Classic Olympiad Techniques Lesson

ROLE: Act as an olympiad trainer who teaches one classic problem-solving technique so well that students recognise when to use it.

CONTEXT: My gifted students in [CLASS/GRADE] (target [BOARD or EXAM e.g. RMO / INMO / JEE]) know scattered tricks but cannot tell WHICH tool fits a new problem. I want a focused lesson on ONE named technique.

YOUR TASK:
1. Explain the technique [TECHNIQUE e.g. pigeonhole principle / telescoping / invariants / extremal principle / substitution] from first principles, in plain language.
2. State precisely the 'trigger signals' in a problem that tell you to try this technique.
3. Give 5 worked examples, ordered from an obvious application to a well-disguised one, each fully solved.
4. Show one NON-example: a problem that looks like it fits but does not, and explain why.
5. Provide 6 graded drill problems with full solutions.

INPUT: Here is the technique to teach: [PASTE THE TECHNIQUE NAME HERE]. If I add a topic to focus the examples on, use it.

HOW TO WORK IT OUT: Work all examples and drills yourself first. VERIFY each final answer by two independent methods (e.g. the technique's argument PLUS a direct/numerical confirmation). Never invent a result or claim the technique applies where it does not. Be explicit about the technique's hypotheses and limits. Keep all mathematics fully legible in clean LaTeX or clear plain text.

OUTPUT FORMAT (exact headings, in order):
- THE TECHNIQUE IN ONE PARAGRAPH
- WHY AND WHEN IT WORKS (the underlying idea)
- TRIGGER SIGNALS (bullet list: spot these in a problem)
- WORKED EXAMPLES 1-5 (obvious to disguised; each fully solved + verified)
- A REVEALING NON-EXAMPLE (why the technique fails here)
- DRILLS (6 problems, graded easy to hard)
- DRILL SOLUTIONS (with verification noted)

RULES: Teach recognition, not just mechanics. Be precise about when the method is valid. Every answer double-checked. No bluffing.

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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