Deep-Dive Into One Topic

ROLE: Act as an enrichment specialist who builds deep, coherent single-topic study units for gifted students.

CONTEXT: I want a thorough deep-dive into [TOPIC e.g. functional equations / inequalities / modular arithmetic] for strong students in [CLASS/GRADE], aiming at [BOARD or EXAM e.g. RMO / INMO / JEE Advanced]. It should go from intuition to genuinely hard problems, as one connected story.

YOUR TASK:
1. Give the big-picture intuition: what this topic is really about and why it matters.
2. Present the key definitions and results, with short derivations or proofs (not just statements).
3. Show 4 worked examples of rising difficulty, each fully solved.
4. Provide a 'toolbox' summary: the standard moves and tricks for this topic.
5. Give 6 challenge problems (graded) with complete solutions.
6. End with 3 pitfalls and how to avoid them.

INPUT: Here is the topic and any sub-areas to include: [PASTE THE TOPIC HERE].

HOW TO WORK IT OUT: Build the unit yourself and check it forms one logical thread. VERIFY every stated result, worked example, and challenge answer by two independent methods (e.g. proof plus concrete test case). Never state a result without justification; never invent notation without defining it. Distinguish clearly between what is proven and what is merely motivated. Keep all mathematics fully legible in clean LaTeX or clear plain text.

OUTPUT FORMAT (exact headings, in order):
- BIG PICTURE (what this topic is really about)
- KEY DEFINITIONS
- KEY RESULTS (each with a short derivation/proof)
- WORKED EXAMPLES 1-4 (rising difficulty; each verified)
- TOOLBOX (standard moves and tricks)
- CHALLENGE PROBLEMS (6, graded)
- CHALLENGE SOLUTIONS (with verification noted)
- THREE PITFALLS AND FIXES

RULES: One coherent narrative, not a disconnected list. Justify every result. Separate proved from motivated. All answers double-checked.

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