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witzar

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3. 3 x 4 Connect-The-Dots Game

Good idea but does that guarantee a win?

5. 3 x 4 Connect-The-Dots Game

If I understand correctly, the following pattern is not considered "crossing": __Ω_Ω_Ω_Ω_Ω__ The loops are closed, but it is "touching" not "crossing".

7. 3 x 4 Connect-The-Dots Game

So you can win instantly in first move connecting two dots and cutting off each of the remaining dots with a loop around it?

Me too.
9. No First-Hand-Neighbors!!

Now it's a different story. BTW "best answer" sequence starts with 3 not with 1.

11. No First-Hand-Neighbors!!

Clearly empty sequence (length 0) wins the contest..

14. Isosceles triangle inscribed in red-blue circle

Equilateral triangle is an isosceles triangle. I guess my last answer makes no sense. What I meant to say is this: arms can be longer, shorter or equal to the base. As long as arms are of equal length, a triangle is isosceles.
15. Isosceles triangle inscribed in red-blue-green circle

Each point of the circle is painted either red, blue or green. Prove, that there exists isosceles triangle inscribed in the circle with all its vertices of the same color.
16. Isosceles triangle inscribed in red-blue circle

Equilateral triangle is an isosceles triangle.
17. Isosceles triangle inscribed in red-blue circle

Each point of the circle is painted either red or blue. Prove, that there exists isosceles triangle inscribed in the circle with all its vertices of the same color.
18. RGB plane

This is not true. Pick a random real number from interval [0,100]. What is the probability that picked number is not 7? Clearly 100%. Does it follow, that picking 7 is impossible or that number 7 does not exists? Clearly not. Divide unit circle into 6 equal arcs. Paint those arc blue and green alternately (so that six end points of the arcs are painted alternately too). You have an "impossible" blue-green circle. In fact you can take just one arc (1/6 of circle) and paint it any way you want. Then for each point P of the arc take a regular hexagon H inscribed in the unit circle such that P is a vertex of H, and paint the other 5 vertices of H alternately (regarding color of P). Again you have an impossible blue-green circle.
19. Rectangle on red-blue plane

Each point of the plane is painted either red or blue. Prove, that there exists a rectangle with all vertices of the same color.

21. RGB plane

This is not true.

23. RGB plane

Each point of the plane is painted either red, green or blue. Prove, that there exists segment of length 1 with both ends of the same color.
24. Black and white plane

Each point of the plane is painted either red or blue. Prove, that there exists equilateral triangle with all vertices of the same color.
25. Nine points

Nine points are given on the plane, no three of which are collinear. Each pair of points is connected with segment, that is painted either white or black. Prove that there exists either quadrilateral with white sides and diagonals or triangle with black sides.
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