RU-ALLRUSSIAN 1979 Problem 13

Find a sequence $\mathbf{a}_{1}$ , $\mathbf{a}_{2}$ , ..., $\mathbf{a}_{25}$ of 0s and 1s such that the following sums are all odd:

$\begin{array}{r l} & {\mathrm{a}_{1}\mathrm{a}_{1} + \mathrm{a}_{2}\mathrm{a}_{2} + \ldots +\mathrm{a}_{25}\mathrm{a}_{25}}\\ & {\mathrm{a}_{1}\mathrm{a}_{2} + \mathrm{a}_{2}\mathrm{a}_{3} + \ldots +\mathrm{a}_{24}\mathrm{a}_{25}}\\ & {\mathrm{a}_{1}\mathrm{a}_{3} + \mathrm{a}_{2}\mathrm{a}_{4} + \ldots +\mathrm{a}_{23}\mathrm{a}_{25}}\\ & {\ldots}\\ & {\mathrm{a}_{1}\mathrm{a}_{24} + \mathrm{a}_{2}\mathrm{a}_{25}} \end{array}$

Show that we can find a similar sequence of n terms for some $\mathrm{n} > 1000$ .