What is the International Mathematical Olympiad?

The IMO began in 1959 and brings together teams nominated through different countries’ selection programmes. Competitors work on unusually demanding proof-based mathematical problems, not routine school multiple-choice quizzes. The two contest days normally contain three problems each, with a total possible score of 42 points; the relevant year’s official rules and problem list determine details.

Typical preparation areas include number theory, combinatorics, Euclidean geometry and algebra. The hallmark is explaining why a conclusion is true for every permitted case. Mathematical maturity, originality, careful reasoning and clear written proofs are more important than memorising long lists of formulas.

The official IMO makes selection and country participation separate from informal school competitions. Even a high score on an independent SCO Maths paper is not a substitute for a country’s approved IMO qualifying pathway.

IMO 2026 in Shanghai: verified countries, results and medals

The 67th official International Mathematical Olympiad took place in Shanghai, People’s Republic of China, 10–21 July 2026. The IMO’s final record gives 117 countries and 666 contestants. The closing ceremony took place on 20 July 2026 at Shanghai High School, where students, leaders, volunteers and organisers celebrated the problem-solving achievements of their international community.

IMO’s official 2026 medal cutoffs were 29 points for gold, 23 for silver and 16 for bronze. There were 141 honourable mentions. These are official IMO medals, not SCO medals; another year’s cutoffs will depend on that year’s contestant results and official award rules. See the complete IMO 2026 edition record for individual results, teams, problems and award documentation.

Examples of global 2026 team achievements

IMO 2026 official results placed China first, the United States second, Russia third, Singapore fourth, Vietnam fifth and South Korea sixth on the team score listing. These rankings reflect this particular event and should not be treated as a permanent ranking of national school education systems.

India’s Press Information Bureau announced on 21 July 2026 that its six-member team earned two gold and four silver medals, placing seventh in the unofficial country-score ordering. The UK Mathematical Olympiad committee reported its own team won two gold, one silver and three bronze medals. The stories show how different national selection programmes produce international proof-solving achievements; they do not promise future medals to pupils entering any particular school exam.

IMO 2027: host city and confirmed dates

The official IMO website currently lists the 68th IMO in Budapest, Hungary, 16–26 July 2027. The dates are for the international gathering; students must still qualify through their own country’s valid national selection process, whose application windows and eligibility can differ considerably.

How does a student qualify for the official IMO?

IMO participation is not normally an open paid individual registration. Countries select competitors through their respective recognised Olympiad systems. A parent should first identify their country’s approved national organiser, current eligibility and progression; read dated official instructions rather than an old general blog or commercial registration page.

India’s 2026–27 mathematical Olympiad programme: important actual dates

The current HBCSE cycle page gives RMO on 15 November 2026, 13:00–16:00, with six proof-based questions. It gives INMO on 17 January 2027, 12:00–16:30, also with six written-proof questions. HBCSE directs candidates to MTA(I)’s 2026 IOQM information for earlier-stage enrolment and results. Always revisit the official portals before acting because notices, cutoffs and deadlines may change.

These national selection stages are distinct from the independently organised School Connect Maths Olympiad. Similar acronyms or a date appearing on two websites must not be interpreted as shared administration, eligibility or qualification.

What kinds of mathematical problems appear at IMO?

The official IMO problem archive holds actual problem statements and year information. The 2026 set spans number theory, geometry, combinatorics and functional reasoning. Because mathematical notation on web pages can be incomplete and official paper wording matters, students should consult the official download or archive rather than copied social media transcriptions.

Skill progression: start by proving small statements, then move to novel proof challenges. Practise mathematical induction, invariant ideas, parity arguments, counting principles, inequalities, geometric construction, and clear communication of every step.

A school learner can use original exercises below to build foundations. They are not reproductions of actual IMO 2026 problems, not authorised selection papers, and should not be presented as identical in difficulty to national-team examinations.

Where School Connect Maths fits: a separate free learning pathway

School Connect Olympiad provides its own International Maths Olympiad learning and 2026–27 exam guide, suitable for the school-age syllabus it covers, plus free Class 1–12 chapter-wise learning, linked educational videos and age-appropriate practice questions.

Its current Winter 2027-labelled independent SCO calendar lists Maths for Classes 1–12 and actual exam dates in November–December 2026. This is not a stage of official national IMO selection. For this page’s primary topic—the official international competition—always use the authentic IMO website and country selector.

Teachers can make good use of both kinds of resource for mathematical education: SCO’s class-wise learning for the foundational concepts, and original IMO problem archives for older learners seeking challenging proofs. Neither should be advertised as a guaranteed route to a medal or university offer.

