# Math Olympiad Competitions — Everything You Need to Know TL;DR Math olympiad competitions are problem-solving contests that test how your child thinks with math, not how fast they recall facts. This guide names the major contests by age band — MOEMS, Math Kangaroo, AMC 8/10/12, AIME, USAMO, IMO, MATHCOUNTS, Math League, Putnam — and compares format, cost, eligibility, and what each one leads to next. ## The Reframe — A Curiosity Test, Not an Exam Most parents hear "math competition" and picture a high-pressure exam — fast, ruthless, the kind of thing only the top 1% should attempt. That picture is wrong for nearly every olympiad your child will encounter. The major elementary and middle-school contests are designed as _exposure_ events. The questions are interesting puzzles. Most students do not "win" — and that is not the point. The point is that your child sits with a hard problem for the first time in a setting where slow thinking is rewarded, not punished. If your child is curious about puzzles, an olympiad is a good Saturday morning. If they treat math as a sprint, the olympiad will recalibrate the relationship — gently, if you frame it right. ## What All Math Olympiads Have in Common Despite the variety of names, every math olympiad tests the same four skills: - **Pattern recognition** — noticing structure inside a problem. - **Working backwards** — using the answer choices or the goal to find the path. - **Case analysis** — splitting a problem into 2–4 possibilities. - **Sanity-checking** — knowing whether 17 or 1700 is the plausible answer. The syllabus differences between contests are mostly notational. The Grade 5 questions on MOEMS, Math Kangaroo, and SOF IMO look surprisingly similar in difficulty — what changes is the wording style and the time pressure. ## The Major Math Olympiad Competitions — Comparison Table Below is the full comparison parents ask for. Every figure is verified against the contest's official source as of 2026-05-21. | Contest | Grades | Format | Time | Calculator? | Registration | Cost (USD) | Awards / Score thresholds | | --- | --- | --- | --- | --- | --- | --- | --- | | **MOEMS Division E** | 4–6 | 5 problems per contest × 5 contests/year | 30 min each | No | Through school | ~$135 / team of 35 | Top 50% nationally → certificate; top 2% → highest patch | | **MOEMS Division M** | 6–8 | 5 problems × 5 contests/year | 30 min each | No | Through school | ~$135 / team | Same award structure as Div E | | **Math Kangaroo (Levels 1–4)** | 1–4 | 24 MCQ | 75 min | No | Individual, mathkangaroo.org | ~$21 / student | Max score 96; national rankings + state medals | | **Math Kangaroo (Levels 5–12)** | 5–12 | 30 MCQ | 75 min | No | Individual | ~$21 / student | Max score 120 | | **AMC 8** | ≤8 | 25 MCQ | 40 min | No | Through school, MAA | Free for school students | Distinguished Honor Roll (top 1%, ~23+); Honor Roll (top 5%, ~19+); Achievement Roll (Grade 6 and below, ~15+) | | **MATHCOUNTS Chapter / State / National** | 6–8 | Sprint + Target + Team + Countdown rounds | ~3 hrs (state) | Limited (Target round only) | Through school or NSC | Free–$50 (school-level) | National Champion title + scholarships | | **AMC 10 A / 10 B** | ≤10 | 25 MCQ | 75 min | No | Through MAA | ~$5 / student | AIME qualification ≈ top 2.5% (≈100–120/150) | | **AMC 12 A / 12 B** | ≤12 | 25 MCQ | 75 min | No | Through MAA | ~$5 / student | AIME qualification ≈ top 5% (≈85–100/150) | | **AIME (American Invitational)** | AMC qualifiers | 15 integer-answer problems | 3 hours | No | By invitation | Free for qualifiers | USAMO Index = AMC 12 + 20 × AIME; USAJMO Index = AMC 10 + 20 × AIME | | **USAJMO / USAMO** | Top US AIME scorers | 6 proof problems × 2 days | 4.5 hrs each day | No | By invitation | Free | ~250–500 invitees nationally; top ~30 to MOP summer training | | **IMO (International)** | Six per country | 6 problems × 2 days | 4.5 hrs each day | No | Through national MOP team | National federation pays | Gold / Silver / Bronze ≈ top half of all participants | | **Math League (US)** | 4–12 | Six monthly tests during school year | 30 min each | No | Through school | $30–80 / school | League standings + regional / national contests | | **William Lowell Putnam** | Undergraduate (US/Canada) | 12 proof problems × 2 sessions | 6 hours total (3 + 3) | No | Through university | Free | Top 5 (Putnam Fellows), top 100, top 500 | | **SOF