MathDB

7

Part of 1990 IMO Longlists

Problems(2)

At least sqrt(n) distinct values - ILL 1990 CUB1

Source:

9/18/2010
AA and BB are two points in the plane α\alpha, and line rr passes through points A,BA, B. There are nn distinct points P1,P2,…,PnP_1, P_2, \ldots, P_n in one of the half-plane divided by line rr. Prove that there are at least n\sqrt n distinct values among the distances AP1,AP2,…,APn,BP1,BP2,…,BPn.AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.
geometrypoint seteuclidean distancecombinatorial geometryIMO ShortlistIMO Longlist
S is the incenter of triangle ABC

Source:

9/19/2010
Let SS be the incenter of triangle ABCABC. A1,B1,C1A_1, B_1, C_1 are the intersections of AS,BS,CSAS, BS, CS with the circumcircle of triangle ABCABC respectively. Prove that SA1+SB1+SC1≥SA+SB+SC.SA_1 + SB_1 + SC_1 \geq SA + SB + SC.
geometryincentercircumcircleinequalities