MathDB

4

Part of 2022 Iranian Geometry Olympiad

Problems(3)

Internal angle bisector problem

Source: IGO 2022 Elementary P4

12/14/2022
Let ADAD be the internal angle bisector of triangle ABCABC. The incircles of triangles ABCABC and ACDACD touch each other externally. Prove that ∠ABC>120∘\angle ABC > 120^{\circ}. (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)
Proposed by Volodymyr Brayman (Ukraine)
geometryangle bisectoriranian geometry olympiad
Compatible polygons

Source: IGO 2022 Intermediate P4

12/13/2022
We call two simple polygons P,QP, Q <spanclass=′latex−italic′>compatible</span><span class='latex-italic'>compatible</span> if there exists a positive integer kk such that each of P,QP, Q can be partitioned into kk congruent polygons similar to the other one. Prove that for every two even integers m,n≥4m, n \geq 4, there are two compatible polygons with mm and nn sides. (A simple polygon is a polygon that does not intersect itself.)
Proposed by Hesam Rajabzadeh
combinatorics
IGO 2022 advanced/free P4

Source: Iranian Geometry Olympiad 2022 P4 Advanced, Free

12/13/2022
Let ABCDABCD be a trapezoid with AB∥CDAB\parallel CD. Its diagonals intersect at a point PP. The line passing through PP parallel to ABAB intersects ADAD and BCBC at QQ and RR, respectively. Exterior angle bisectors of angles DBADBA, DCADCA intersect at XX. Let SS be the foot of XX onto BCBC. Prove that if quadrilaterals ABPQABPQ, CDQPCDQP are circumcribed, then PR=PSPR=PS.
Proposed by Dominik Burek, Poland
geometry