MathDB

3

Part of 2013 Kurschak Competition

Problems(1)

Construction of a sequence, with a lower bound on |a_i-a_j|

Source: Kürschak 2013, problem 3

7/6/2014
Is it true that for integer n≥2n\ge 2, and given any non-negative reals ℓij\ell_{ij}, 1≤i<j≤n1\le i<j\le n, we can find a sequence 0≤a1,a2,…,an0\le a_1,a_2,\ldots,a_n such that for all 1≤i<j≤n1\le i<j\le n to have ∣ai−aj∣≥ℓij|a_i-a_j|\ge \ell_{ij}, yet still ∑i=1nai≤∑1≤i<j≤nℓij\sum_{i=1}^n a_i\le \sum_{1\le i<j\le n}\ell_{ij}?
combinatorics unsolvedcombinatorics