OF THE EUROPEAN MATHEMATICAL SOCIETY

Problem corner
XIX Olimpiada Iberomaericana de Matemáticas
Castellón, Spain, 2004.
Problem 1
We colour some cells of a 1001 × 1001 board according to
the following rules:
1) Whenever two cells share a common edge, at least one
of them must be coloured.
2) Every six consecutive cells (in a column or a row) must
have at least two neighbouring cells coloured.
Find the minimum numbers of cells to be coloured.
Problem 2
Let O be the centre of a circle of radius r and let A be an
exterior point. Let M be any point on the circle and N be
diametrically opposite to M. Find the locus of the centres
of the circles passing through A, M and N as M moves
along the circle.
Problem 3
For n and k positive integers such that n is odd or n and k
are even, prove that there exist integers a and b such that
gcd(a, n) = gcd(b, n) = 1 and k = a+b.
Problem 4
Find all couples (a,b) of 2-digit positive integers such that
100a + b and 201a + b are 4-digit perfect squares.
Problem 5
Given a scalene triangle ABC, let A’ be the intersection
point of the angle bisector of A with the segment BC. B’
and C’ are defi ned similarly.
We consider the following points: A” is the intersection
of BC and the perpendicular bisector of AA’; B” is the
intersection of AC and the perpendicular bisector of BB’;
and C” is the intersection of AB and the perpendicular
bisector of CC’. Prove that A”, B” and C” lie on a line.
Problem 6
For a set H of points in the plane, we say that a point P is
a cut point of H if there are four different points A, B, C
and D belonging to H such that lines AB and CD are different
and meet at P. Given a fi nite set A 0 , we consider
the sequence of sets {A 1 , A 2 , A 3 , …}; A j+1 being the union
A j with the set of its cut points. If the union of all the
sets of the sequence is fi nite, prove that A j = A 1 for all j.
To know more
Royal Spanish Mathematical Society:
http://www.rsme.es
Ibero-American Organization:
http://www.campus-oei.org/oim/
Spanish Federation of Mathematics Teachers:
http://www.fespm.org/
Miguel de Guzman’s project:
http://www.estalmat.org
Marco Castrillón López
[mcastri@mat.ucm.es] received his
PhD from the Universidad Complutense
de Madrid in 1999. After
a period as assistant at École Polythechnique
Fédérale de Lausanne
(Switzerland) he is now Associate
Professor at the Department of Geometry
and Topology of Universidad
Complutense de Madrid. He is a member of the Olympic
Committee of the Royal Spanish Mathematical Society
and has been the tutor of the Spanish Olympic teams under
several international competitions.
Reto Müller (ETH Zürich, Switzerland)
Differential Harnack Inequalities and
the Ricci Flow
ISBN 3-03719-030-2
2006. 100 pages. Softcover. 17 cm x 24 cm
28.00 Euro
In 2002, Grisha Perelman presented a new kind
of differential Harnack inequality which involves
both the (adjoint) linear heat equation and the
Ricci flow. This led to a completely new approach to the Ricci flow that
allowed interpretation as a gradient flow which maximizes different
entropy functionals. The goal of this book is to explain this analytic tool
in full detail for the two examples of the linear heat equation and the
Ricci flow. It begins with the original Li–Yau result, presents Hamilton’s
Harnack inequalities for the Ricci flow, and ends with Perelman’s
entropy formulas and space-time geodesics. The book is a self-contained,
modern introduction to the Ricci flow and the analytic methods
to study it. It is primarily addressed to students who have a basic introductory
knowledge of analysis and of Riemannian geometry and who
are attracted to further study in geometric analysis. No previous knowledge
of differential Harnack inequalities or the Ricci flow is required.
European Mathematical Society Publishing House
Seminar for Applied Mathematics, ETH-Zentrum FLI C4
Fliederstrasse 23, CH-8092 Zürich, Switzerland
orders@ems-ph.org
www.ems-ph.org
44 EMS Newsletter September 2006