MathDB

3

Part of 2005 Federal Competition For Advanced Students, Part 2

Problems(2)

Altitudes and circles

Source: Austrian MO 2005 round 2

6/27/2005
Triangle DEFDEF is acute. Circle c1c_1 is drawn with DFDF as its diameter and circle c2c_2 is drawn with DEDE as its diameter. Points YY and ZZ are on DFDF and DEDE respectively so that EYEY and FZFZ are altitudes of triangle DEFDEF . EYEY intersects c1c_1 at PP, and FZFZ intersects c2c_2 at QQ. EYEY extended intersects c1c_1 at RR, and FZFZ extended intersects c2c_2 at SS. Prove that PP, QQ, RR, and SS are concyclic points.
geometryAustriaAUT
AQ = BQ [intersections of lines through Q with cube surface]

Source: Austrian MO 2005 round 2

6/27/2005
Let QQ be a point inside a cube. Prove that there are infinitely many lines ll so that AQ=BQAQ=BQ where AA and BB are the two points of intersection of ll and the surface of the cube.
geometry3D geometrygeometric transformationreflectionPutnamgeometry solved