INMO — Indian National Mathematical Olympiad — Answer Writing
Olympiad marking is close to binary on rigour: a brilliant idea with a gap in the argument scores far below a plodding proof that closes every case. Writing the proof is the skill, and it is separable from solving the problem.
Attempt all problems. Justify every step; a correct answer without a complete argument earns no credit. Calculators are not permitted.
- 1.Find all positive integers n such that n² + 1 divides n³ + 3.17
- 2.Let ABC be a triangle with circumcircle ω. Prove that the reflection of the orthocentre in BC lies on ω.17
- 3.Determine all functions f : ℝ → ℝ satisfying f(x + f(y)) = f(x) + y for all real x, y.17
One INMO answer, from question to marked report
A real question, an attempt with the weaknesses a real attempt has, and the report an AI examiner hands back — in the marking unit INMO genuinely uses. This is the whole product; there is nothing else to sign up for.
Scoped to the paper and topic you pick, at the marks it is genuinely set at.
Find all positive integers n such that n squared plus 1 divides n cubed plus 3.
Type it, dictate it, or write on paper and photograph the page — your handwriting is read.
Photographed, read, and marked in red — every expected step scored down the right margin and the omissions written in.
We have n^3 + 3 = n(n^2 + 1) - n + 3. So n^2 + 1 divides -n + 3, that is n^2 + 1 divides n - 3. For large n, n^2 + 1 is bigger than n - 3, so we need n - 3 = 0 or small n.? Checking, n = 3 works and n = 1 works. So the answers are n = 1 and n = 3. First note that n^2 + 1 is positive for every positive integer n, so the divisibility statement makes sense throughout. Writing the division the other way, n^3 + 3 = n(n^2 + 1) + (3 - n), which is the same relation with the sign changed, and a divisor of a number also divides its negative, so it makes no difference whether we work with 3 - n or with n - 3. Since n^2 + 1 grows quadratically while n - 3 grows linearly, the divisibility can only hold when the linear part is zero or is small in comparison. Trying small values, n = 1 gives 2 dividing 4 which is true, and n = 3 gives 10 dividing 30 which is also true.
- The explicit inequality and the value of n from which it holds
- Checking every remaining case, n equals 1, 2 and 3, visibly
- A closing sentence asserting that these are all solutions
Model answer for this question
Reduce: n cubed plus 3 equals n times (n squared plus 1) minus n plus 3, so n squared plus 1 divides n minus 3. Bound: if n is at least 4 then n squared plus 1 exceeds the absolute value of n minus 3, forcing n minus 3 equals 0, giving n equals 3. Check the remaining cases n equals 1, 2, 3 directly. Conclude: the complete solution set is n in {1, 3}, and state that no other n can occur by the bound.
Photograph the page from your phone and the Studio reads your handwriting — the only way to practise the speed and layout the paper actually tests.
Fast enough to write, read the report and rewrite the same answer in one sitting — which is where the improvement actually comes from.
The scoring behind those pen marks: every expected step, what it was worth, what you earned, and a model answer.
- Reduction to a divisibility of lower degree5/5
Correct and complete.
- Size argument stated as an inequality2/5
Asserted informally. Write: for n greater than or equal to 4, n squared plus 1 is greater than the absolute value of n minus 3, so the quotient must be zero.
- Exhaustive check of the remaining finite cases2/4
n equals 1 and 3 are found; n equals 2 is never checked and the reader cannot see that the list is complete.
- Conclusion stated as a complete solution set2/3
The set is stated but not asserted as exhaustive.
INMO at a glance
| Conducting body | Homi Bhabha Centre for Science Education (HBCSE) |
|---|---|
| Mode | Handwritten proofs |
| Papers | 4 |
| Total | Six problems over four and a half hours |
| Marking | INMO — step marking |
What INMO sets
The papers you sit, and what each is worth.
Functional equations, polynomials, inequalities.
Counting, extremal arguments, invariants.
Synthetic and computational, with full justification.
Divisibility, congruences, Diophantine equations.
The rest of your INMO preparation
- Marked to the INMO scheme
- Model answer, every time
- Handwriting read from a photo
- AI viva on any topic
- Full timed papers
- Every attempt kept in your archive
- WriteA question, answered and marked.
- VivaQuestioned on Functional equations and more.
- Full paperA timed paper, one scorecard.