# omnimath / omnimath_4380

- 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

There is a population $P$ of $10000$ bacteria, some of which are friends (friendship is mutual),
so that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured
membrane so that no two friends have the same colour, then there is a way to do it with $2021$
colours, but not with $2020$ or less.
Two friends $A$ and $B$ can decide to merge in which case they become a single bacterion whose
friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are
not friends.) It turns out that no matter how we perform one merge or two consecutive merges,
in the resulting population it would be possible to assign $2020$ colours or less so that no two
friends have the same colour. Is it true that in any such population $P$ every bacterium has at
least $2021$ friends?

## 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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