IMO shortlist problems 1959-2009 (new)
IMO Shortlist 2002
... (2m − 1)×2
2 × 2 m − 22× 3
(m − 2)+2 (2m + 1)×2
(m − 1)+2 m
2
+ m +(m + 1)
Comment
1. Covering problems are not uncommon, but this problem seems to be rather unusual. What
is nice is that when
IMO Shortlist 2003
... 10f(2m − 1).
44th International
Mathematical Olympiad
Short-listed
Problems and
Solutions
Tokyo Japan
July 2003
CONTENTS v
Contents
I Problems 1
Algebra 3
Combinatorics 5
Geometry 7
Number Theory ... Solution 2 shows that this problem can be solved without the knowledge of
Simson’s theorem.
Part I
Problems
1
19
Since f(xy) = A
g(xy)
and A is bijective, it follows that either g(xy) =...
IMO Shortlist 2004
... ABM.
Duˇsan Djuki´c
Vladimir Jankovi´c
Ivan Mati´c
Nikola Petrovi´c
IMO Shortlist 2004
From the book The IMO Compendium,
www .imo. org.yu
Springer
Berlin Heidelberg NewYork
Hong Kong London
Milan ... quadrilateral
EF GH is a rectangle if and only if ∠ACB −∠ABC = 60
◦
.
1
Problems
1.1 The Forty-Fifth IMO
Athens, Greece, July 7–19, 2004
1.1.1 Contest Problems
First Day (July 12)
1....
IMO Shortlist 2006
... c) with all coordinates distinct.
47
th
International Mathematical Olympiad
Slovenia 2006
Shortlisted Problems with Solutions
48
G9. Points A
1
, B
1
, C
1
are chosen on the sides BC, CA, AB of
IMO Shortlist 2008
... !
=
2
k +
k +
k
nice sets.
49
th
International Mathematical Olympiad
Spain 2008
Shortlisted Problems with Solutions
15
A6. Let f : R → N be a function which satisfies
f
x +
1
f(y)
= ... Part (a) alone might also be considered as a possible contest problem (in the category
of easy problems) .
18
Note that
MN >
ac(a + c) + bd(b + d)
s ≥ |W |· s. (4)
Now (2) and (4) yield
|...
IMO Shortlist 2009
... insistently ask everybody to consider the following IMO Regulations rule:
These Shortlist Problems
have to be kept strictly confidential
until IMO 2010.
The Problem Selection Committee
Konrad Engel, ... into that bucket and produces an overflow since b < 2 − 2
1−R
.
35
Combinatorics Problem Shortlist 50th IMO 2009
Combinatorics
C1 NZL (New Zealand)
Consider 2009 cards, each havin...
IMO Shortlist 2010
... Confidentiality
The Shortlis t ed Problems sh ould be kept
strictly confidential until IMO 2011.
Contributing Countries
The Organizing Committee and the Problem Selection Committee of IMO 2010 thank the
following
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