MathDB

G1

Part of 2016 Costa Rica - Final Round

Problems(2)

equilateral wanted, DE//AC, AE: BE = 2: 1

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G1 day1

9/23/2021
Let △ABC\vartriangle ABC be isosceles with AB=ACAB = AC. Let ω\omega be its circumscribed circle and OO its circumcenter. Let DD be the second intersection of COCO with ω\omega. Take a point EE in ABAB such that DE∥ACDE \parallel AC and suppose that AE:BE=2:1AE: BE = 2: 1. Show that △ABC\vartriangle ABC is equilateral.
geometrycircumcircle
collinear wanted, 2 circles and orthocenter related

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G1 day2

9/25/2021
Let △ABC\vartriangle ABC be acute with orthocenter HH. Let XX be a point on BCBC such that B−X−CB-X-C. Let Γ\Gamma be the circumscribed circle of △BHX\vartriangle BHX and Γ2\Gamma_2 be the circumscribed circle of △CHX\vartriangle CHX. Let EE be the intersection of ABAB with Γ\Gamma , and DD be the intersection of ACAC with Γ2\Gamma_2. Let LL be the intersection of line HDHD with Γ\Gamma and JJ be the intersection of line EHEH with Γ2\Gamma_2. Prove that points LL, XX, and JJ are collinear.
Notation: A−B−CA-B-C means than points A,B,CA,B,C are collinear in that order i.e. B B lies between A A and CC.
geometrycollinear