MathDB

three "old" circles and four concurrent lines

Source: IMO Shortlist 2006, Geometry 6, AIMO 2007, TST 3, P3

6/28/2007

Circles

w_{1}

and

w_{2}

with centres

O_{1}

and

O_{2}

are externally tangent at point

D

and internally tangent to a circle

w

at points

E

and

F

respectively. Line

t

is the common tangent of

w_{1}

and

w_{2}

at

D

. Let

AB

be the diameter of

w

perpendicular to

t

, so that

A, E, O_{1}

are on the same side of

t

. Prove that lines

AO_{1}

,

BO_{2}

,

EF

and

t

are concurrent.

ratiogeometryprojective geometryhomothetyIMO Shortlist

tiling holey triangles with diamonds

Source: IMO Shortlist 2006, Combinatorics 6

6/28/2007

A holey triangle is an upward equilateral triangle of side length

n

with

n

upward unit triangular holes cut out. A diamond is a

60^\circ-120^\circ

unit rhombus. Prove that a holey triangle

T

can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length

k

in

T

contains at most

k

holes, for

1\leq k\leq n

.
Proposed by Federico Ardila, Colombia

rhombuscombinatoricstilingsIMO ShortlistHall s marriage theorem

local champion integers

Source: IMO Shortlist 2006, Number Theory 6, AIMO 2007, TST 1, P3

6/28/2007

Let

a > b > 1

be relatively prime positive integers. Define the weight of an integer

c

, denoted by

w(c)

to be the minimal possible value of |x| \plus{} |y| taken over all pairs of integers

x

and

y

such that ax \plus{} by \equal{} c. An integer

c

is called a local champion if

w(c) \geq w(c \pm a)

and

w(c) \geq w(c \pm b)

.
Find all local champions and determine their number.
Proposed by Zoran Sunic, USA

number theoryrelatively primegreatest common divisorIMO Shortlist

6

Problems(3)