MathDB

1

Part of 2008 IMO Shortlist

Problems(2)

IMO ShortList 2008, Combinatorics problem 1

Source: IMO ShortList 2008, Combinatorics problem 1, German TST 1, P1, 2009

7/9/2009
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest n n for which there exist n n boxes B1 B_1, … \ldots, Bn B_n such that Bi B_i and Bj B_j intersect if and only if i≢j±1(modn) i\not\equiv j\pm 1\pmod n. Proposed by Gerhard Woeginger, Netherlands
geometryrectanglecombinatoricscombinatorial geometryIMO ShortlistExtremal combinatorics
IMO ShortList 2008, Number Theory problem 1

Source: IMO ShortList 2008, Number Theory problem 1

7/9/2009
Let nn be a positive integer and let pp be a prime number. Prove that if aa, bb, cc are integers (not necessarily positive) satisfying the equations an+pb=bn+pc=cn+pa a^n + pb = b^n + pc = c^n + pa then a=b=ca = b = c.
Proposed by Angelo Di Pasquale, Australia
algebranumber theoryIMO Shortlistequation