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Part of 2020 Novosibirsk Oral Olympiad in Geometry

Problems(3)

areas by lines in regular 12-gon (2020 Novosibirsk Oral Geo Oly 7.1)

Source:

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png

geometrydodecagonarea

3 squares inscribed in a triangle (2020 Novosibirsk Oral Geo Oly 8.1)

Source:

Three squares of area

4, 9

36

are inscribed in the triangle as shown in the figure. Find the area of the big triangle https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png

geometrySquares

tangent semicircles inside a rectangle (2020 Novosibirsk Oral Geo Oly 9.1)

Source:

Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled? https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png

geometryrectangletangent circlessemicircles