Just started solving and came across this example:-
Consider any four integers $a,b,c,d $ where not all are equal. It is allowed to change this sequence to $a-b,b-c,c-d,d-a$. Prove that at least one of the numbers from sequence can be arbitrarily large by repetitions of the legal move.
I haven't yet seen the official solution but have one of my own please verify:
First observe that the first number in the sequence is of form: $k(a-d)+ K(c-b)$ or $h(a-b)+ H(c-d)$ or of some other form.(where $k,K,h,H$ increase as we apply move again and again) Similar pattern is there for all the terms. Now considering some number to be largest we can find a term that remains positive and increases over time as we have terms like $h(a-b)+H(c-d)$ and opposite of that i.e. $g(b-a)+G(d-c)$