Problem 2228

To solve the problem, we need to identify all sequences $(x_1, x_2, \ldots, x_{2011})$ of positive integers such that for every positive integer $n$, there exists an integer $a$ satisfying:

$$\sum_{j=1}^{2011} j \, x^n_j = a^{n+1} + 1$$

### Step-by-Step Solution:

1. **Consider the Case $n = 1$:**

Starting with $n = 1$, the condition becomes:

$$\sum_{j=1}^{2011} j \, x_j = a^2 + 1$$

2. Explore the Structure of the Sequence:

To satisfy the condition for every $n$, observe the symmetry of the powers and sums. If we choose $x_1 = 1$, and the rest of the sequence as constant $x_j = k$ for $j = 2, \ldots, 2011$, we can potentially simplify the problem. Define the sequence as:

$$(x_1, x_2, \ldots, x_{2011}) = (1, k, k, \ldots, k)$$

3. Analyze the Expression:

Substituting $x_1 = 1$ and $x_2 = x_3 = \cdots = x_{2011} = k$, the sum becomes:

$$\sum_{j=1}^{2011} j \, x_j^n = 1^n + \sum_{j=2}^{2011} j \, k^n$$

$$= 1 + k^n \sum_{j=2}^{2011} j = 1 + k^n \left(2 + 3 + \cdots + 2011 \right)$$

4. Sum Calculation:

Evaluate the sum $2 + 3 + \cdots + 2011$. Using the formula for the sum of consecutive integers:

$$S = \frac{2011 \times 2012}{2} - 1 = \frac{2011 \times 2012}{2} - 1$$

Solving gives:

$$S = 2023065$$

5. **Define $k$ and Verify:**

If we let $k = 2023065$, the sum becomes:

$$\sum_{j=1}^{2011} j \, x_j^n = 1 + 2023065^n \cdot 2023065 = 1 + (2023065)^{n+1}$$

This ensures for any $n$, there exists an integer $a = 2023065$ such that:

$$a^{n+1} + 1$$

6. Conclusion:

Hence, the sequence that satisfies the given condition for every positive integer $n$ is:

$$(x_1, x_2, \ldots, x_{2011}) = (1, 2023065, \ldots, 2023065)$$

Thus, the sequence is:

$$\boxed{(x_1, x_2, \ldots, x_{2011}) = (1, 2023065, \ldots, 2023065)}$$