MathDB

3

Part of 2010 IberoAmerican

Problems(2)

Problem 3, Iberoamerican Olympiad 2010

Source:

9/24/2010
The circle Γ \Gamma is inscribed to the scalene triangle ABCABC. Γ \Gamma is tangent to the sides BC,CABC, CA and ABAB at D,ED, E and FF respectively. The line EFEF intersects the line BCBC at GG. The circle of diameter GDGD intersects Γ \Gamma in RR (R≠D R\neq D ). Let PP, QQ (P≠R,Q≠R P\neq R , Q\neq R ) be the intersections of Γ \Gamma with BRBR and CRCR, respectively. The lines BQBQ and CPCP intersects at XX. The circumcircle of CDECDE meets QRQR at MM, and the circumcircle of BDFBDF meet PRPR at NN. Prove that PMPM, QNQN and RXRX are concurrent.
Author: Arnoldo Aguilar, El Salvador
geometrycircumcircleincenterprojective geometrygeometry proposed
Problem 6, Iberoamerican Olympiad 2010

Source:

9/25/2010
Around a circular table sit 1212 people, and on the table there are 2828 vases. Two people can see each other, if and only if there is no vase lined with them. Prove that there are at least two people who can be seen.
combinatorics proposedcombinatorics