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Part of 2011 Tuymaada Olympiad
Problems(3)
Red, blue, and green children raising their hands
Source: XVIII Tuymaada Mathematical Olympiad (2011)
7/29/2011
Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, children raised their hands. Prove that some child that raised her hand had two green neighbors.
combinatorics unsolvedcombinatorics
Reals a,b such that a+b and ab are different colours
Source: XVIII Tuymaada Mathematical Olympiad (2011), Junior Level
7/31/2011
Each real number greater than is coloured red or blue with both colours being used. Prove that there exist real numbers and such that the numbers and are of different colours.
combinatorics unsolvedcombinatorics
Two-coloring (1,infty) implies a+1/b & b+1/a are different
Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level
7/29/2011
Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers and such that the numbers and are different colors.
combinatorics unsolvedcombinatorics