For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq  0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class.

[i]Proposed by Romania[/i]
To solve this problem, we need to determine the smallest integer \( h(r) \geq 1 \) such that for any partition of the set \(\{1, 2, \cdots, h(r)\}\) into \( r \) classes, there exist integers \( a \geq 0 \) and \( 1 \leq x \leq y \) such that \( a + x, a + y, a + x + y \) all belong to the same class.

1. **Reformulate the Condition:**
   Given \( u \leq v < w \) are three such integers in the same class, we can set:
   \[
   x = w - v, \quad y = w - u, \quad a = u + v - w
   \]
   This reformulates the condition to finding the largest \( g(r) \) (where \( g(r) = h(r) - 1 \)) such that there is a partition of the set \(\{1, 2, \cdots, g(r)\}\) into \( r \) classes where for all \( u \leq v < w \) in the same class, \( u + v < w \).

2. **Class Constraints:**
   If \( u \) belongs to a class, no element \( x \) with \( u < x \leq 2u \) can belong to the same class. This is because if \( u \leq v < w \) and \( u + v \geq w \), then \( u, v, w \) cannot all be in the same class.

3. **Bounding \( g(r) \):**
   Consider the elements \( g, g-1, \ldots, \left\lfloor \frac{g+1}{2} \right\rfloor \). These elements must all be in different classes because if any two of them were in the same class, their sum would be greater than or equal to the third element, violating the condition \( u + v < w \).

4. **Calculate \( g(r) \):**
   The number of elements from \( g \) to \(\left\lfloor \frac{g+1}{2} \right\rfloor \) is:
   \[
   g - \left\lfloor \frac{g+1}{2} \right\rfloor + 1
   \]
   For these elements to be in different classes, we need:
   \[
   r \geq g - \left\lfloor \frac{g+1}{2} \right\rfloor + 1
   \]
   Simplifying, we get:
   \[
   r \geq \left\lceil \frac{g}{2} \right\rceil + 1
   \]
   Therefore:
   \[
   g \leq 2r - 1
   \]

5. **Sharpness of the Bound:**
   To show that this bound is sharp, consider the partition:
   \[
   S_i = \{i, i + r\} \quad \text{for} \quad i = 1, \ldots, r-1 \quad \text{and} \quad S_r = \{r\}
   \]
   This partition ensures that no three elements \( u \leq v < w \) in the same class satisfy \( u + v \geq w \).

6. **Conclusion:**
   Thus, the smallest integer \( h(r) \) such that for any partition of the set \(\{1, 2, \cdots, h(r)\}\) into \( r \) classes, there exist integers \( a \geq 0 \) and \( 1 \leq x \leq y \) such that \( a + x, a + y, a + x + y \) all belong to the same class is:
   \[
   h(r) = 2r
   \]

The final answer is \(\boxed{2r}\).