To solve this problem, we need to explore the operations provided and determine the conditions under which any integer can eventually be written on the board starting from a pair $(u,v)$.

1. Initial Setup:
We begin with two distinct integers $u$ and $v$ on the board. At each operation, we can:
- Add two distinct integers $a$ and $b$ to form $a + b$, or
- Use three distinct integers $a$, $b$, and $c$ to find roots $x$ of the quadratic equation $ax^2 + bx + c = 0$.

2. Operation (i) Insight:
- Using the first operation, we can generate any linear combination of integers on the board. Thus, if we can reach any integer through linear combinations, then having all integers equals the ability to construct them through the sum operation.

3. Operation (ii) Insight:
- The second operation broadens our potential to generate integers on the board beyond linear combinations. Specifically, solving quadratic equations brings the prospects of writing quadratic roots, which can include integers beyond immediate integer sums.

4. Key Restrictions Analysis:
- Consider the integers on the board. If all integers can be obtained, we must have a way to generate both positive and negative integers. Thus, initial integers must allow us to cross zero in some fashion.

5. Finding Robust Initial Conditions:
- If $u$ and $v$ are non-zero and at least one of them is positive, then using additions and positive/negative combinations, we can eventually reach any integer, positive or negative.
- However, starting with opposite signs and both non-zero, say $(-1, 1)$ or $(1, -1)$, notice that:
- The only sums directly achievable from these are 0 or integers closer to each other symmetrically reducing chances of crossing zero for positive convergence or broader exploration.

6. Deriving the Condition:
Therefore, the pairs $(u, v)$ that allow us to eventually write any integer on the board must satisfy:
- Both numbers are non-zero $u \neq 0$, $v \neq 0$,
- At least one of them is positive i.e. $\max(u, v) > 0$,
- Exclude the pairs $(-1, 1)$ and $(1, -1)$ since they restrict reaching all integers.

Thus, the required pairs $(u, v)$ are:

$$\boxed{\text{All pairs } (u, v) \text{ of distinct integers for which } u \neq 0, v \neq 0, \max(u, v) > 0, \text{ and } (u, v) \notin \{(-1, 1), (1, -1)\}.}$$