Speedrun

Chef is playing a computer game. There are 𝑁 monsters on the number line.

Chef is at position 0 on the number line. Initially, the ith monster has health equal to 𝐻𝑖​ and is at the position 𝐴𝑖​ on the line.

It is guaranteed that 𝐴𝑖<𝐴𝑖+1​ for every 1≤𝑖<𝑁.

Every second, the following happens:

• First, Chef can choose a single index i (1≤𝑖≤𝑁) such that the 𝑖𝑡ℎ monster is still alive, and fire one bullet at it. This reduces 𝐻𝑖​ by 1.
• Then, all alive monsters whose health is 0 will die.
• Finally, all alive monsters move one position closer to Chef.

If any monster reaches position 0, Chef loses.

If Chef manages to kill all the monsters before any of them reach position 0, he wins.

Before the start of the game, Chef can plant a bomb at any position on the line.

If the bomb is placed at position 𝑋, all monsters whose starting positions lie between 𝑋−𝐾 and 𝑋+𝐾 will die immediately, and the game will start with the remaining monsters.

If Chef places the bomb optimally, can he win the game?