Tiered Olympiad-Style Problem Set

ROLE: Act as an Olympiad problem-setter and trainer (RMO/INMO/IMO style) who writes original, classroom-ready problems.

CONTEXT: I teach gifted students preparing for math olympiads and tough entrance tests. Build for [CLASS/GRADE] aiming at [BOARD or EXAM e.g. CBSE / ICSE / State Board / JEE / RMO]. The single topic is [TOPIC e.g. number theory: divisibility]. Keep it strictly within tools a student at this level has met, but push the thinking hard.

YOUR TASK:
1. Create exactly 12 ORIGINAL problems on [TOPIC], split into 3 tiers of 4: Tier 1 (warm-up, a clever observation), Tier 2 (multi-step, needs a key idea), Tier 3 (genuine olympiad difficulty).
2. For each problem give: the statement, the main idea/technique it tests, and a complete step-by-step solution.
3. Make every problem self-contained and unambiguous. No problem may reuse the same trick as another.

INPUT: Here is the topic and level: [PASTE TOPIC AND ANY CONSTRAINTS HERE]. If I paste a sample problem I like, match its flavour.

HOW TO WORK IT OUT: Solve every problem fully yourself BEFORE writing anything. Then VERIFY each final answer by at least two independent methods (e.g. a direct computation plus a small-case check, or a substitution plus a bounding argument). Never invent formulas or values; if a problem turns out flawed or has no clean answer, discard it and write a fresh one. Keep all mathematics fully legible using clean LaTeX or clear plain text. State any non-obvious assumption explicitly.

OUTPUT FORMAT (use these exact headings, in order):

FORMAT THE OUTPUT — PRIMARY DELIVERABLE:
Return the complete material as a clean, print-ready A4 handout that can be pasted into Word or Google Docs and saved as PDF. Use a clear title, logical sections, readable mathematics and page-break markers where needed. Keep student material separate from the answer key or teacher notes.


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.
- TITLE BLOCK: "Olympiad Problem Set: [TOPIC] - Compiled by [YOUR NAME]"
- HOW TO USE THIS SET (2 lines for the teacher)
- TIER 1 (PROBLEMS 1-4): each as "Problem N | Idea tested | Statement"
- TIER 2 (PROBLEMS 5-8): same layout
- TIER 3 (PROBLEMS 9-12): same layout
- FULL SOLUTIONS: numbered 1-12, each with "Key idea", "Solution", "Verification (two checks)", "Final answer"
- DIFFICULTY MAP: one line rating each problem 1-5

RULES: Original problems only - do not copy famous contest problems verbatim. Every final answer must be double-checked. No hand-waving in proofs; each logical step must follow. If unsure, say so rather than bluff.

At the very end add a footer line: "Prepared by [YOUR NAME]", and put "Compiled by [YOUR NAME]" in the title block.

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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