MathDB

5

Part of 2022 Iranian Geometry Olympiad

Problems(3)

Quadrilateral geo

Source: IGO 2022 Intermediate P5

12/13/2022
Let ABCDABCD be a quadrilateral inscribed in a circle ω\omega with center OO. Let PP be the intersection of two diagonals ACAC and BDBD. Let QQ be a point lying on the segment OPOP. Let EE and FF be the orthogonal projections of QQ on the lines ADAD and BCBC, respectively. The points MM and NN lie on the circumcircle of triangle QEFQEF such that QM∥ACQM \parallel AC and QN∥BDQN \parallel BD. Prove that the two lines MEME and NFNF meet on the perpendicular bisector of segment CDCD.
Proposed by Tran Quang Hung, Vietnam
geometry
Equilateral triangles and squares

Source: IGO 2022 Elementary P5

12/14/2022
a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? (Note that in both parts, there is no assumption on the intersection of interior of polygons.)
Proposed by Hesam Rajabzadeh
iranian geometry olympiadgeometryEquilateral Trianglesquare
IGO 2022 advanced/free P5

Source: Iranian Geometry Olympiad 2022 P5 Advanced, Free

12/13/2022
Let ABCABC be an acute triangle inscribed in a circle ω\omega with center OO. Points EE, FF lie on its side ACAC, ABAB, respectively, such that OO lies on EFEF and BCEFBCEF is cyclic. Let RR, SS be the intersections of EFEF with the shorter arcs ABAB, ACAC of ω\omega, respectively. Suppose KK, LL are the reflection of RR about CC and the reflection of SS about BB, respectively. Suppose that points PP and QQ lie on the lines BSBS and RCRC, respectively, such that PKPK and QLQL are perpendicular to BCBC. Prove that the circle with center PP and radius PKPK is tangent to the circumcircle of RCERCE if and only if the circle with center QQ and radius QLQL is tangent to the circumcircle of BFSBFS.
Proposed by Mehran Talaei
geometry