# Technical Document Extraction: Growing Spiral Problems and Solutions
## Overview
The image contains two distinct mathematical problems and their solutions related to counting points in a "growing spiral" configuration. Each problem defines a spiral as a sequence of integer-coordinate points and asks for the of points that **cannot** be the last point of any such spiral within specified bounds.
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### Problem 1 (Left Column)
**Problem Statement**
Define a *growing spiral* in the plane as a sequence of points with integer coordinates \( P_0 = (0, 0), P_1, \ldots, P_n \) such that \( n \geq 2 \).
**Question**: How many points \((x, y)\) with integer coordinates \( 0 \leq x \leq 2011, 0 \leq y \leq 2011 \) **cannot** be the last point \( P_n \) of any growing spiral?
**Solution**
The solution identifies the set of excluded points as:
- \((0, y)\) for \( 0 \leq y \leq 2011 \)
- \((x, 0)\) and \((x, 1)\) for \( 1 \leq x \leq 2011 \)
- \((x, 2)\) for \( 2 \leq x \leq 2011 \)
- \((x, 3)\) for \( 3 \leq x \leq 2011 \)
**Calculation**:
\[
2012 \, (\text{for } y=0\text{ to }2011) + 2011 \, (\text{for } x=1\text{ to }2011) + 2011 \, (\text{for } x=1\text{ to }2011) + 2010 \, (\text{for } x=2\text{ to }2011) + 2009 \, (\text{for } x=3\text{ to }2011) = 10053
\]
**Final Answer**:
\[
\boxed{10053}
\]
---
### Problem 2 (Right Column)
**Problem Statement**
Define a *growing spiral* in the plane as a sequence of points with integer coordinates \( L_0 = (0, 0), L_1, \ldots, L_n \) such that \( n \geq 2 \).
**Question**: How many points \((w, v)\) with integer coordinates \( 0 \leq w \leq 4680, 0 \leq v \leq 4680 \) **cannot** be the last point \( L_n \) of any growing spiral?
**Solution**
The solution identifies the set of excluded points as:
- \((0, v)\) for \( 0 \leq v \leq 4680 \)
- \((w, 0)\) and \((w, 1)\) for \( 1 \leq w \leq 4680 \)
- \((w, 2)\) for \( 2 \leq w \leq 4680 \)
- \((w, 3)\) for \( 3 \leq w \leq 4680 \)
**Calculation**:
\[
4681 \, (\text{for } v=0\text{ to }4680) + 4680 \, (\text{for } w=1\text{ to }4680) + 4680 \, (\text{for } w=1\text{ to }4680) + 4679 \, (\text{for } w=2\text{ to }4680) + 4678 \, (\text{for } w=3\text{ to }4680) = 23398
\]
**Final Answer**:
\[
\boxed{23398}
\]
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### Metadata
- **Year**: 2011
- **ID**: A1
- **Final Answers**:
- Problem 1: 10053
- Problem 2: 23398
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### Notes
1. **Structure**: The image is divided into two columns, each containing a problem and its solution.
2. **Textual Content**: All text is in English. No non-English content is present.
3. **Formatting**: Final answers are boxed using LaTeX syntax (\(\boxed{}\)).
4. **Calculations**: Both solutions involve summing arithmetic sequences based on the bounds of the spiral coordinates.
5. **Key Observations**:
- The excluded points follow a pattern where the first coordinate (or second) is fixed at 0, 1, 2, or 3, with the other coordinate varying within the problem's bounds.
- The total count of excluded points is derived by summing the lengths of these sequences.
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### Conclusion
The image provides two combinatorial problems with explicit solutions. The solutions rely on counting excluded points in a systematic manner, leveraging arithmetic series to arrive at the final answers. No visual elements (e.g., charts, diagrams) are present, so the focus is entirely on textual and numerical data extraction.
