MathDB

g7

Part of Indonesia MO Shortlist - geometry

Problems(4)

<AMD=<AND wanted, starting with isosceles rapezoid

Source: Indonesia INAMO Shortlist 2008 G7

8/25/2021
Given an isosceles trapezoid ABCDABCD with base ABAB. The diagonals ACAC and BDBD intersect at point SS. Let MM the midpoint of BCBC and the bisector of the angle BSCBSC intersect BCBC at NN. Prove that ∠AMD=∠AND\angle AMD = \angle AND.
geometryequal anglestrapezoidisosceles
CQ bisects <ACD , OA =(OB x OD)(OC + CD), O intersection of diagonals of ABCD

Source: Indonesia INAMO Shortlist 2009 G7 https://artofproblemsolving.com/community/c1101409_

12/11/2021
Given a convex quadrilateral ABCDABCD, such that OA=OB⋅ODOC+CDOA = \frac{OB \cdot OD}{OC + CD} where OO is the intersection of the two diagonals. The circumcircle of triangle ABCABC intersects BDBD at point QQ. Prove that CQCQ bisects ∠ACD\angle ACD
angle bisectorgeometry
min |(cot A + cot B)(cot B + cot C)(cot C + cot A)| in ABC

Source: Indonesia INAMO Shortlist 2010 G7

8/27/2021
In triangle ABCABC, find the smallest possible value of ∣(cot⁡A+cot⁡B)(cot⁡B+cot⁡C)(cot⁡C+cot⁡A)∣|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|
geometrygeometric inequalitytrigonometryinequalities
angle bisector wanted, tangents to semicircle (O)

Source: Indonesia INAMO Shortlist 2017 G7 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

11/15/2021
A semicircle (O)(O) is drawn with the center OO, where OO lies on a line ℓ\ell. CC and DD lie on the circle (O)(O), and the tangent lines of (O)(O) at points CC and DD intersects the line ℓ\ell at points BB and AA, respectively, such that OO lies between points BB and AA. Let EE be the intersection point between ACAC and BDBD, and FF the point on ℓ\ell so that EFEF is perpendicular to line ℓ\ell. Prove that EFEF bisects the angle ∠CFD\angle CFD.
geometryangle bisector