MathDB

Problem 2

Part of 1998 Ukraine National Math Olympiad

Problems(4)

1998 hexagons, 666 don't share a vertex (Ukraine 1998 Grade 8 P2)

Source:

6/3/2021
The plane is partitioned into congruent regular hexagons. Of these hexagons, some 19981998 are marked. Show that one can select 666666 of the marked hexagons in such a way that no two of them share a vertex.
combinatoricsgeometrycombinatorial geometry
construct triangle with same area as quadrilateral (Ukraine 1998 Grade 9 P2)

Source:

6/4/2021
A quadrilateral ABCDABCD is inscribed in a circle with diameter ADAD. Using a ruler and a compass, construct a triangle inscribed in the same circle and having the same area as ABCDABCD.
geometryconstruction
minimal distance to circumcenter from point outside triangle

Source: Ukraine 1998 Grade 10 P2

6/5/2021
Point MM is arbitrarily taken on side ACAC of a triangle ABCABC. Let OO be the intersection of the perpendiculars from the midpoints of segments AMAM and MCMC to lines BCBC and ABAB, respectively. For which position of MM is the distance OMOM minimal?
geometrycircumcircleTriangles
altitude cap bisector, prove triangle isosceles

Source: Ukraine 1998 Grade 11 P2

6/6/2021
The altitude CDCD of triangle ABCABC meets the bisector BKBK of this triangle at MM and the altitude KLKL of △BKC\triangle BKC at NN. The circumcircle of triangle BKNBKN meets the side ABAB at point P≠BP\ne B. Prove that the triangle KPMKPM is isosceles.
geometrycircumcircleTriangles