IMO Shortlist 2009 - Problem C4

Source:

7/5/2010

For an integer

m\geq 1

, we consider partitions of a

2^m\times 2^m

chessboard into rectangles consisting of cells of chessboard, in which each of the

2^m

cells along one diagonal forms a separate rectangle of side length

1

. Determine the smallest possible sum of rectangle perimeters in such a partition.
Proposed by Gerhard Woeginger, Netherlands

geometryrectanglecombinatoricsdissectionIMO ShortlistChessboardperimeter

IMO Shortlist 2009 - Problem G4

Source:

7/5/2010

Given a cyclic quadrilateral

ABCD

, let the diagonals

AC

and

BD

meet at

E

and the lines

AD

and

BC

meet at

F

. The midpoints of

AB

and

CD

are

G

and

H

, respectively. Show that

EF

is tangent at

E

to the circle through the points

E

,

G

and

H

.
Proposed by David Monk, United Kingdom

geometryIMO Shortlistgeometry solvedcyclic quadrilateralprojective geometrytangentpower of a point

IMO Shortlist 2009 - Problem N4

Source:

7/5/2010

Find all positive integers

n

such that there exists a sequence of positive integers

a_1

,

a_2

,

\ldots

,

a_n

satisfying:

a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1

for every

k

with

2\leq k\leq n-1

.
Proposed by North Korea

modular arithmeticnumber theoryIMO ShortlistSequenceDivisibility

4

Problems(4)