The problem is the IOI 2007 competition's Sails problem. It has also appeared in Pranav A. Sriram Olympiad Combinatorics notes (Chapter 1 Exercise 14). I copied the problem description from Pranav. You are given $n$ integers $a_1, a_2, \dots, a_n$ and another set of n integers $b_1, b_2, \dots, b_n$ such that for each $i, b_i \leq a_i$. For each $i = 1, 2, \dots, n$, you must choose a set of $b_i$ distinct integers from the set $\{1, 2, \dots, a_i\}$. In total, $(b_1 + b_2 + \dots + b_n)$ integers are selected, but not all of these are distinct. Suppose $k$ distinct integers have been selected, with multiplicities $c_1, c_2, c_3, \dots, c_k$. Your score is defined as $\sum_{i=1}^{k} c_i(c_i-1)$ . Give an efficient algorithm to select numbers in order to minimize your score.
The greedy algorithm is that you sort by $a_i$ into increasing order. After this you go through one by one and for a given $i$ you choose the set of numbers from $\{1, \dots, a_i\}$ so that the increase in the objective function is minimal.
Now the problem I have is the following. I understand the greedy algorithm. I also understand why it is locally optimal (trivial), but I have been unable to prove for the past $3$ days that the greedy algorithm is optimal. I have searched the internet for a proof but did not find one. I have also given this problem to everybody I know, but no solution has been shown to me so far. I hope somebody here can prove that the greedy algorithm is optimal.
I have tried the following things among many others.
Going from an easier problem to this problem, like starting from this problem.
$$
\begin{equation*}
\begin{array}{ll@{}ll}
\text{minimize} & \displaystyle\sum\limits_{i=1}^{k} c^2_i &\\
\text{subject to}& \displaystyle\sum\limits_{i=1}^{k} c_i = \displaystyle\sum\limits_{j=1}^{N} b_j \\
&c_i \in \mathbb{R_+} \\
&b_j \geq 0
\end{array}
\end{equation*}
$$
Here you can show that the optimal solution is when $c_i=c_j$. Then if you exchange the condition of $c_i \in \mathbb{R_+}$ to $c_i \in \mathbb{N} = \{1,2,\dots\}$ then your optimal solution need to have the following property $\forall i,j$ $|c_i-c_j| \leq 1$. After this, I was able to show that for optimal and greedy solutions it is true that
If $\exists i,j$ such that $c_i>c_j+1$ and $\exists l$ such that $\max(i,j) \leq a_l$ and $i \in S_l, j \notin S_l$ then we can exchange a sail on the $l$th mast between levels of $i$ and $j$ such that the objective value strictly decreased.
But one can show with an example that this does not imply optimality. I have also tried induction and that also failed. Other exchange-like arguments failed also.