To solve the given system of equations:

$$x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,$$

$$x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2},$$

with $x_i \geq 0$ for all $i$ and $x_i$ being nonnegative real numbers, we will demonstrate that the only solution is

$$(x_1, x_2, \ldots, x_n) = (1, 1, \ldots, 1).$$

### Step-by-Step Solution

1. Understanding the System:

We have two equations involving powers and weighted sums of the variables. Notice that the sequence $1, 2, \ldots, n$ is used in both equations, highlighting the hierarchical nature of indices in their contributions to the overall sum.

2. Interpretation of the First Equation:

The left side of the first equation can be interpreted as a sum of powers of the variables. The simplest way to satisfy $x_1 + x_2^2 + \cdots + x_n^n = n$ while respecting nonnegative constraints is by setting each power term to contribute equally if possible.

3. Equitable Setting:

Let's explore the setting $x_i = 1$ for $1 \le i \le n$:

$$x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = 1^1 + 1^2 + 1^3 + \cdots + 1^n = n,$$

which matches the first equation exactly.

4. Checking with the Second Equation:

Substitute $x_i = 1$ into the second equation:

$$x_1 + 2x_2 + 3x_3 + \cdots + nx_n = 1 \cdot 1 + 2 \cdot 1 + 3 \cdot 1 + \cdots + n \cdot 1 = \frac{n(n+1)}{2},$$

the sum of the first $n$ integers, which matches the right-hand side of the second equation.

5. Uniqueness of the Solution:

Since each $x_i$ contributing $1$ satisfies both equations concurrently and any deviation in one of these terms must be counteracted to maintain the balance in both sums, maintaining $x_i = 1$ is crucial. Any attempt to increase or decrease $x_i$ would disrupt equality since the corresponding powers and coefficients magnify the changes in other terms, leading inexorably away from balancing both expressions symmetrically.

Therefore, the only configuration of values for $x_1, x_2, \ldots, x_n$ that simultaneously solves both equations is:

$$\boxed{(1, 1, \ldots, 1)}.$$