MathDB

g4

Part of Indonesia MO Shortlist - geometry

Problems(3)

PQ^3 /PN^2 = PS x RS /NS wanted, starting with 2 internally tangent circles

Source: Indonesia INAMO Shortlist 2008 G4

8/25/2021
Given that two circles σ1\sigma_1 and σ2\sigma_2 internally tangent at NN so that σ2\sigma_2 is inside σ1\sigma_1. The points QQ and RR lies at σ1\sigma_1 and σ2\sigma_2, respectively, such that N,R,QN,R,Q are collinear. A line through QQ intersects σ2\sigma_2 at SS and intersects σ1\sigma_1 at OO. The line through NN and SS intersects σ1\sigma_1 at PP. Prove that PQ3PN2=PS⋅RSNS.\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.
geometrytangent circles
AP/AD >= 1 - BC/(AB + CA), touchpoints with incircle

Source: Indonesia INAMO Shortlist 2009 G4 https://artofproblemsolving.com/community/c1101409_

12/10/2021
Let D,E,FD, E, F, be the touchpoints of the incircle in triangle ABCABC with sides BC,CA,ABBC, CA, AB, respectively, . Also, let ADAD and EFEF intersect at PP. Prove that APAD≥1−BCAB+CA\frac{AP}{AD} \ge 1 - \frac{BC}{AB + CA}.
geometric inequalitygeometryincircle
TA, TB,TC are sidelengths of triangle if T interior of equilateral ABC

Source: Indonesia INAMO Shortlist 2017 G4 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

11/15/2021
Inside the equilateral triangle ABCABC lies the point TT. Prove that TATA, TBTB and TCTC are the lengths of the sides of a triangle.
geometryEquilateraltriangle inequality