A cyclic sum S N ( x ) = ∑ x i / ( x i + 1 + x i + 2 ) {S_N}({\mathbf {x}}) = \sum {{x_i}/({x_{i + 1}} + {x_{i + 2}})} is formed with N components of a vector x, where in the sum x N + 1 = x 1 {x_{N + 1}} = {x_1} , x N + 2 = x 2 {x_{N + 2}} = {x_2} , and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which S N ( x ) > N / 2 {S_N}({\mathbf {x}}) > N/2 if N ≥ 14 N \geq 14 and even, and if N ≥ 24 N \geq 24 . It has been proved that the inequality S N ( x ) ≥ N / 2 {S_N}({\mathbf {x}}) \geq N/2 holds for N ≤ 13 N \leq 13 . Although it has been conjectured repeatedly that the inequality also holds for odd N between 15 and 23, this has apparently never been proved. Here we will confirm that the inequality indeed holds for all odd N ≤ 23 N \leq 23 . This settles the question for all N.