MathDB

3

Part of 1999 IberoAmerican

Problems(2)

14th ibmo - cuba 1999/q3.

Source: Spanish Communities

4/16/2006
Let P1,P2,…,PnP_1,P_2,\dots,P_n be nn distinct points over a line in the plane (n≥2n\geq2). Consider all the circumferences with diameters PiPjP_iP_j (1≤i,j≤n1\leq{i,j}\leq{n}) and they are painted with kk given colors. Lets call this configuration a (n,kn,k)-cloud.
For each positive integer kk, find all the positive integers nn such that every possible (n,kn,k)-cloud has two mutually exterior tangent circumferences of the same color.
linear algebramatrixinductionpigeonhole principlecombinatorics unsolvedcombinatorics
14th ibmo - cuba 1999/q6.

Source: Spanish Communities

4/16/2006
Let AA and BB points in the plane and CC a point in the perpendiclar bisector of ABAB. It is constructed a sequence of points C1,C2,…,Cn,…C_1,C_2,\dots, C_n,\dots in the following way: C1=CC_1=C and for n≥1n\geq1, if CnC_n does not belongs to ABAB, then Cn+1C_{n+1} is the circumcentre of the triangle △ABCn\triangle{ABC_n}.
Find all the points CC such that the sequence C1,C2,…C_1,C_2,\dots is defined for all nn and turns eventually periodic.
Note: A sequence C1,C2,…C_1,C_2, \dots is called eventually periodic if there exist positive integers kk and pp such that Cn+p=cnC_{n+p}=c_n for all n≥kn\geq{k}.
inductionnumber theorygeometry unsolvedgeometry