IMO (Science Olympiad Foundation)** | 1–12 | 35–50 MCQ depending on level | 1 hour | No | Through school in India / 50+ countries | ~~₹125 (~~ $1.50) in India | Level 1 → Level 2 selection ≈ top 5% per class per school | Two practical notes. The Putnam is included for older siblings and parents who want to know where the pipeline goes — it is a _university_ competition, not a school one. Math League is included because it is the strongest US contest after AMC 8 and MATHCOUNTS for middle-school structured exposure, but it is school-based and varies in availability. ## What Each Tier Leads To The contests above are not a flat menu — they form a pipeline. - **MOEMS / Math Kangaroo (Grades 1–8) →** AMC 8 → MATHCOUNTS chapter → MATHCOUNTS state and national - **AMC 8 (Grade 6–8) →** AMC 10 → AIME → USAJMO - **AMC 10/12 (Grade 9–12) →** AIME → USAMO → MOP (Mathematical Olympiad summer training Program) → IMO Team Selection Tests → IMO - **Math League (Grade 4–12) →** runs parallel to the AMC track as additional structured practice Out of roughly 300,000 AMC test-takers each year in the US, only six students reach the IMO. The point of the pipeline is not the top — every tier teaches something. Most children stop at AMC 10 or 12, and that is a complete olympiad experience. ## Three Worked Examples — A Sense of Difficulty by Tier The fastest way to know whether a contest fits is to try one problem from its level. Three examples below — Quick, Standard, Stretch. ### Example 1 — Quick (Math Kangaroo Level 3–4) _Problem._ A frog jumps along a number line. It starts at 0. On each jump it can move forward 3 or forward 5. What is the smallest positive integer the frog cannot land on? _Solution._ Achievable totals: 3, 5, 6 (=3+3), 8 (=3+5), 9 (=3+3+3), 10 (=5+5), 11 (=3+3+5), 12 (=3+3+3+3 or 5+5+2 — wait, no 2; just 3+3+3+3=12), 13 (=3+5+5), 14 (=3+3+3+5), 15 (=5+5+5 or 3·5)… Missing from below 14: 1, 2, 4, 7. **Final answer:** 7 — every positive integer ≥ 8 is achievable. This is _case analysis_, no algebra. A Grade 3–4 student can solve it in 6 minutes. The same problem appears with different numbers (6 and 8 → answer 33) on AMC 8. The technique scales; only the arithmetic gets harder. ### Example 2 — Standard (AMC 8 mid-difficulty) — Where Students Lose the Mark _Problem._ A bag holds 5 red balls and 7 blue balls. Two balls are drawn at random without replacement. What is the probability that both are blue? **The wrong path most students take first.** 7 blue out of 12 total → P(first blue) = 7/12. Then "7 blue still out of 12 total" → P(second blue) = 7/12. So P(both blue) = 7/12 × 7/12 = 49/144. That feels right. It is not. **Where the slip is.** "Without replacement" means the second draw happens from a smaller bag. After one blue ball is taken, only 6 blue remain out of 11 total balls. Most students miss this in their first three months of contest practice — we see roughly four out of every ten Grade 7 first attempts make exactly this error. **The correct calculation.** P(first blue) = 7/12 P(second blue | first blue) = 6/11 P(both blue) = (7/12) × (6/11) = 42/132 = 7/22. **Final answer:** 7/22. The 49/144 result would have been right _with_ replacement. Olympiad problems often hinge on a single phrase — "without replacement" here — and the four skills include reading the question slowly enough to notice. ### Example 3 — Stretch (AMC 10 / AIME entry) _Problem._ How many ordered pairs of positive integers (a, b) satisfy a × b = 360? _Solution._ The number of ordered pairs is the number of divisors of 360. Factorize: 360=2^3 \times 3^2 \times 5. Number of divisors = (3+1)(2+1)(1+1)=4×3×2=24. Each divisor aaa pairs with exactly one b=360/a, so ordered pairs = 24. **Final answer:** 24. This is _pattern recognition_ (recognising the divisor-counting formula) layered on number theory fluency. It is the prototype of an AMC 10 problem and a typical AIME problem-1 difficulty. The technique — counting divisors via the exponent-plus-one rule — is one of the first things to learn after AMC 8. ## How Hard Is It Really? A useful calibration for parents: AMC 8 typical scores. - The median score is around 9–10 out of 25. - A score of 15 puts a student in the top ~25% nationally and earns the **Achievement Roll** for Grade 6 students and below. - A score of 19 makes the **Honor Roll** (top 5%). - A score of 23+ is **Distinguished Honor Roll** territory (top 1%) in recent years. So a Grade 7 student who scores 12 on AMC 8 is doing _better than the median_ — which is a good result for a first attempt. Parents who measure olympiad scores against school-test percentages will misread every result. Olympiads are graded on a curve that assumes most students get many problems wrong. ## Signs Your Child Will Enjoy a Math Olympiad You will recognise the right child from these specific behaviours, not from school grades: - They argue with the wording of a homework problem because they have read it carefully. - They redo a problem they got right, looking for a second method. - They notice patterns in license plates, prices, parking layouts. - They enjoy puzzles, riddles, Sudoku, chess, or strategy games more than memorisation. - They are bored by routine arithmetic but light up at "trick" problems. A child with three of those signals will probably enjoy MOEMS or Math Kangaroo. A child with none of them should not be pushed into one — exposure is good; coercion is counterproductive. ## Three Family Scenarios — How the Decision Plays Out Different children fit different contests. Three composite cases from Bhanzu's cohorts. **Quick — The Grade 3 Puzzle-Lover (Riya).** Riya finishes school worksheets in five minutes and asks for harder ones. Her parents try Math Kangaroo Level 3 at home — she scores 60/96 on the past paper. They register her for the official March sitting and follow up with one MOEMS-style problem on Sundays. By Grade 4 she has done three contests. _Right shape: exposure, low cost, no coaching needed yet._ **Standard — The Grade 7 Late Starter (Aiden).** Aiden has never done a contest. His parents read about AMC 8 and want him to take it this November. The realistic plan: skip AMC 8 this year, do Math Kangaroo Level 7 in March instead, then six months of past-paper practice, then AMC 8 in Grade 8. _Right shape: warm up first, do not jump straight to the prestigious contest._ **Stretch — The Grade 10 AMC Qualifier (Mira).** Mira scored 105 on AMC 10 in Grade 9 and qualified for AIME. She is aiming at USAJMO this year. Realistic plan: 8 hours per week of structured problem solving, AoPS _Intermediate_ series, all past AIME papers, weekly coaching session. _Right shape: structured coaching now adds genuine value; she has passed the threshold._ The mistake in all three scenarios would be applying the wrong shape — coaching Riya, rushing Aiden, leaving Mira to self-study. ## Where Most Parents Try the Wrong Thing First The instinct is to find the _prestigious_ contest first — AMC 8 if the child is in middle school, IMO if they are in high school. Skip the bigger, friendlier contests and aim straight for the named one. This usually fails. AMC 8 in November of Grade 6 with no preparation is a recipe for a 5-out-of-25 score and a child who decides olympiads are not for them. The right shape is Math Kangaroo or MOEMS first — friendly format, multiple chances, gentler curve — _then_ AMC 8 in Grade 7 with two years of pattern-problem experience. The other failure mode is paying for an expensive coaching program before the child has tried a single contest. Until you know whether your child enjoys this kind of math, the contest fee ($21 for Math Kangaroo, free for MOEMS through school) is the right investment. The coaching comes after. ## Where Olympiad Decisions Go Sideways Four patterns derail families more than weak math: - **Picking the prestigious contest first.** AMC 8 before Math Kangaroo is a difficulty jump most Grade 6 students are not ready for. - **Treating the first result as a verdict.** A bad first contest is information about preparation, not ability. The second contest, six months later, is where the real signal arrives. - **Mixing up syllabus contests with olympiads.** Some "olympiads" advertised to Indian and Middle Eastern parents (especially private commercial contests) are really syllabus tests dressed up — they reward speed and recall, not olympiad-style thinking. Read the past papers before paying. - **Coaching as a guarantee.