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3

Part of 2022 Assara - South Russian Girl's MO

Problems(2)

closed polyline by veritces of regular n-gon

Source: 2022 South Russian Girls MO Assara Juniors p3

2/28/2024
For what nn can the vertices of a regular nn-gon be connected in a nn-link closed polyline so that such a polyline does not have three equal links?
geometrycombinatorial geometry
concurrent, 4 right angles related

Source: 2022 South Russian Girls MO - Assara Seniors p3

3/1/2024
In a convex quadrilateral ABCDABCD, angles BB and DD are right angles. On on sides ABAB, BCBC, CDCD, DADA points KK, LL, MM, NN are taken respectively so that KN⊥ACKN \perp AC and LM⊥ACLM \perp AC. Prove that KMKM, LNLN and ACAC intersect at one point.
geometryconcurrencyconcurrent