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Part of 2005 Oral Moscow Geometry Olympiad

Problems(2)

KN + LM >= AC wanted, AK = BL, CN = BM

Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p3

10/16/2020
In triangle ABCABC, points K,PK ,P are chosen on the side ABAB so that AK=BLAK = BL, and points M,NM,N are chosen on the side BCBC so that CN=BMCN = BM. Prove that KN+LM≥ACKN + LM \ge AC.
(I. Bogdanov)
geometrygeometric inequalityequal segments
cover plane with specific non convex pentagons

Source: 2005 Oral Moscow Geometry Olympiad grades 10-11 p3

10/21/2020
ABCBEABCBE is a regular pentagon. Point B′B' is symmetric to point BB wrt line ACAC (see figure). Is it possible to pave the plane with pentagons equal to AB′CBEAB'CBE?
(S. Markelov) https://cdn.artofproblemsolving.com/attachments/9/2/cbb5756517e85e56c4a931e761a6b4da8fe547.png
combinatorial geometrygeometrypentagon