MathDB

C-B=60 <degrees>

Source: Moldova TST 2005, IMO Shortlist 2004 geometry problem 3

4/10/2005

Let

O

be the circumcenter of an acute-angled triangle

ABC

with

{\angle B<\angle C}

. The line

AO

meets the side

BC

at

D

. The circumcenters of the triangles

ABD

and

ACD

are

E

and

F

, respectively. Extend the sides

BA

and

CA

beyond

A

, and choose on the respective extensions points

G

and

H

such that

{AG=AC}

and

{AH=AB}

. Prove that the quadrilateral

EFGH

is a rectangle if and only if

{\angle ACB-\angle ABC=60^{\circ }}

.
Proposed by Hojoo Lee, Korea

geometrycircumcirclehomothetyTriangleIMO Shortlist

[f^2(m)+f(n)]|(m^2+n)^2

Source: IMO ShortList 2004, number theory problem 3

2/19/2005

Find all functions

f: \mathbb{N^{*}}\to \mathbb{N^{*}}

satisfying

\left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}

for any two positive integers

m

and

n

.
Remark. The abbreviation

\mathbb{N^{*}}

stands for the set of all positive integers:

\mathbb{N^{*}}=\left\{1,2,3,...\right\}

. By

f^{2}\left(m\right)

, we mean

\left(f\left(m\right)\right)^{2}

(and not

f\left(f\left(m\right)\right)

).
Proposed by Mohsen Jamali, Iran

functionnumber theoryalgebraDivisibilityfunctional equationIMO Shortlist

Number theory or function ?

Source: IMO ShortList 2004, algebra problem 3

3/18/2005

Does there exist a function

s\colon \mathbb{Q} \rightarrow \{-1,1\}

such that if

x

and

y

are distinct rational numbers satisfying

{xy=1}

or

{x+y\in \{0,1\}}

, then

{s(x)s(y)=-1}

? Justify your answer.
Proposed by Dan Brown, Canada

functionnumber theorycontinued fractionalgebrafunctional equationIMO Shortlist

Colombia TST

Source: IMO ShortList 2004, combinatorics problem 3

6/7/2005

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer

{n\ge 4}

, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on

n

vertices (where each pair of vertices are joined by an edge).
Proposed by Norman Do, Australia

graph theorycombinatoricsalgorithmgameIMO Shortlist

3

Problems(4)