Question 82
Let $N$ be a positive integer. On a board, we initially have two numbers: a red $0$ and a blue $0$. We define the following procedure:
At each step, a natural number $k$ is chosen (not necessarily distinct from previous choices). Let $x$ be the blue number and $y$ the red number. Replace them with:
$$
x \rightarrow x + k + 1, \quad y \rightarrow y + k^2 + 2,
$$
and color the new values blue and red, respectively. This process continues until the blue number is at least $N$.
Determine the minimum possible value of the red number at the end of the process.
At each step, a natural number $k$ is chosen (not necessarily distinct from previous choices). Let $x$ be the blue number and $y$ the red number. Replace them with:
$$
x \rightarrow x + k + 1, \quad y \rightarrow y + k^2 + 2,
$$
and color the new values blue and red, respectively. This process continues until the blue number is at least $N$.
Determine the minimum possible value of the red number at the end of the process.