Problem

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Tags: topology, geometry, circumcircle, calculus, integration, invariant, combinatorics proposed



11 people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and 11 cards with numbers 1 to 11 are given to them. Some may have no card and some may have more than 1 card. In each round, one [and only one] can give one of his cards with number i to his adjacent person if after and before the round, the locations of the cards with numbers i1,i,i+1 don’t make an acute-angled triangle. (Card with number 0 means the card with number 11 and card with number 12 means the card with number 1!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.