Free School Connect class-wise learning links (independent of IMO team selection)

The following existing School Connect Maths/class resource links appeared on this page before its upgrade and are retained here for learners. They belong to SCO’s separate school-level learning ecosystem. Primary links open general class learning gateways; higher-class links open Maths topics. They are not official IMO syllabi, and class access does not mean a young learner may sit the national-team IMO.

Primary classes 1–5

Middle and secondary classes 6–12

For exact SCO Maths subject and registration terms, use the official School Connect Maths Olympiad subject guide and its live independent exam calendar. The old 19 November 2021, 20 November 2021, 4 December 2021 and 11 December 2021 table formerly on this page was removed because it is historical, not the current calendar.

Global mathematics celebrations: official medals, event videos and lessons

Explore the global achievement behind mathematical Olympiads without mistaking different organisers for one another. This mobile-friendly slider includes two direct official IMO 2026 award/ceremony resources, a source-linked IMSO 2026 international medal ceremony video for a distinct primary-school event, and a short mathematics talk from TED. School Connect does not own or claim any of these awards or publisher videos.

We deliberately link to the IMO organiser’s own Shanghai 2026 ceremony material instead of inventing an unofficial YouTube embed. The host and IMO websites show the real 2026 medal presentation. IMSO’s embedded award ceremony is a different competition in Indonesia, for younger school students; its presence is educational inspiration, not a route into IMO.

Browse: use the topic pills, previous/next buttons or swipe on phones; each full-size player has a direct original-publisher link if embedding is unavailable. Video captions and language choices depend on source providers.

2026 global awards to explore: the official IMO 2026 medals/results, Shanghai closing celebration, the distinct IMSO international primary-school medal ceremony and India’s official 2026 medal report. These examples show what students may achieve in different national and global mathematics pathways, not what an individual SCO enrolment guarantees.

16 original mathematical Olympiad proof exercises with solutions

These are progressively structured original teaching exercises designed to build the proof-writing habits used in national Olympiad pathways. They are neither official 2026 IMO paper questions nor equivalent to the most difficult IMO selection problems. Try each for 5–20 minutes at a suitable level, then read the proof and extension question.

Four families of Olympiad thinking: number theory and divisibility, combinatorial counting and invariants, algebra and inequalities, and Euclidean geometry. A correct proof must cover every allowed case rather than a single example.

Problem 1 — Number theory · foundations

Challenge: Prove that the square of any odd integer is odd.

Worked proof: Write an odd integer as 2k+1. Its square is 4k²+4k+1 = 2(2k²+2k)+1, which is odd. This proves the claim for every integer k.

Extend your reasoning: Try expressing an even integer as 2k and checking its square.

Problem 2 — Number theory · divisibility

Challenge: Prove that n(n−1)(n+1) is divisible by 6 for every integer n.

Worked proof: These are three consecutive integers. At least one is a multiple of 3; at least one is even. The product therefore has factors 3 and 2, so it is divisible by 6.

Extend your reasoning: Explain why the argument works even when n is negative or zero.

Problem 3 — Number theory · coprimality

Challenge: Prove that any two consecutive positive integers are coprime.

Worked proof: If an integer d divides both n and n+1, it divides their difference (n+1)−n = 1. Thus their only common positive divisor is 1.

Extend your reasoning: Try n=104 and n+1=105 before writing the general proof.

Problem 4 — Number theory · congruences

Challenge: Show that the square of every odd integer leaves remainder 1 when divided by 8.

Worked proof: Let n=2k+1. Then n² = 4k(k+1)+1. Since one of k and k+1 is even, k(k+1)=2m. Hence n²=8m+1.

Extend your reasoning: Find the remainder of 35² divided by 8 without multiplying out in full.

Problem 5 — Combinatorics · pigeonhole

Challenge: Show that among any 13 people, two must have birthdays in the same month.

Worked proof: There are only 12 months. If each month contained at most one person’s birthday, at most 12 people could be assigned. Thirteen people force some month to contain at least two.

Extend your reasoning: What can you guarantee if there are 25 people?

Problem 6 — Combinatorics · remainders

Challenge: Prove that any six integers include two that have the same remainder upon division by 5.

Worked proof: Possible remainders are only 0, 1, 2, 3 and 4. With six integers in five remainder classes, at least two occupy the same class.

Extend your reasoning: Show that the difference of that pair is divisible by 5.

Problem 7 — Combinatorics · graphs

Challenge: Prove that any finite friendship graph has an even number of vertices with odd degree.

Worked proof: Each edge contributes 1 to the degree of each of its two endpoints; total degree is twice the edge count and therefore even. A sum of odd numbers is even only when their count is even.

Extend your reasoning: Does the result depend on how many disconnected groups there are?

