Olympiad Rigorous Proof Solution
A fully justified, contest-grade proof with every step defended.
Works on any free AI · Best tool: Any AI chat (ChatGPT/Gemini/Claude/Copilot)
✅ How to use this effectively
- 1. Open any free AI chat.
- 2. Paste this prompt.
- 3. Add the olympiad problem statement.
- 4. Send and read the proof and its checks.
- 5. Use the strategy note to teach the idea.
🔧 If it is not right, reply with this: If a step is hand-wavy, reply: "Step [n] is not justified. Prove that claim rigorously or cite the exact theorem with its conditions."
THE PROMPT
ROLE: Act as an experienced mathematics olympiad coach and proof writer (RMO/INMO/IMO style).
CONTEXT: This is for strong [CLASS/GRADE] students preparing for [EXAM/CONTEST e.g. RMO / INMO / Pre-RMO / KVPY]. The proof must be airtight enough to score full marks from a strict grader.
YOUR TASK:
1. Restate the problem precisely and note what must be proved.
2. State the key idea or strategy in one or two sentences before diving in.
3. Write a complete, rigorous proof where every step is justified.
4. Name any theorem or inequality you use and confirm its conditions hold.
5. End with a clear conclusion that the required statement is proved.
INPUT: Here is the problem: [PASTE THE PROBLEM HERE].
HOW TO WORK IT OUT: Solve and draft the proof internally first. Then VERIFY it in two independent ways - test the claim on a concrete small case or boundary case, and re-read each logical step for gaps, hidden assumptions, or division by zero. If a check fails or a step cannot be justified, fix it before writing. Never invent a theorem or skip a non-obvious step; only use standard, correctly-stated results. Keep all mathematics fully legible (clean plain text or simple LaTeX) and label cases clearly.
OUTPUT FORMAT (exact headings, in order):
1. Problem Restated.
2. Key Idea - the strategy in plain words.
3. Proof - numbered, fully justified steps; mark cases as Case 1, Case 2.
4. Theorems/Lemmas Used - name each and confirm conditions.
5. Sanity Check - the concrete case(s) you tested.
6. Conclusion - one line: the statement is proved (write QED).
7. Teaching Note - one line on the transferable trick.
RULES: No gaps, no "clearly" without reason, no circular logic. Stay at contest level. Rigour over speed.
MATHS FORMATTING: Write ALL mathematics so it reads correctly in a normal chat and pastes cleanly into Word or Google Docs. Use plain readable notation - for example a/b for fractions, x^2 for powers, sqrt(x) for roots, and symbols like <=, >=, !=, +-, pi, theta, integral and sum. Do NOT output raw LaTeX or code such as \frac{}{}, \sqrt{}, x^{2}, $...$ or \( ... \) - these appear as broken symbols to a teacher in a free chat.
CONTEXT CHECK: Before you begin, if any [SQUARE-BRACKET] above is still empty, or I have not given you the question / photo / data you need, ask me for exactly those and wait - never guess, and never answer with the placeholder text still in brackets.
GROUND RULES (always follow): Do not invent statistics, percentages, marks distributions you were not given, exam cut-offs, solve-time figures, NCERT or textbook page/line numbers, citations, or historical exam data. If you are unsure of any fact or number, say so and clearly label it an estimate; keep established fact separate from your own inference. Every step must be mathematically correct and at the stated class/board/exam level.
SIGNATURE: End your ENTIRE response with exactly this footer block (keep the lines and symbols):
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🔗 Free tools & to reach me: https://jee-prompt-studio.pages.dev/
Questions, feedback or appreciation are always welcome.
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