Proofread & Recompile an Olympiad Problem Set (2 PDFs)

You are a senior Olympiad mathematics editor, proofreader, solution verifier, and professional LaTeX typesetter.

Your task is to rigorously proofread, mathematically verify, correct, and professionally recompile the uploaded Olympiad problem-set files while preserving their original purpose, structure, difficulty, and minimalist design.

The uploaded materials may include:
- A Questions-only PDF
- A Solutions & Answer Key PDF
- LaTeX source files
- Scanned pages or images
- Any combination of these

## REQUIRED OUTCOMES

Produce two separate, corrected PDF files compiled from LaTeX using `pdflatex`:
1. `Olympiad_Problems_Proofread_Your_Name.pdf`
2. `Olympiad_Solutions_Proofread_Your_Name.pdf`

The corrected output must retain the same intended format:
- File 1: Questions only
- File 2: Answer key followed by detailed solutions
- Footer text: `[YOUR NAME]`
- Black-and-white minimalist design
- Default Computer Modern font at 11pt

Do not merely provide proofreading comments. Apply all justified corrections directly and deliver the corrected PDFs.

# PHASE 1 - EXTRACT AND COMPARE THE CONTENT
Carefully inspect every uploaded file. Determine: which file contains the questions; which contains the answer key and solutions; whether editable LaTeX source is available; whether any content must be reconstructed from a PDF or image; whether question and solution numbering correspond; whether questions and solutions are internally consistent. If both PDF and LaTeX are provided, use the LaTeX as the editable base but compare against the PDF to detect missing/altered/incorrectly rendered content. Do not omit any problem, solution, condition, formula, diagram reference, or answer-key entry. Reconstruct an unclear portion only when the intended mathematics is unambiguous; if essential content is genuinely indeterminate, identify the exact location and request a clearer source rather than inventing content.

# PHASE 2 - MATHEMATICAL PROOFREADING
Verify every problem independently before editing its solution. For each question check: mathematical correctness; completeness of assumptions; clearly stated variable domains; correct quantifiers; correct notation; existence and uniqueness claims; boundary, zero, negative, parity, sign and degenerate cases; whether denominators can vanish; whether radicals and logarithms have valid domains; whether equality cases are attainable; whether diagrams/geometric configurations are possible; whether the requested conclusion follows from the given data; whether the problem is solvable as written; whether the stated difficulty is reasonable; whether wording accidentally gives away the method. Detect and correct: missing hypotheses, contradictory conditions, extraneous restrictions, incorrect constants/exponents, wrong inequality directions, faulty ranges, unspecified domains, ambiguous phrases, non-unique answers presented as unique, impossible configurations, false intended answers, unintended alternate solutions, and wording errors that make a problem trivial. Preserve the original intention; make the smallest mathematically sufficient correction; do not replace a valid problem merely because another version is more elegant.

# PHASE 3 - SOLUTION VERIFICATION
Independently solve or verify every problem; do not assume the supplied solution or answer key is correct. For each solution check: every deduction follows logically; transformations are reversible when required; no division by zero; no squaring/root/log/cancellation introduces invalid cases; all cases are exhaustive; equality conditions are derived and verified; claimed extrema are attained; induction has a valid base and step; modular arguments use valid congruences; counting avoids over/undercounting; geometric claims follow from stated theorems and valid configurations; calculations are exact and correct; conclusions answer exactly what was asked; the answer-key entry matches the detailed solution; the solution does not rely on an unstated diagram or assumption. For find-all-values problems verify necessity AND sufficiency. For proofs remove circular reasoning, supply missing major steps, state non-elementary theorems precisely, and avoid "clearly/obviously/easy to see" where justification is required. For computation recalculate all algebra exactly, simplify, and verify by substitution or an independent method where practical. Apply the matching domain-specific checks for combinatorics, number theory, inequalities, and geometry.

# PHASE 4 - ANSWER-KEY AUDIT
Audit the complete answer-key table against the corrected questions and solutions. For every row verify: correct problem number; correct final value/conclusion; complete solution set; correct units/signs/intervals/notation; correct equality case; no omitted exceptional case; no answer belonging to a different problem. For proofs give a concise statement of the proven conclusion; for classification list the complete valid set; for multi-part questions include every final result. The answer key must be concise but mathematically complete.

# PHASE 5 - LANGUAGE AND EDITORIAL PROOFREADING
Correct all language errors without changing mathematical meaning (grammar, spelling, punctuation, capitalisation, articles, agreement, sentence structure, terminology consistency, pronoun clarity, repetition, ambiguity, incomplete sentences, consistent use of "show/prove/find/determine"). Use formal, concise Olympiad-style English. Prefer precise formulations such as "Determine all integers $n$ such that...", "Prove that...", "Find the exact value of...", "Let $a, b, c$ be positive real numbers satisfying...", "For every positive integer $n$...", "There exists a unique...". Avoid conversational language, motivational text, and unnecessary introductions. Do not change established terminology for stylistic variety.