Problem 8 — Combinatorics · counting

Challenge: How many different unordered pairs can be selected from n distinct students? Prove the formula.

Worked proof: Count n choices for the first student and n−1 for the second. This counts each unordered pair twice, once in each order; divide by 2 to obtain n(n−1)/2.

Extend your reasoning: Check n=5 by writing all ten pairs.

Problem 9 — Combinatorics · invariant

Challenge: An 8×8 chessboard has two opposite corners removed. Explain why the remaining squares cannot be tiled with 31 dominoes covering two adjacent squares each.

Worked proof: Colour the board alternately black and white. Opposite corner squares on an even-sized chessboard have the same colour. Removing them leaves 30 of that colour and 32 of the other. Each domino covers exactly one square of each colour, so 31 dominoes would cover 31 of each; impossible.

Extend your reasoning: Why is counting area alone insufficient to prove non-tileability?

Problem 10 — Algebra · inequalities

Challenge: Prove a²+b²≥2ab for all real a and b.

Worked proof: A square is never negative: (a−b)²≥0. Expand to a²−2ab+b²≥0, then rearrange to a²+b²≥2ab. Equality holds exactly when a=b.

Extend your reasoning: Explain why checking a few numerical examples cannot replace the proof.

Problem 11 — Algebra · AM–GM

Challenge: For positive real a and b, prove (a+b)/2 ≥ √(ab).

Worked proof: Because (√a−√b)²≥0, we have a+b−2√(ab)≥0. Divide by 2. Equality holds if and only if a=b.

Extend your reasoning: Try a=1 and b=9; identify which average is larger.

Problem 12 — Algebra · sums

Challenge: Prove 1+2+···+n = n(n+1)/2 for positive integer n.

Worked proof: Let S=1+2+···+n. Reverse the sum: S=n+(n−1)+···+1. Adding termwise gives 2S = (n+1)+(n+1)+···+(n+1), with n terms. Therefore S=n(n+1)/2.

Extend your reasoning: Check n=4 and then explain why the paired proof applies to every n.

Problem 13 — Geometry · angles

Challenge: Prove that the interior angles of a Euclidean triangle add up to 180°.

Worked proof: Through one vertex, draw a line parallel to the opposite side. By alternate interior angles, the other two vertex angles equal adjacent angles on that line. Those three adjacent angles form a straight angle of 180°.

Extend your reasoning: Identify exactly where Euclidean parallel-line geometry is used.

Problem 14 — Geometry · polygons

Challenge: Show that the exterior angles of any simple convex polygon, taking one exterior turning angle at each vertex, add to 360°.

Worked proof: Walking once around a convex polygon turns the direction of travel by one full revolution. The signed exterior turns make 360°. Equivalently, subtract the interior angle sum (n−2)180° from n×180° to obtain 360°.

Extend your reasoning: Explain why the sum does not depend on the number of polygon sides.

Problem 15 — Number theory · modular arithmetic

Challenge: Prove that n⁵−n is divisible by 5 for every integer n.

Worked proof: Every integer has one of five residues modulo 5: 0,1,2,3,4. Their fifth powers have the same residues respectively (0,1,32≡2,243≡3,1024≡4). Thus n⁵−n≡0 (mod 5).

Extend your reasoning: Try the proof using Fermat’s little theorem if that result is already known.

Problem 16 — Number theory · products

Challenge: Show that the product of any four consecutive integers is divisible by 24.

Worked proof: Among four consecutive integers, one is divisible by 3. At least two are even, and one of those is a multiple of 4. Thus the product has factors 3×2×4=24, giving divisibility by 24.

Extend your reasoning: Discuss why the argument is valid for zero and negative integers too.

Eight steps to prepare for mathematical proof-based Olympiads

1. Know your competition. Use the national organiser for true IMO team selection, rather than an unrelated commercial “IMO” registration link.
2. Build solid foundations. Ensure algebra, divisibility, geometry and counting rules from school learning are genuinely understood.
3. Read a solution critically. Every claim needs justification; if the solution says “obvious”, test why it must hold for every valid case.
4. Try small cases first. Draw figures, list early values and look for an invariant or common pattern before attempting a general proof.
5. Choose the right technique. Consider parity, modular arithmetic, pigeonhole principle, double counting, inequalities or auxiliary geometric construction.
6. Write full proofs. Define variables, justify transformations, handle equality and boundary cases, and write a clear conclusion.
7. Compare authorised past problems. Study the official IMO problem archive and your national organiser’s published training problems where permitted; attribute the official source accurately.
8. Seek precise feedback. Teachers and peers should check logical gaps. Track which types of problem remain difficult, not merely the number attempted.

