Problem

Source: Iran MO 3rd round 2023 , Day 1 P3

Tags: combinatorics



For each $k$ , find the least $n$ in terms of $k$ st the following holds: There exists $n$ real numbers $a_1 , a_2 ,\cdot \cdot \cdot , a_n$ st for each $i$ : $$0 < a_{i+1} - a_{i} < a_i - a_{i-1}$$And , there exists $k$ pairs $(i,j)$ st $a_i - a_j = 1$.