Can you create a subset of (1, 2, 3, ..., 3k) such that none of its 2k-1 members is twice the value of another?
Either provide such a set or show none exists.
Inspired by: Austrian-Polish Math. Competition.
Split the integers into two sets based on the number of 2's in the prime factorization. Set A will have an even number of 2's and Set B will have an odd number of 2's.
More explicitly, Set A will have all odd numbers, 4x odd numbers, 16x odd numbers, etc. And Set B will have 2x odd numbers, 8x odd numbers, 32x odd numbers, etc.
It should be obvious if z is in one set then 2z is in the other.
Now there will be approximately twice as many odd numbers as 2x odd numbers, twice as many 4x odd numbers as 8x odd numbers, etc.
So the asymptotic behavior is that Set A will have twice as many members as Set B. So the subset sought is Set A.
This isn't rigorous so I can't call it a proof.
Another Idea
