MathDB

6

Part of 2008 IMO Shortlist

Problems(3)

IMO ShortList 2008, Algebra problem 6

Source: IMO ShortList 2008, Algebra problem 6

7/9/2009
Let f:R→N f: \mathbb{R}\to\mathbb{N} be a function which satisfies f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right) for all x x, y∈R y\in\mathbb{R}. Prove that there is a positive integer which is not a value of f f.
Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania
functionalgebrafunctional equationrangeIMO Shortlist
IMO Shortlist 2008, Geometry problem 6

Source: IMO Shortlist 2008, Geometry problem 6, German TST 7, P3, 2009, Exam set by Christian Reiher

7/9/2009
There is given a convex quadrilateral ABCD ABCD. Prove that there exists a point P P inside the quadrilateral such that \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} if and only if the diagonals AC AC and BD BD are perpendicular.
Proposed by Dusan Djukic, Serbia
geometrycircumcirclesymmetryhomothetyquadrilateralIMO Shortlist
IMO ShortList 2008, Combinatorics problem 6

Source: IMO ShortList 2008, Combinatorics problem 6

7/9/2009
For n≥2 n\ge 2, let S1 S_1, S2 S_2, … \ldots, S2n S_{2^n} be 2n 2^n subsets of A \equal{} \{1, 2, 3, \ldots, 2^{n \plus{} 1}\} that satisfy the following property: There do not exist indices a a and b b with a<b a < b and elements x x, y y, z∈A z\in A with x<y<z x < y < z and y y, z∈Sa z\in S_a, and x x, z∈Sb z\in S_b. Prove that at least one of the sets S1 S_1, S2 S_2, … \ldots, S2n S_{2^n} contains no more than 4n 4n elements. Proposed by Gerhard Woeginger, Netherlands
combinatoricsSubsetsExtremal combinatoricsIMO Shortlist