MathDB

9.6

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

max no of circles in regular hexagon (I Soros Olympiad 1994-95 R1 9.6 10.4 11.3)

Source:

7/31/2021
Given a regular hexagon, whose sidelength is 1 1 . What is the largest number of circles of radius 34\frac{\sqrt3}{4} can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)
combinatorial geometrycircleshexagongeometry
tangent circles

Source: I Soros Olympiad 1994-95 Round 2 9.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/25/2024
A circle can be drawn around the quadrilateral ABCDABCD. KK is a point on the diagonal BDBD . The straight line CKCK intersects the side ADAD at the point MM. Prove that the circles circumscribed around the triangles BCKBCK and ACMACM are tangent.
geometrytangent circles
orthocenter lies on incircle (I Soros Olympiad 1994-95 Ukraine R2 9.6)

Source:

6/6/2024
In the triangle ABCABC, the orthocenter HH lies on the inscribed circle. Is this triangle necessarily isosceles?
geometryorthocenterincircle