witzar

No other conditions, although if you want something really interesting try having a continuous function...

I agree with Rainman. The proper version comes from "What is the Name of this Book" by Raymond Smullyan and is called 'The Drinking Principle'. Smullyan also states a dual paradox: There is at least one person such that if anybody drinks, then he does, and more dramatic version of TDP: There is a woman on earth such that if she becomes sterile, the whole human race will die out.

Well done!

(1b) solved. Well done.

i believe that this is the sum of a different sequence The sequence is the same, but there is an error in reasoning. True. 3 x False. True.

i like this approach. What is your reasoning?

(1a) solved

(1a) Paint the plane using 3 colors in such way that no 3-colored straight line exists. (1b) Prove that if plane is painted with 4 colors, then a 3-colored straight line exists. (2a) Paint the plane using as many colors as you can in such way that no 4-colored straight line exists. (2b) Let m be the number of colors you used in (2a). Prove that if plane is painted with (m+1) colors, then a 4-colored straight line exists. (3) What about 5-colored straight lines? (4) What about n-colored straight lines? Note: I don't have solutions to all of the problems above. Partial solutions are welcome.