Question 41
The elements of the set $\{1, 2, \dots, 21\}$ are written randomly on a circle, in the order $a_1, a_2, \dots, a_{21}$ (see figure). Consider the sums
$$
\begin{align*}
S_1 &= a_1 + a_2 + a_3 + a_4 + a_5, \\
S_2 &= a_2 + a_3 + a_4 + a_5 + a_6, \\
\vdots \\
S_{17} &= a_{17} + a_{18} + a_{19} + a_{20} + a_{21}, \\
S_{18} &= a_{18} + a_{19} + a_{20} + a_{21} + a_1.
\end{align*}
$$
Show that at least two of the 18 sums leave different remainders when divided by 5.
$$
\begin{align*}
S_1 &= a_1 + a_2 + a_3 + a_4 + a_5, \\
S_2 &= a_2 + a_3 + a_4 + a_5 + a_6, \\
\vdots \\
S_{17} &= a_{17} + a_{18} + a_{19} + a_{20} + a_{21}, \\
S_{18} &= a_{18} + a_{19} + a_{20} + a_{21} + a_1.
\end{align*}
$$
Show that at least two of the 18 sums leave different remainders when divided by 5.