Let $\{a_n\}$ be a sequence of integers satisfying the following conditions. 
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[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. 
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Determine the $2021^{2022}$th term of the sequence $\{a_n\}$. 
1. **Define the sequence and initial conditions:**
   Let $\{a_n\}$ be a sequence of integers with the following properties:
   - $a_1 = 2021^{2021}$
   - $0 \le a_k < k$ for all integers $k \ge 2$
   - The alternating sum $a_1 - a_2 + a_3 - a_4 + \cdots + (-1)^{k+1}a_k$ is a multiple of $k$ for all positive integers $k$.

2. **Introduce notation and initial observations:**
   Denote $N = 2021^{2021}$. Define the partial sums $S_k = \sum_{i=1}^{k} (-1)^{i+1} a_i$. By the problem's conditions, $S_k$ is a multiple of $k$, i.e., $S_k = kb_k$ for some integer $b_k$.

3. **Analyze the sequence properties:**
   If there exists an integer $1 \leq k \leq 2021N$ such that $a_k = b_k = 0$, then it can be shown that $a_i = b_i = 0$ for all $k \leq i \leq 2021N$. This would imply that the sequence stabilizes at zero.

4. **Assume all $b_k$ are positive and derive a contradiction:**
   Assume all $b_k$ in the range are positive. We will examine the relationship between $b_k$ and $b_{k+1}$.
   - If $S_k = kb_k$ is divisible by $k+1$, then $b_{k+1} = \frac{k}{k+1} b_k < b_k$.
   - Otherwise, there exists an integer $t$ such that $(k+1)t < kb_k < (k+1)(t+1)$. This implies:
     - If $k$ is odd, $b_{k+1} = t < \frac{k}{k+1} b_k < b_k$.
     - If $k$ is even, $b_{k+1} = t + 1 < \frac{k}{k+1} b_k + 1 < b_k + 1$.

5. **Conclude that $b_{k+1} \leq b_k$:**
   Hence, in both cases, $b_{k+1} \leq b_k$. This implies that $b_{k+2} \leq b_k - 1$.

6. **Calculate the bound for $b_{2021N}$:**
   Since $b_1 = N$, we have:
   \[
   b_{2021N} \leq b_1 - \frac{2021N - 1}{2} = N - \frac{2021N - 1}{2} < 0
   \]
   This is a contradiction because $b_{2021N}$ must be a non-negative integer.

7. **Conclude that $a_{2021N} = 0$:**
   Therefore, the assumption that all $b_k$ are positive is false, and we must have $a_{2021N} = b_{2021N} = 0$.

The final answer is $\boxed{0}$.