MathDB

3

Part of 2011 IberoAmerican

Problems(2)

Problem 3, Iberoamerican Olympiad 2011

Source:

10/2/2011
Let ABCABC be a triangle and X,Y,ZX,Y,Z be the tangency points of its inscribed circle with the sides BC,CA,ABBC, CA, AB, respectively. Suppose that C1,C2,C3C_1, C_2, C_3 are circle with chords YZ,ZX,XYYZ, ZX, XY, respectively, such that C1C_1 and C2C_2 intersect on the line CZCZ and that C1C_1 and C3C_3 intersect on the line BYBY. Suppose that C1C_1 intersects the chords XYXY and ZXZX at JJ and MM, respectively; that C2C_2 intersects the chords YZYZ and XYXY at LL and II, respectively; and that C3C_3 intersects the chords YZYZ and ZXZX at KK and NN, respectively. Show that I,J,K,L,M,NI, J, K, L, M, N lie on the same circle.
geometrygeometric transformationhomothetytrapezoidpower of a pointradical axisgeometry proposed
Problem 6, Iberoamerican Olympiad 2011

Source:

10/2/2011
Let kk and nn be positive integers, with k≥2k \geq 2. In a straight line there are knkn stones of kk colours, such that there are nn stones of each colour. A step consists of exchanging the position of two adjacent stones. Find the smallest positive integer mm such that it is always possible to achieve, with at most mm steps, that the nn stones are together, if: a) nn is even. b) nn is odd and k=3k=3
floor functioncombinatorics proposedcombinatorics