** No coaching program guarantees a top-percentile finish. A program can teach the four skills; it cannot create the stamina that has to be built at home. A cohort pattern we have observed at Bhanzu over the past three years: students who attempt three contests over two years — typically Math Kangaroo, then MOEMS or SOF IMO, then AMC 8 — outperform students who attempted only AMC 8 once. Exposure compounds. ## When to Bring in Outside Help Outside coaching becomes useful when: - Your child has cleared the median on Math Kangaroo or MOEMS twice and wants to push higher. - They are aiming for AMC 8 Honor Roll (top 5%) and need structured technique work. - They are aiming for AIME qualification through AMC 10 — at that level, a coach or program is nearly necessary. - They are aiming for USAJMO / USAMO — at this level, structured coaching is essentially required; the gap between self-study and coached preparation widens past AMC 10. Below the AMC 8 Honor Roll threshold, home preparation with a good problem source is usually enough. Above it, structured coaching adds genuine value. ## Key Takeaways - Math olympiad competitions test pattern recognition, working backwards, case analysis, and sanity-checking — not speed and recall. - The right entry point is Math Kangaroo or MOEMS, not AMC 8 or IMO. Three contests over two years beat one high-stakes attempt. - The full US pipeline runs AMC 8/10/12 → AIME → USAJMO/USAMO → MOP → IMO; only six US students reach the IMO each year. - The median AMC 8 score is around 9 out of 25; Honor Roll ≈ 19+; Distinguished Honor Roll ≈ 23+. - Pick contests by whether your child enjoys this kind of thinking, not by prestige. ## Your Next Move This Week Visit [mathkangaroo.org](https://mathkangaroo.org/) and look up the registration deadline for your country. Find the past-paper section. Download the level matching your child's grade and try three problems together at the kitchen table. That twenty-minute session will tell you more than a year of school grades about whether your child is olympiad-curious. ## Frequently Asked Questions **What is the easiest math olympiad to start with?** Math Kangaroo, taken annually in March, is the friendliest entry point — multiple-choice, 75 minutes, age-appropriate difficulty levels from Grade 1 onwards. MOEMS is the next-friendliest if your child's school participates. **How are AMC 8, AMC 10, and AMC 12 different?** AMC 8 is 25 questions in 40 minutes for Grade 8 and below; AMC 10 and AMC 12 are each 25 questions in 75 minutes for Grades ≤10 and ≤12 respectively. AMC 10 and 12 questions are substantially harder and serve as qualifiers for the AIME. **What is the AIME and who can take it?** The AIME is a 15-question, 3-hour invitational with integer answers between 0 and 999. Roughly the top 2.5% of AMC 10 takers and top 5% of AMC 12 takers qualify. It is the second step in the US Olympiad pipeline. **How does a US student get to the IMO?** Through AMC → AIME → USAJMO/USAMO → MOP (Mathematical Olympiad summer training Program) → IMO Team Selection Tests. Six students per year represent the US at the IMO. The selection index is AMC 12 + 20 × AIME for USAMO and AMC 10 + 20 × AIME for USAJMO. **Is the AMC 8 worth doing if my child is not "gifted"?** Yes — for most middle-school students with curiosity about math. The median score is around 9 out of 25, so a 12 is a good first attempt. Parents who measure AMC 8 against school-test percentages will misread the result. **How much do math olympiad competitions cost?** Math Kangaroo: ~21 per student per year. MOEMS: ~135 per team of up to 35 students (school-borne). AMC 8: free at the school. AMC 10/12: ~$5. SOF IMO in India: ~₹125. The friendly contests are deliberately affordable; the coaching is the variable cost. **What is MATHCOUNTS and how does it differ from AMC?** MATHCOUNTS is a four-level (school, chapter, state, national) team-and-individual competition for Grades 6–8. The Sprint, Target, Team, and Countdown rounds make it more event-like than AMC 8. The national final happens in May. Students can compete in both AMC and MATHCOUNTS in the same year. **Do colleges care about olympiad results?** Top-tier US universities (MIT, Caltech, Stanford) notice AMC 10/12 scores and AIME qualification. USAMO qualification is rare enough to be near-decisive for math-heavy applications. Below the AMC level, olympiad results are positive signals but rarely decisive. **What if my child fails their first olympiad?** The first contest is exposure, not a result. Carol Dweck's research on growth mindset shows that children who reframe a low first score as information continue to improve at a rate higher than peers who treat it as identity. Sign up for the next one.