MathDB

3

Part of 2024 Saint Petersburg Mathematical Olympiad

Problems(3)

KZ=KT implies XT \perp YZ

Source: Saint Petersburg olympiad 2024, 11.3

9/22/2024
In unequal triangle ABCABC bisector AKAK was drawn. Diameter XYXY of its circumcircle is perpendicular to AKAK (order of points on circumcircle is B−X−A−Y−CB-X-A-Y-C). A circle, passing on points XX and YY, intersect segments BKBK and CKCK in points TT and ZZ respectively. Prove that if KZ=KTKZ=KT, then XT⊥YZXT \perp YZ.
geometrycircumcircle
DE+AC>2BM

Source: Saint Petersburg olympiad 2024, 10.3

9/22/2024
On the side BCBC of acute triangle ABCABC point PP was chosen. Point EE is symmetric to point BB onto line APAP. Segment PEPE meets circumcircle of triangle ABPABP in point DD. MM is midpoint of side ACAC. Prove that DE+AC>2BMDE+AC>2BM.
geometrycircumcircle
Line, passing 2 ants, tangent to a fixed circle

Source: Saint Petersburg olympiad 2024, 9.3

9/21/2024
The triangle ABCABC is inscribed in a circle. Two ants crawl out of points BB and CC at the same time. They crawl along the arc BCBC towards each other so that the product of the distances from them to point AA remains unchanged. Prove that during their movement (until the moment of meeting), the straight line passing through the ants touches some fixed circle.
geometry