Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable.

[I]Proposed by Alexandar Ivanov, Bulgaria.[/i]
We need to find the least positive integer \( D \) such that for any \( n \)-diverse collection of \( D \) coins, there are at least \( k \) numbers that are \( n \)-reachable. 

1. **Claim**: The smallest such value of \( D \) is \( \boxed{n+k-1} \).

2. **Verification**: Consider a configuration with \( n \) coins worth \( 0 \) and \( k-2 \) coins worth \( 0 \). This configuration produces only \( k-1 \) different values (it produces all the integers from \( 0 \) to \( k-2 \)). Thus, we must have \( D \le n+k-1 \).

3. **Proof**: We now show that \( D = n+k-1 \) is indeed \( n \)-diverse. Suppose the coins have values
   \[
   x_1 \le x_2 \le \cdots \le x_{n+k-1}.
   \]
   By taking the complement, it suffices to show that there are at least \( n \) different values that can be made from collections of \( k-1 \) coins.

4. **Consider the expression**:
   \[
   \mathcal{C}_a: \quad (x_1 + \cdots + x_{a-1}) + x_a + (x_{n+a+1} + \cdots + x_{n+k-1}) 
   \]
   We need to show that there are at least \( k \) distinct sums. 

5. **Inequality chain**:
   \[
   (x_1 + \cdots + x_{a-1}) + x_a + (x_{n+a+1} + \cdots + x_{n+k-1}) 
   \le (x_1 + \cdots + x_{a-1}) + x_{a+1} + (x_{n+a+1} + \cdots + x_{n+k-1}) 
   \le \cdots 
   \le (x_1 + \cdots + x_{a-1}) + x_{n+a} + (x_{n+a+1} + \cdots + x_{n+k-1}).
   \]
   We see that one of the inequality signs in
   \[
   x_a \le x_{a+1} \le \cdots \le x_{n+a}
   \]
   is strict, so one of the inequality signs in \( \mathcal{C}_a \) is strict.

6. **Combining inequalities**:
   \[
   \mathcal{D}: \quad \mathcal{C}_{k-1} \le \mathcal{C}_{k-2} \le \cdots \le \mathcal{C}_1.
   \]
   Since there is a strict inequality in each individual \( \mathcal{C}_a \), the expression \( \mathcal{D} \) has at least \( k-1 \) strict inequalities. Thus, it has at least \( k \) distinct values.

7. **Conclusion**: We have at least \( k \) values made from groups of \( k-1 \) coins, so \( D = n+k-1 \) is the least positive integer such that for any \( n \)-diverse collection of \( D \) coins, there are at least \( k \) numbers that are \( n \)-reachable.