MathDB

Not hard but cute (points lying on a circle)

Source: IMO Shortlist 2005; Polish second round 2006; Costa Rica final round 2006

2/24/2006

Given a triangle

ABC

satisfying

AC+BC=3\cdot AB

. The incircle of triangle

ABC

has center

I

and touches the sides

BC

and

CA

at the points

D

and

E

, respectively. Let

K

and

L

be the reflections of the points

D

and

E

with respect to

I

. Prove that the points

A

,

B

,

K

,

L

lie on one circle.
Proposed by Dimitris Kontogiannis, Greece

geometrycircumcirclehomothetyIMO Shortlisttriangle -incenter

polynomial with coefficients in {1,-1}

Source: Polish MO 2006

4/8/2006

Find all pairs of integers

a,b

for which there exists a polynomial

P(x) \in \mathbb{Z}[X]

such that product

(x^2+ax+b)\cdot P(x)

is a polynomial of a form

x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0

where each of

c_0,c_1,\ldots,c_{n-1}

is equal to

1

or

-1

.

algebrapolynomialrootsquadraticscoefficientsIMO Shortlist

shortlist2005

Source: IMO Shortlist 2005 Problem C1, but also Aus MO 1990

7/8/2006

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.
Proposed by Australia

combinatoricsIMO Shortlistgraph theoryinvariant

1

Problems(3)