Problem 1743

To determine whether there exist integers $a_1, a_2, \ldots$ that are distinct and satisfy the given conditions, we analyze the problem as follows:

1. For all $k \in \mathbb{N}_+$, $a_{k^2} > 0$ and $a_{k^2 + k} < 0$.
2. For all $n \in \mathbb{N}_+$, $|a_{n+1} - a_n| \leq 2023 \sqrt{n}$.

Assume such a sequence $\{a_n\}$ exists. Let $f(k)$ denote an integer in the interval $[k^2, k^2 + k - 1]$ such that $a_{f(k)} > 0$ and $a_{f(k) + 1} < 0$. Similarly, let $g(k)$ denote an integer in the interval $[k^2 + k, (k+1)^2 - 1]$ such that $a_{g(k)} < 0$ and $a_{g(k) + 1} > 0$.

By the triangle inequality and the given condition $|a_{n+1} - a_n| \leq 2023 \sqrt{n}$, we can bound the values of $a_{f(k) \pm C}$ and $a_{g(k) \pm C}$ for any integer $C$ as follows:
$$|a_{f(k) \pm C}| \leq 2023 (C + 1) (k + 1),$$
$$|a_{g(k) \pm C}| \leq 2023 (C + 1) (k + 1).$$

Consider a large integer $N$ and the number of terms $t$ such that $|a_t| \leq N^2$. On one hand, this number must be at most $2N^2 + 1$.

On the other hand, if $j$ is finite and very small compared to $N$, for each $t \in \left[ \frac{jN^2}{2023}, \frac{(j+1)N^2}{2023} \right]$, we need:
$$|a_t| = |a_{f(\lfloor \sqrt{t} \rfloor) \pm C}| \leq 2023 (C + 1) (\sqrt{t} + 1),$$
or
$$|a_{g(\lfloor \sqrt{t} \rfloor) \pm C}| \leq 2023 (C + 1) (\sqrt{t} + 1).$$

This implies that $C < \frac{N}{4046 \sqrt{j}}$ works for sure. There are $\frac{N (\sqrt{j+1} - \sqrt{j})}{2023} < \frac{N}{4100 \sqrt{j}}$ intervals, so we can pick $\frac{N^2}{10^9 j}$ terms that are guaranteed to be at most $N^2$.

By choosing $j = \exp(2 \cdot 10^9)$ and $N > \exp(j)$, we get that $2N^2 + 1 < 3N^2 < N^2 (H_j - 1) 10^{-9}$. As the former is the number of possible terms that have absolute value at most $N^2$ and the latter is the number of terms that must have absolute value at most $N^2$, we reach a contradiction.

Therefore, no such sequence $\{a_n\}$ exists.

The answer is: \boxed{\text{No}}.