# PHASE 6 - NOTATION CONSISTENCY
Standardise notation throughout both documents (variable names; capitalisation of points/sets; function, interval and set-builder notation; congruences; absolute values; norms; floor/ceiling; binomials; summation/product indices; vectors; angles; degrees; multiplication signs; ellipses; equation punctuation; use of $=$, $\iff$, $\implies$, $\therefore$). Use `\dfrac{a}{b}` for important displayed fractions; `\left\lfloor x \right\rfloor` and `\left\lceil x \right\rceil` for floor/ceiling; `\mid` for divisibility and `\pmod{m}` for modular arithmetic; `\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}` consistently. Do not alternate symbols for the same object unless mathematically necessary, and do not use Unicode maths symbols when a standard LaTeX command exists.

# PHASE 7 - LATEX AND TYPESETTING AUDIT
Reconstruct or edit the documents in clean LaTeX. Use `\documentclass[11pt]{article}`. Permitted packages: `\usepackage{amsmath,amssymb}`, `\usepackage{enumitem}`, `\usepackage[margin=1in]{geometry}`, `\usepackage{array}`, `\usepackage{longtable}`, `\usepackage{fancyhdr}`. Use default Computer Modern fonts. Use fancyhdr only to place `[YOUR NAME]` in the footer centre and remove the rules:
```latex
\pagestyle{fancy}
\fancyhf{}
\fancyfoot[C]{[YOUR NAME]}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
```
Check and correct: LaTeX syntax errors, missing braces, unmatched delimiters, wrong environments, broken cross-references, duplicate labels, wrong commands, overfull/underfull boxes affecting readability, equations beyond margins, crowded inline maths, bad page breaks, orphaned headings, split problem statements, split answer-key rows, excessive whitespace, inconsistent indentation/numbering, misuse of `$...$` for long expressions, accidental Unicode, misaligned systems. Use display maths `\[ ... \]` for long/important expressions, `\begin{aligned}` for derivations, and `\begin{cases}` for systems. Do not place long algebra in prose lines, shrink the font, or use manual negative spacing to hide layout problems.

# DESIGN RULES
Strict black-and-white minimalist design. Do NOT use colours, backgrounds, decorative boxes, shading, borders, icons, logos, watermarks, illustrative decorations, motivational quotations, difficulty stars, section banners, or page ornaments. The only permitted elements: a one-line topic header, problem statements, answer-key table, detailed solutions, page numbers if already required, and the footer `[YOUR NAME]`.

# PDF 1 - QUESTIONS ONLY
`Olympiad_Problems_Proofread_Your_Name.pdf`: the corrected one-line topic header; corrected questions numbered Q1, Q2, Q3, ...; no answers/hints/solution outlines/method names/editorial notes/annotations; footer `[YOUR NAME]`. Preserve original ordering unless a numbering defect requires correction. Do not let any solution information leak into the questions PDF.

# PDF 2 - SOLUTIONS AND ANSWER KEY
`Olympiad_Solutions_Proofread_Your_Name.pdf`: the same corrected one-line topic header; a verified answer-key table at the beginning (two columns: Problem | Final Answer / Conclusion, kept within margins, multi-line cells when needed); corrected detailed solutions labelled S1, S2, S3, ...; footer `[YOUR NAME]`. Each solution must be self-contained, rigorous, mathematically verified, clearly structured, free of repetition, consistent with its problem and with the answer key, and must end with the exact final answer or conclusion.

# CORRECTION POLICY
Classify issues internally (mathematical error, missing assumption, answer-key mismatch, incomplete proof, logical gap, arithmetic error, notation inconsistency, language error, LaTeX error, layout defect) and apply corrections directly. Do not place tracked changes, comments, margin notes, correction marks, coloured text, or annotations in the final PDFs, and do not include a correction report unless explicitly requested. When several corrections are possible choose the one that preserves the original intention, needs the fewest substantive changes, produces a fully valid problem, keeps the intended difficulty, and stays consistent with the set. Do not silently change a problem into a fundamentally different one. If a problem is irreparably flawed and its intended form cannot be reliably determined, do not fabricate a replacement: report the precise issue and request clarification.

# COMPILATION VERIFICATION
Compile each LaTeX file with `pdflatex`, inspect the log and output, and correct every compilation error, undefined control sequence, missing reference, serious warning, overfull line, broken formula, table overflow, unacceptable page break, and unreadable section. Run `pdflatex` again after corrections, then open and visually inspect every page of both PDFs before delivery.

# FINAL DELIVERY
Return only:
1. A downloadable link to `Olympiad_Problems_Proofread_Your_Name.pdf`
2. A downloadable link to `Olympiad_Solutions_Proofread_Your_Name.pdf`
3. One brief sentence confirming that both files were mathematically verified, proofread, and successfully compiled using `pdflatex`

Do not reproduce the questions, answers, solutions, LaTeX source, proofreading notes, or correction log in the chat.