MathDB

5

Part of 2024 Saint Petersburg Mathematical Olympiad

Problems(3)

AD,CD \cap [ABC] is empty

Source: Saint Petersburg olympiad 2024, 11.5

9/22/2024
There are 100100 points of general position marked on the plane (i.e. no three lie on the same straight line). Prove that it is possible to select three marked points A,B,CA, B, C so that for any point DD of the remaining 9797 marked points, the lines ADAD and CDCD would not contain points lying inside the triangle ABCABC.
geometrycombinatorial geometry
Segments with lengths 97, 100, 103

Source: Saint Petersburg olympiad 2024, 10.5

9/22/2024
2 000 0002 \ 000 \ 000 points with integer coordinates are marked on the numeric axis. Segments of lengths 9797, 100100 and 103103 with ends at these points are considered. What is the largest number of such segments?
number theorycombinatorics
BD+BE=2CH

Source: Saint Petersburg olympiad 2024, 9.5

9/22/2024
Let AHAH be altitude in acute trinagle ABCABC, inscribed in circle ss. Points DD and EE are chosen on segment BHBH. Points XX and YY are chosen on rays ADAD and AEAE, respectively, such that midpoints of segments DXDX and EYEY lies on ss. Suppose that points BB, XX, YY and CC are concyclic. Prove that BD+BE=2CHBD+BE=2CH.
geometry