2023 Putnam B4

For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:

(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$;

(b) $f(t_0) = 1/2$;

(c) $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$;

(d) For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$.

Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?

对于非负整数 $n$ 和严格递增的实数序列 $t_0,t_1,\dots,t_n$，令 $f(t)$ 为通过以下属性为 $t \geq t_0$ 定义的相应实值函数：

(a) $f(t)$ 对于 $t \geq t_0$ 是连续的，并且对于除 $t_1,\dots,t_n$ 之外的所有 $t>t_0$ 是两次可微的；

(b) $f(t_0) = 1/2$；

(c) $\lim_{t \to t_k^+} f'(t) = 0$ 对于 $0 \leq k \leq n$；

(d) 对于 $0 \leq k \leq n-1$，当 $t_k < t< t_{k+1}$ 时，我们有 $f''(t) = k+1$，当 $t>t_n$ 时，$f''(t) = n+1$。

考虑 $n$ 和 $t_0,t_1,\dots,t_n$ 的所有选择，使得 $1 \leq k \leq n$ 的 $t_k \geq t_{k-1}+1$ ，当 $f(t_0+T) = 2023$ 时，$T$ 的最小可能值是多少？