MathDB

Problem 6

Part of 2024 Ukraine National Mathematical Olympiad

Problems(4)

We tried our best coming up with easy geo

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 8.6

3/20/2024
Cyclic quadrilateral ABCDABCD is such that ∠BAD=2∠ADC\angle BAD = 2\angle ADC and CD=2BCCD = 2BC. Let HH be the projection of CC onto ADAD. Prove that BH∥CDBH \parallel CD.
Proposed by Fedir Yudin, Anton Trygub
geometrycyclic quadrilateral
Modern sangaku

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 9.6

3/20/2024
You are given a convex hexagon with parallel opposite sides. For each pair of opposite sides, a line is drawn parallel to these sides and equidistant from them. Prove that the three lines thus obtained intersect at one point if and only if the lengths of the opposite sides are equal.
Proposed by Nazar Serdyuk
geometry
Author of IMO 2021 P3 claims this is his best problem ever!

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 10.6

3/20/2024
Inside a quadrilateral ABCDABCD with AB=BC=CDAB=BC=CD, the points PP and QQ are chosen so that AP=PB=CQ=QDAP=PB=CQ=QD. The line through the point PP parallel to the diagonal ACAC intersects the line through the point QQ parallel to the diagonal BDBD at the point TT. Prove that BT=CTBT=CT.
Proposed by Mykhailo Shtandenko
geometry
6 equal angles, 2 unequaled authors

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.6

3/20/2024
The points A,B,C,DA, B, C, D lie on the line ℓ\ell in this order. The points PP and QQ are chosen on one side of the line ℓ\ell, and the point RR is chosen on the other side so that:
∠APB=∠CPD=∠QBC=∠QCB=∠RAD=∠RDA\angle APB = \angle CPD = \angle QBC = \angle QCB = \angle RAD = \angle RDA
Prove that the points P,Q,RP, Q, R lie on the same line.
Proposed by Mykhailo Shtandenko, Fedir Yudin
geometry