MathDB

2

Part of 2008 Kurschak Competition

Problems(1)

Differences equal to a power of 2

Source: Kürschak 2008, problem 2

7/6/2014
Let n≥1n\ge 1 and a1<a2<⋯<ana_1<a_2<\dots<a_n be integers. Let SS be the set of pairs 1≤i<j≤n1\le i<j\le n for which aj−aia_j-a_i is a power of 22, and TT be the set of pairs 1≤i<j≤n1\le i<j\le n with j−ij-i a power of 22. (Here, the powers of 22 are 1,2,4,…1,2,4,\dots.) Prove that ∣S∣≤∣T∣|S|\le |T|.
combinatorics unsolvedcombinatorics