1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer.
2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.
### Part 1:
1. Given two nonnegative real numbers \( x \) and \( y \) such that \( x + y = a \), we need to find the minimum value of \( x^m + y^m \) where \( m \) is a given positive integer.
2. By the Power Mean Inequality, for nonnegative real numbers \( x \) and \( y \) and for \( m \geq 1 \):
   \[
   \sqrt[m]{\frac{x^m + y^m}{2}} \geq \frac{x + y}{2}
   \]
3. Since \( x + y = a \), we have:
   \[
   \sqrt[m]{\frac{x^m + y^m}{2}} \geq \frac{a}{2}
   \]
4. Raising both sides to the power of \( m \):
   \[
   \frac{x^m + y^m}{2} \geq \left(\frac{a}{2}\right)^m
   \]
5. Multiplying both sides by 2:
   \[
   x^m + y^m \geq 2 \left(\frac{a}{2}\right)^m = 2 \cdot \frac{a^m}{2^m} = \frac{a^m}{2^{m-1}}
   \]
6. Equality holds if and only if \( x = y \). Given \( x + y = a \), this occurs when \( x = y = \frac{a}{2} \).

Therefore, the minimum value of \( x^m + y^m \) is:
\[
\boxed{\frac{a^m}{2^{m-1}}}
\]

### Part 2:
1. Let \( x_1, x_2, \ldots, x_n \) be nonnegative real numbers with a constant sum \( k \). We need to prove that the minimum value of \( x_1^m + x_2^m + \cdots + x_n^m \) occurs when \( x_1 = x_2 = \cdots = x_n \).
2. By the Power Mean Inequality, for nonnegative real numbers \( x_1, x_2, \ldots, x_n \) and for \( m \geq 1 \):
   \[
   \sqrt[m]{\frac{x_1^m + x_2^m + \cdots + x_n^m}{n}} \geq \frac{x_1 + x_2 + \cdots + x_n}{n}
   \]
3. Since \( x_1 + x_2 + \cdots + x_n = k \), we have:
   \[
   \sqrt[m]{\frac{x_1^m + x_2^m + \cdots + x_n^m}{n}} \geq \frac{k}{n}
   \]
4. Raising both sides to the power of \( m \):
   \[
   \frac{x_1^m + x_2^m + \cdots + x_n^m}{n} \geq \left(\frac{k}{n}\right)^m
   \]
5. Multiplying both sides by \( n \):
   \[
   x_1^m + x_2^m + \cdots + x_n^m \geq n \left(\frac{k}{n}\right)^m = n \cdot \frac{k^m}{n^m} = \frac{k^m}{n^{m-1}}
   \]
6. Equality holds if and only if \( x_1 = x_2 = \cdots = x_n \). Given \( x_1 + x_2 + \cdots + x_n = k \), this occurs when \( x_1 = x_2 = \cdots = x_n = \frac{k}{n} \).

Therefore, the minimum value of \( x_1^m + x_2^m + \cdots + x_n^m \) is:
\[
\boxed{\frac{k^m}{n^{m-1}}}
\]