Stage 3 national Olympiad guide
INMO 2027 Preparation: Deep Proofs, Long Problems and Clear Writing
For the 2026–27 Olympiad cycle, INMO is scheduled for 17 January 2027 from 12:00 to 16:30. The 4.5-hour paper has six proof problems and demands depth, endurance and precise mathematical exposition.
INMO problems may resist routine approaches for a long time. Preparation should develop conjecture testing, lemma building, proof architecture, strategic persistence and the ability to present several pages of correct reasoning without hidden assumptions.
Official 2026–27 cycle
INMO 2027 date and paper format
| Item | Official information |
|---|---|
| Date | 17 January 2027 |
| Time | 12:00–16:30 |
| Duration | 4.5 hours |
| Questions | 6 |
| Response style | Detailed proof for every question |
The official HBCSE announcement controls the schedule and participation instructions. Selection from RMO and registration rules are notified separately; qualifying students must follow those notices.
RMO → INMO
Increase depth without abandoning fundamentals
Longer exploration
Useful structure may appear only after examples, failed approaches and a carefully chosen lemma.
Stronger synthesis
A solution can combine ideas from more than one domain or require an unfamiliar viewpoint.
Higher exposition standard
Several correct pages must remain logically ordered, readable and complete.
Do not interpret “advanced” as a license to skip elementary facts. Many strong solutions use basic tools in an unusually sharp way.
Core preparation areas
Build a repertoire of proof ideas across four domains
Number theory
Congruences, valuations, Diophantine equations, multiplicative structure, descent and constructive arguments.
Algebra
Inequalities, polynomials, sequences, recurrences, functional equations and algebraic transformations with equality analysis.
Geometry
Classical Euclidean geometry, transformations, inversion where appropriate, projective or coordinate viewpoints and precise diagrams.
Combinatorics
Double counting, extremal principle, invariants, colouring, graph arguments, recurrences and constructive methods.
HBCSE identifies these as the major Mathematical Olympiad areas and notes increasing difficulty from RMO to INMO to IMO.
How to work when stuck
Turn exploration into a sequence of justified advances
- Restate the target in your own precise language.
- Generate small examples, special cases or a cleaner equivalent formulation.
- List what each condition prevents or guarantees.
- Search for a quantity, configuration or extremal object that organises the problem.
- Write and prove a smaller lemma.
- Test whether the lemma actually moves toward the target.
- If an approach stalls, preserve valid discoveries and change representation.
Proof architecture
Write so the evaluator can see the argument
| Part | What good writing shows |
|---|---|
| Setup | Definitions, notation and the exact hypotheses being used |
| Main lemmas | Clear statements followed by complete justification |
| Case structure | Why cases are disjoint and exhaustive |
| Boundary conditions | Zero, sign, equality, degeneracy and exceptional values |
| Conclusion | An explicit return from the proved statements to the required claim |
After solving, revise the proof once for mathematics and once for communication. A correct idea can become difficult to evaluate if notation changes or dependencies are hidden.
Between RMO and INMO
Use the available weeks for depth and full-paper endurance
- First phase: diagnose RMO proofs, rewrite incomplete solutions and repair one weak domain.
- Middle phase: solve a small number of substantial INMO problems each week with detailed feedback.
- Integration phase: use mixed long-problem sessions and practise choosing which problem deserves sustained time.
- Simulation phase: take complete 4.5-hour papers sparingly and review them deeply.
- Final phase: consolidate recurring lemmas, writing errors and paper strategy; do not chase an entirely new advanced library.
The exact interval depends on official results and registration. Begin proof improvement immediately rather than waiting for a perfect schedule.
Official papers and solutions
Study how national-level solutions are constructed
The official HBCSE archive provides many INMO papers and available solutions.
- Attempt one problem seriously, often across more than one sitting.
- Record partial lemmas before opening a solution.
- Identify the decisive idea and why your approach missed it.
- Close the source and reconstruct the proof.
- Compare alternate solutions when available to broaden representation.
Do not measure progress only by how many full papers are “completed”. Measure whether the student can later reproduce and adapt the ideas.
4.5-hour strategy
Train endurance without turning every session into an exam
During preparation
Most sessions can focus deeply on one or two problems. Use occasional longer blocks to train persistence and writing stamina.
During a paper
Scan all six, begin where progress is concrete, preserve clean partial results and reserve time to organise and check proofs.
Full simulations are demanding. Schedule recovery and review rather than stacking them so closely that the student only accumulates unread solutions.
Mentor feedback
Ask for diagnosis, not a replacement solution
Useful feedback identifies the earliest invalid inference, a missing case, an imprecise statement or a more economical lemma. The student should then rewrite the proof independently.
A good question is “Where does my argument first stop being justified?” This preserves ownership while making feedback precise.
Final checklist
Audit every proof for silent assumptions
- Did I use a converse that was never proved?
- Did I divide by an expression that may be zero?
- Did I assume a diagram is in a special orientation?
- Did I verify equality or boundary cases?
- Did I prove existence but not uniqueness, or vice versa?
- Did I use induction without a valid base and transition?
- Can a reader identify the main idea and final conclusion?
Questions answered
Frequently asked questions
When is INMO 2027?
For the 2026–27 cycle, HBCSE schedules INMO on 17 January 2027 from 12:00 to 16:30.
How long is the INMO paper?
The official duration is 4.5 hours.
How many questions are in INMO?
The official 2027 pattern lists six questions, each requiring a detailed proof.
How is INMO different from RMO?
Both are proof-based, but HBCSE notes that difficulty increases from RMO to INMO; INMO also has a longer 4.5-hour format.
What topics should I prepare?
The major areas are algebra, combinatorics, geometry and number theory, studied through deeper proof methods and synthesis.
Where can I find INMO previous papers?
Use the official HBCSE past-papers archive for papers and available solutions.
How many full INMO mocks should I take?
Use a small number that can be reviewed thoroughly. Most training should still include deep work on selected long problems.
What should mentor feedback focus on?
It should identify logical gaps, missing cases, theorem conditions and proof structure, after which the student rewrites the argument independently.
Next step
Train depth, then make the depth readable.
Use official INMO papers, build lemmas patiently and revise every proof until the mathematical structure is visible.
MENTOR PERSPECTIVE
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