MathDB

1

Part of 2005 India IMO Training Camp

Problems(4)

A well known one

Source: Indian IMOTC 2005 Day 2 Problem 1

9/17/2005
Consider a nn-sided polygon inscribed in a circle (n≥4n \geq 4). Partition the polygon into n−2n-2 triangles using non-intersecting diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.
geometry solvedgeometry
Rationals and irrationals again

Source: Indian IMOTC 2005 Day 3 Problem 1

9/23/2005
Let 0<a<b0 <a <b be two rational numbers. Let MM be a set of positive real numbers with the properties: (i) a∈Ma \in M and b∈Mb \in M; (ii) if xx ∈M\in M and y∈My \in M, then xy∈M\sqrt{xy} \in M. Let M∗M^*denote the set of all irrational numbers in MM. prove that every c,dc,d such that a<c<d<ba <c <d<b, M∗M^* contains an element mm with property c<m<dc<m<d
number theory unsolvednumber theory
One on quads

Source: Indian IMOTC 2005 Day 4 Problem 1

9/23/2005
Let ABCDABCD be a convex quadrilateral. The lines parallel to ADAD and CDCD through the orthocentre HH of ABCABC intersect ABAB and BCBC Crespectively at PP and QQ. prove that the perpendicular through HH to th eline PQPQ passes through th eorthocentre of triangle ACDACD
geometryindia
Radical axis of two incircles passes through a fixed point

Source: IMO Shortlist 2004 geometry problem G7 (extended)

6/12/2005
For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X. This is a slight extension of the [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=41033]IMO Shortlist 2004 geometry problem 7 and can be found, together with the proposed solution, among the files uploaded at http://www.mathlinks.ro/Forum/viewtopic.php?t=15622 . Note that the problem was proposed by Russia. I could not find the names of the authors, but I have two particular persons under suspicion. Maybe somebody could shade some light on this... Darij
geometrypower of a pointradical axisIMO Shortlistgeometry proposed