[freemind]

A non-empty set of positive integers is said to be good if there is a coloring with colors of all positive integers so that no number in is the sum of two different positive integers (not necessarily in ) of the same color. Find the largest value can take so that the set is good, for any positive integer .

P.S.

I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...

[bilarev]

http://www.artofproblemsolving.com/viewtopic.php?s ... 2&t=149160

[SpongeBob]

This really isn't some very hard problem, I'm surprised that nobody has posted some solution already...
Answer is .
Look at the set for some even . This set contains elements, so two of them have the same color. In other hand, if with then , so must be less then because there is no good set when is even..
When we color the numbers like this: numbers have color , number is in color , have color , ..., have color , and numbers from and grater have color . You see that if you pick two different numbers with same color, their sum is either smaller then or bigger then , and we're done

Bye