MathDB

4

Part of 2018 Iranian Geometry Olympiad

Problems(3)

2018 IGO Elementary Level P4

Source:

9/20/2018
There are two circles with centers O1,O2O_1,O_2 lie inside of circle ω\omega and are tangent to it. Chord ABAB of ω\omega is tangent to these two circles such that they lie on opposite sides of this chord. Prove that ∠O1AO2+∠O1BO2>90∘\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ.
Proposed by Iman Maghsoudi
IGO2018 igoIrangeometrygeometric ineq
connecting points with centroids of faces of a polyhedron, in a sequence

Source: Iranian Geometry Olympiad 2018 IGO Intermediate p4

9/19/2018
We have a polyhedron all faces of which are triangle. Let PP be an arbitrary point on one of the edges of this polyhedron such that PP is not the midpoint or endpoint of this edge. Assume that P0=PP_0 = P. In each step, connect PiP_i to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point Pi+1P_{i+1}. Continue this process with Pi+1P_{i+1} and the other face containing Pi+1P_{i+1}. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)
Proposed by Mahdi Etesamifard - Morteza Saghafian
geometry3-Dimensional Geometry3D geometrypolyhedronCentroid
ABCD IS tangential, if KLMN is cyclic then ABCD is cyclic

Source: Iranian Geometry Olympiad 2018 IGO Advanced p4

9/19/2018
Quadrilateral ABCDABCD is circumscribed around a circle. Diagonals AC,BDAC,BD are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments AB,BC,CDAB,BC,CD and DADA at points K,L,MK,L,M and NN. Given that KLMNKLMN is cyclic, prove that so is ABCDABCD.
Proposed by Nikolai Beluhov (Bulgaria)
tangential quadrilateralgeometrycyclic quadrilateralangle bisector