MathDB

5

Part of 2011 IMO Shortlist

Problems(3)

IMO Shortlist 2011, Algebra 5

Source: IMO Shortlist 2011, Algebra 5

7/11/2012
Prove that for every positive integer n,n, the set {2,3,4,…,3n+1}\{2,3,4,\ldots,3n+1\} can be partitioned into nn triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
Proposed by Canada
trigonometryinequalitiestriangle inequalityalgebraTriangleIMO Shortlist
IMO Shortlist 2011, Combinatorics 5

Source: IMO Shortlist 2011, Combinatorics 5

7/12/2012
Let mm be a positive integer, and consider a m×mm\times m checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time 00, each ant starts moving with speed 11 parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn 90∘90^{\circ} clockwise and continue moving with speed 11. When more than 22 ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.
Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist.
Proposed by Toomas Krips, Estonia
combinatoricsIMO ShortlistalgorithmHi
IMO Shortlist 2011, G5

Source: IMO Shortlist 2011, G5

7/13/2012
Let ABCABC be a triangle with incentre II and circumcircle ω\omega. Let DD and EE be the second intersection points of ω\omega with AIAI and BIBI, respectively. The chord DEDE meets ACAC at a point FF, and BCBC at a point GG. Let PP be the intersection point of the line through FF parallel to ADAD and the line through GG parallel to BEBE. Suppose that the tangents to ω\omega at AA and BB meet at a point KK. Prove that the three lines AE,BDAE,BD and KPKP are either parallel or concurrent.
Proposed by Irena Majcen and Kris Stopar, Slovenia
geometryincentercircumcirclegeometric transformationhomothetyIMO Shortlist