CF1428E Carrots for Rabbits

There are some rabbits in Singapore Zoo. To feed them, Zookeeper bought $ n $ carrots with lengths $ a_1, a_2, a_3, \ldots, a_n $ . However, rabbits are very fertile and multiply very quickly. Zookeeper now has $ k $ rabbits and does not have enough carrots to feed all of them. To solve this problem, Zookeeper decided to cut the carrots into $ k $ pieces. For some reason, all resulting carrot lengths must be positive integers. Big carrots are very difficult for rabbits to handle and eat, so the time needed to eat a carrot of size $ x $ is $ x^2 $ . Help Zookeeper split his carrots while minimizing the sum of time taken for rabbits to eat the carrots.

The first line contains two integers $ n $ and $ k $ $ (1 \leq n \leq k \leq 10^5) $ : the initial number of carrots and the number of rabbits. The next line contains $ n $ integers $ a_1, a_2, \ldots, a_n $ $ (1 \leq a_i \leq 10^6) $ : lengths of carrots. It is guaranteed that the sum of $ a_i $ is at least $ k $ .

Output one integer: the minimum sum of time taken for rabbits to eat carrots.

For the first test, the optimal sizes of carrots are $ \{1,1,1,2,2,2\} $ . The time taken is $ 1^2+1^2+1^2+2^2+2^2+2^2=15 $ For the second test, the optimal sizes of carrots are $ \{4,5,5,5\} $ . The time taken is $ 4^2+5^2+5^2+5^2=91 $ .