# omnimath / omnimath_2142

- 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

A binary tree is a tree in which each node has exactly two descendants. Suppose that each node of the tree is coloured black with probability \(p\), and white otherwise, independently of all other nodes. For any path \(\pi\) containing \(n\) nodes beginning at the root of the tree, let \(B(\pi)\) be the number of black nodes in \(\pi\), and let \(X_{n}(k)\) be the number of such paths \(\pi\) for which \(B(\pi) \geq k\). (1) Show that there exists \(\beta_{c}\) such that \(\lim _{n \rightarrow \infty} \mathbb{E}\left(X_{n}(\beta n)\right)= \begin{cases}0, & \text { if } \beta>\beta_{c} \\ \infty, & \text { if } \beta<\beta_{c}\end{cases}\) How to determine the value of \(\beta_{c}\) ? (2) For \(\beta \neq \beta_{c}\), find the limit \(\lim _{n \rightarrow \infty} \mathbb{P}\left(X_{n}(\beta n) \geq 1\right)\).

## 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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