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6

Part of 2016 Oral Moscow Geometry Olympiad

Problems(2)

bending a square paper of side 2016, less than 10 times, build segment 1?

Source: 2016 Oral Moscow Geometry Olympiad grades 8-9 p6

7/30/2019
Given a square sheet of paper with a side of 20162016. Is it possible to bend its not more than ten times, construct a segment of length 11?
paperSegmentgeometryfoldingsquare
ratio of circumradii, symmetric points wrt sides and midpoints related

Source: 2016 Oral Moscow Geometry Olympiad grades 10-11 p6

7/29/2019
Given an acute triangle ABCABC. Let A′A' be a point symmetric to AA with respect to BC,OABC, O_A is the center of the circle passing through AA and the midpoints of the segments A′BA'B and A′C.OBA'C. O_B and OCO_C points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles ABCABC and OAOBOCO_AO_BO_C.
ratiogeometrycircumradiuscircumcirclesymmetry