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Part of 2013 Tournament of Towns

Problems(4)

2013 ToT Spring Junior O p1 6 points split into 2 triples and create triangle

Source:

There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?

combinatoricscombinatorial geometrypoints

2013 ToT Spring Junior A p1 sum of any 2 is some power of 2

Source:

Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example,

2, 4, 8,...

). What is the maximal possible number of different integers on the blackboard?

number theorypower of 2Sum

2013 ToT Fall Junior O p1 100 participants in a wrestling tournament

Source:

In a wrestling tournament, there are

100

participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?

2013 ToT Fall Junior A p1 0 red, 100 yellow and 100 green sticks

Source:

100

100

100

green sticks. One can construct a triangle using any three sticks all of different colours (one red, one yellow and one green). Prove that there is a colour such that one can construct a triangle using any three sticks of this colour.

combinatoricsColoringcombinatorial geometry