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Part of 2005 May Olympiad

Problems(2)

12 men and 12 women at the dance

Source: XI May Olympiad (Olimpiada de Mayo) 2005 L2 P4

At a dance there are

12

1

12

12

1

12

. Each man is assigned a “secret friend” among the

11

others. They all danced all the pieces. In the first piece each man danced with the woman who has the same number. From then on, each man danced the new piece with the woman who had danced the piece earlier with his secret friend. In the third piece the couples were: https://cdn.artofproblemsolving.com/attachments/c/d/f5ea0931e5751739c1ba556f84ab5736f2d11a.png Find the number of each man's secret friend.

paper constructions, equilateral + rectangle (May Olympiad 2005 L1)

Source:

There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

geometryrectangleEquilateral Trianglepaper