KöMaL 2020 B5115

Ali has $n$ coins in his purse, and Baba has $n-1$ purses, initially all empty. Baba is playing the following game: he divides the coins (all in the same purse at start) into two purses, with $a_1$ coins in one of them and $b_1$ coins in the other ($a_1,b_1>0$), and then he writes the product $a_1b_1$ on a blackboard. Then he continues in the same way: in the $k$th move ($k=2,3,\ldots$) he selects a purse containing at least two coins, divides them between two empty purses, with $a_k$ coins in one of them and $b_k$ in the other ($a_k,b_k>0$), and writes the product $a_kb_k$ on the board. The game terminates when there is 1 coin in each purse. Then Ali gives as many coins to Baba as the sum of all the products $a_k b_k$ on the board. What is the maximum number of coins that Baba may get?