A suggested seven-day maths reasoning practice plan

Younger learners can adapt this schedule to simpler questions from School Connect’s free school Maths resources, and should not be pressured to attempt advanced IMO papers before learning the fundamentals.

What a strong mathematical explanation includes

• A precise statement of what is given and what must be shown
• Reasons for every substitution, inequality or inference
• A method that covers all allowed values or geometric cases
• Recognition of special and equality cases where appropriate
• A concise conclusion that actually answers the question

Frequently asked questions: International Mathematical Olympiad

Is the official IMO the same as SCO International Maths Olympiad?

No. The official International Mathematical Olympiad is a national-team mathematical proof competition. SCO Maths is a separately organised school-level learning and examination programme.

Where was the 2026 International Mathematical Olympiad held?

The 67th IMO was held in Shanghai, China, from 10 to 21 July 2026, with the closing ceremony on 20 July.

How many countries and contestants joined IMO 2026?

The final official edition record lists 117 countries and 666 contestants. Early pre-event forecasts and some host news used different estimates.

How many gold, silver and bronze medals were awarded in IMO 2026?

The official 2026 results list 55 gold, 105 silver and 189 bronze medals. Medal thresholds that year were 29, 23 and 16 points respectively.

What is the maximum score at a standard IMO?

There are six problems, each worth up to seven points, giving a maximum of 42. Check the relevant edition’s rules for the official format.

Where will IMO 2027 take place?

The official IMO currently lists Budapest, Hungary, for 16–26 July 2027. This is not a registration deadline for an individual student.

Can a Class 1 pupil register directly for the national-team IMO?

No. The official contest is for selected secondary-school national delegates, subject to country-level criteria; the SCO Maths programme and free younger-grade resources are separate.

Can SCO Olympiad participation select me for the official IMO national team?

Not automatically. Eligibility and team selection depend exclusively on the recognised national mathematics Olympiad organisation for the student’s country.

How does a student in India begin the official IMO selection pathway?

Use the official MTA(I) IOQM 2026–27 portal and HBCSE mathematical Olympiad programme information. Progression can involve IOQM, RMO, INMO and later training/selection.

When is RMO 2026 in India?

HBCSE currently announces RMO for 15 November 2026, 13:00–16:00, with six proof-based questions. Confirm latest information on the official HBCSE page.

When is INMO 2027 in India?

HBCSE currently lists 17 January 2027, 12:00–16:30, with six proof problems in its 2026–27 cycle.

Where can I read real IMO previous-year problems?

Visit the official IMO problems archive and choose the desired year and official problem statement. Do not confuse a third-party paper bearing ‘IMO’ with this competition.

Are the 16 problems in this article real IMO past-paper problems?

No. They are original, guided proof exercises to develop number theory, geometry, algebra and combinatorics skills. They are not copies of IMO 2026 problems.

What is mathematical proof?

A proof is a logically valid explanation showing that a statement holds in all specified cases, not merely that it works in a few examples.

Which school topics are useful before attempting IMO-level problems?

Strong foundations in algebra, number theory, Euclidean geometry and counting are valuable; advanced IMO success also depends on original reasoning and extensive proof practice.

Do IMO medal awards belong to School Connect?

No. Official IMO medals and any IMSO or other independent competition medals belong to those organisers. They are included here only as global inspiration and accurate public information.

What does the IMSO ceremony video show?

It shows award recognition from the separate 2026 International Mathematics and Science Olympiad for younger pupils held in Indonesia. IMSO is not the secondary-school IMO.

Are SCO Maths class-wise videos and resources free?

School Connect provides free chapter-wise mathematics resources, linked educational videos and practice questions. Optional paid SCO exam registration is separate from free studying.

Does completing SCO Maths lessons guarantee an IMO medal or university admission?

No. Learning resources do not guarantee selection, scores, medals, scholarships or admission results.

What should parents and teachers do first?

Check whether the student wants everyday maths learning, a School Connect Maths assessment or a national-team IMO pathway; then use the correct organiser and age-appropriate resources.

Choose a suitable next mathematics learning step

For official national-team preparation: check the IMO’s own website, its problems archive and your national selection organiser before studying an actual Olympiad paper. For an Indian student, the HBCSE 2026–27 selection stages and MTA(I)’s IOQM information are authoritative.

For free Class 1–12 Maths learning: use School Connect’s free learning hub, its chapter-wise videos, age-appropriate independent SCO Maths Olympiad guide and STEM activities. These support understanding without requiring an official international team selection or buying a discontinued subscription plan.

Remember: learning about IMO medals and viewing authentic international ceremonies does not mean School Connect organises the official IMO. All organiser, results and awards links are credited to their own providers.