IMO shortlist problems 1959-2009 (new)

Ngày tải lên: 02/06/2014, 16:33

... (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

Ngày tải lên: 02/06/2014, 16:33

... 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) =...

Ngày tải lên: 02/06/2014, 16:33

... 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....

Ngày tải lên: 02/06/2014, 16:34

... 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

Ngày tải lên: 02/06/2014, 16:35

Ngày tải lên: 02/06/2014, 16:36

... ! = 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 |...

Ngày tải lên: 02/06/2014, 16:37

... 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...

Ngày tải lên: 02/06/2014, 16:41

... 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

Ngày tải lên: 02/06/2014, 16:42

Ngày tải lên: 02/06/2014, 16:45