Subcontests
(3)Bound for number of terms at most m
Let a1,a2,… be a strictly increasing sequence of positive integers, such that for any positive integer n, an is not representable in the for ∑i=1n−1ciai for ci∈{0,1}. For every positive integer m, let f(m) denote the number of ai that are at most m. Show that for any positive integers m,k, we have that f(m)≤ak+k+1m. Combo NT
Let n be a positive integer and let p,q>n be odd primes. Prove that the positive integers 1,2,…,n can be colored in 2 colors, such that for any x=y of the same color, xy−1 is not divisible by p and q.