\section*{5. For ALL APPLICANTS.}
Miriam and Adam agree to relieve the boredom of the school holidays by eating sweets, but their mother insists they limit their consumption by obeying the following rules.

\begin{itemize}
  \item Miriam eats as many sweets on any day as there have been sunny days during the holiday so far, including the day in question.
  \item Adam eats sweets only on rainy days. If day $k$ of the holiday is rainy, then he eats $k$ sweets on that day.
\end{itemize}

For example, if the holiday is eight days long, and begins Rainy, Sunny, Sunny, ... then the tally of sweet consumption might look like this:

\begin{center}
\begin{tabular}{ r | c c c c c c c c c }
Day & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & Total \\
\hline
Weather & R & S & S & R & R & S & S & R &  \\
Miriam & 0 & 1 & 2 & 2 & 2 & 3 & 4 & 4 & 18 \\
Adam & 1 & 0 & 0 & 4 & 5 & 0 & 0 & 8 & 18 \\
\end{tabular}
\end{center}

In this case, Miriam and Adam eat the same number of sweets in total.\\
(i) If the holiday has 30 days, 15 of which are sunny and 15 rainy, what arrangement of sunny and rainy days would lead Miriam to eat the greatest number of sweets in total, and what arrangement would lead to the least number? Give the number of sweets that Miriam eats in each case.\\
(ii) Show that, in the two cases mentioned in part (i), Adam eats the same number of sweets as Miriam.\\
(iii) Suppose, in a sequence of sunny and rainy days, we arrange to swap a rainy day with a sunny day that immediately follows it. How does the total number of sweets eaten by Miriam change when we make the swap? What about the total number of sweets eaten by Adam?\\
(iv) If the holiday has 15 sunny days and 15 rainy days, must Miriam and Adam eat the same number of sweets in total? Explain your answer.



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