# omnimath / omnimath_4173

- taskset: [omnimath](https://harnessreport.com/tasks/omnimath.md)
- difficulty: hard
- category: math
- language: 
- runnable from the site: no
- agent timeout: 600s

## Results by harness

_none yet_

## Instruction

```
# Mathematical Problem

Two different integers $u$ and $v$ are written on a board. We perform a sequence of steps. At each step we do one of the following two operations:

(i) If $a$ and $b$ are different integers on the board, then we can write $a + b$ on the board, if it is not
already there.
(ii) If $a$, $b$ and $c$ are three different integers on the board, and if an integer $x$ satisfies $ax^2 +bx+c = 0$,
then we can write $x$ on the board, if it is not already there.

Determine all pairs of starting numbers $(u, v)$ from which any integer can eventually be written on the board after a finite sequence of steps.

## Instructions

Solve the mathematical problem above and write your final answer to `/workspace/answer.txt`.

**Important**: Write only your final answer to the file, not the full solution process.

### Guidelines

- Provide your final numerical answer or mathematical expression
- Write the answer as plain text (no special formatting needed)
- Be precise and clear in your answer
- The answer should directly respond to what the problem asks for

### Example

If the problem asks "What is 2 + 2?", your answer file should contain:

```
4
```

Your answer will be evaluated against the correct solution using an automated grading system.
```
---
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