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1

Part of 1999 Tuymaada Olympiad

Problems(2)

Tuymaada Olympiad 2004, Question 5

Source: Tuymaada Olympiad 2004, Question 5

7/31/2012
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
Proposed by A. Khrabrov, incorrect translation from Hungarian
geometrygeometric transformationcombinatorics unsolvedcombinatorics
Tuymaada 1999, Q5

Source: Tuymaada 1999, Q5

7/31/2012
In the triangle ABCABC we have ∠ABC=100∘\angle ABC=100^\circ, ∠ACB=65∘\angle ACB=65^\circ, M∈ABM\in AB, N∈ACN\in AC, and ∠MCB=55∘\angle MCB=55^\circ, ∠NBC=80∘\angle NBC=80^\circ. Find ∠NMC\angle NMC.
St.Petersburg folklore
geometrygeometric transformationreflectiontrigonometrygeometry unsolved