Proved, the site's label. The status-defining source is Hegyvári's paper On consecutive sums in sequences (Acta Math. Hungar. 48 (1986), no. 1--2, 193--200, refereed), which the site credits with the affirmative answer and an explicit bound of the shape $f(a,K)\ll ae^{O(K)}$; the site adds that the author thinks the exponential dependence on $K$ can be improved. The claim is recorded on its claim page (Hegyvári, 1986) and accepted on the refereed publication and the site's credit. Theorem 3 (p. 197) of the paper states $f(a,K)<(a+K/2)e^{K+1}+Ke^{2K+2}$, where $f(a,K)$ is "the largest integer with the following property: There exists an increasing sequence $a=a_1<a_2<\ldots<a_s=f(a,K)$ such that $a_{i+1}-a_i\le K$, $i=1,2,\ldots,s-1$ and all $c$-sums are different". Two equal $c$-sums are two distinct intervals of equal sum, overlapping intervals included, so this is the question's $f(a,K)$ with explicit constants.