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International Mathematical Olympiad is one of the most prestigious international Olympiads, and probably the best-known Olympiad in mathematics. Numerous eminent mathematicians, Fields Medalists, and researchers have taken part in it. Hence, it is interesting to explore what predicts success in the IMO and how it relates to national measures of IQ. [1]
Methodology:
A naive methodology would be the following: count how many medals each country receives, and analyze the medals per capita relative to each country's population.
This may seem like a good approach on paper, but it has several issues. One of them is the structure of the IMO:
Each country selects up to six students to represent their nation annually. Usually, students are chosen through regional competitions such as the USAMO (via the AIME) in the United States. Thus, students who qualify for the IMO are likely the best of the best in their country.
However, the glaring problem is that large countries are "bottlenecked" by IMO selection. There may be a large pool of applicants if the population is large enough, but the IMO only allows each country to send six students. This rewards smaller countries in per capita measurements and doesn’t provide an "objective measure" of a country’s mathematical talent. If each country could send one person per million inhabitants, we could argue that we’d have a fairer per capita comparison—but six students is too limiting.
On the other hand, we can’t entirely dismiss the impact of a large population. If a country selects the best students, the larger the population, the more likely it is to have exceptional individuals. Therefore, we must consider another approach…
Let’s assume that the IMO is strongly correlated with some form of mathematical ability (not strictly IQ)—such as math knowledge, working memory, spatial reasoning skills, etc. While most of these are strongly correlated with IQ, for the sake of this discussion, let’s assume that this skill is independent of the g-factor (general intelligence). Let’s also assume the following:
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This mathematical ability is distributed across the population either normally (Gaussian) or following a power law.
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People at the far right tail of the distribution are overwhelmingly more likely to be selected for the IMO team. Given this, the distinction between a normal distribution and a power law becomes negligible for our purposes, as both decay exponentially (or faster) at the extremes.
For simplicity, let’s work with a normal distribution model. We may imagine something similar to the figure below:
Let’s design a probabilistic model for this!
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Innate Mathematical Ability: Each country has an underlying distribution of mathematical ability among its population. This includes both raw cognitive potential and acquired mathematical knowledge.
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National Selection: Each country conducts internal competitions to select its IMO participants. These national contests effectively simulate the IMO and are used to identify the top 6 students, i.e., those who achieve the highest scores or are most likely to earn medals.
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Selection Likelihood: Students with higher mathematical ability are overwhelmingly more likely to be selected for the national team and to win medals. However, this process includes some degree of randomness, luck, test-day performance, and other non-deterministic factors introduce noise.
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IMO Performance Variance: Once selected, students compete at the IMO. Their performance at the international level can vary, someone who ranked first nationally or earned a "gold-equivalent" in local competitions may end up receiving silver or bronze at the IMO. This is partly due to regression toward the mean, especially in cases where selection was influenced by favorable variance. However, for students who are extreme outliers (e.g., 4 or 5 standard deviations above the mean), this regression effect becomes less significant. Their innate ability is so high that variance has a reduced impact on outcomes.
Now, given our hypothetical distribution, we ask:
If we randomly sample a person from a given point on the distribution (e.g., someone 2σ, 3σ, or 4σ above the mean), what is the probability they will receive a medal at the IMO?
As expected, the answer is that the farther to the right you go (i.e., the higher the ability level), the higher the probability of receiving a medal, though this probability function is not deterministic, and some variance must be accounted for. This introduces a probabilistic mapping from ability level to medal likelihood.
Given the eligible population for the IMO (typically students aged 14 to 18), we can construct a representative agent or statistical archetype to represent the IMO team for a given year. In other words, given a top individual drawn from the ability distribution of a country, what is the expected probability distribution over medal outcomes (gold, silver, bronze, or no medal) for that individual?
Now that we have a rough framework for our model, let’s move forward with a concrete example.
let's pick some arbitrary country, ill chose UK (because UK is picked as a median point of 100 IQ in Lynn's studies [2]) and set the distribution mean at 0 For our model, we can therefore standardize the distribution and set the mean at 0 (μ = 0) and standard deviation at 1 (σ = 1).
We now ask the following question:
What is the Z-score threshold corresponding to each medal category (gold, silver, bronze) that best reproduces the historical IMO medal distribution of the UK?
Given the size of the eligible population (i.e., the number of 14–18-year-olds in the UK, I am assuming fixed proportion of roughly 8% for each country, because I am lazy to check every single population pyramid, and 8% of the population works on a rough level), we can estimate the expected Z-scores for the top 6 individuals (those selected for the IMO team). From there, we can derive the probabilities of each of them earning a medal, assuming a mapping from ability level (Z-score) to medal likelihood.
The goal is to find the Z-score thresholds that yield medal probabilities which, when applied to the top representatives from the UK, most closely match the country’s actual medal distribution at the IMO.
The estimated Z-score thresholds are approximately as follows: Gold = 5.18, Silver = 4.89, Bronze = 4.51, and Honorable Mention = 4.25, based on our assumptions. These values are quite extraordinary, even receiving an Honorable Mention at the IMO corresponds to an inferred ability level of approximately 4 standard deviations above the population mean.
To better quantify the distribution of medals, we introduce a numerical measure: the average rank (or AvgRank) associated with each medal category.
Let G be the number of gold medals a country has won, S for silver, B for bronze, and H for honorable mentions. If the AvgRank is low, it means the country predominantly receives gold medals; if it is high, it means the country receives more honorable mentions or no medals. Thus, AvgRank serves as a useful metric to capture the expected medal level for a country’s top performer.
To compare countries in terms of mathematical ability, we need to examine how changes in population size affect the medal distribution, and how much we would need to shift the mean of a country’s ability distribution to match its observed medal outcomes.
This can be done by solving for the mean in the Z-score formula for each country.
Results:
| Country | ObsAvgRank | Pop_millions | EligiblePop | InferredMu |
|--------------------------|-------------|--------------|-------------|--------------|
| Luxembourg | 3.322033898 | 0.634814 | 50785.12 | 0.62313824 |
| Iceland | 3.774193548 | 0.341243 | 27299.44 | 0.605532686 |
| Macau | 3.527272727 | 0.6823 | 54584 | 0.531946876 |
| Bulgaria | 2.291021672 | 6.71456 | 537164.8 | 0.476726748 |
| Hungary | 2.12371134 | 9.660351 | 772828.08 | 0.470368591 |
| Singapore | 2.484693878 | 5.87075 | 469660 | 0.428599937 |
| Estonia | 3.5 | 1.326535 | 106122.8 | 0.381855717 |
| North Korea | 1.869565217 | 25.778815 | 2062305.2 | 0.377675044 |
| Cyprus | 3.6 | 1.207359 | 96588.72 | 0.368483884 |
| Republic of Moldova | 3.125874126 | 2.657637 | 212610.96 | 0.361474912 |
| Armenia | 3.086666667 | 2.963243 | 237059.44 | 0.352660417 |
| North Macedonia | 3.333333333 | 2.083459 | 166676.72 | 0.344323049 |
| Mongolia | 3.050561798 | 3.278292 | 262263.36 | 0.342749084 |
| Romania | 2.107734807 | 19.237691 | 1539015.28 | 0.332032136 |
| Taiwan | 2.040609137 | 23.582 | 1886560 | 0.319837996 |
| Latvia | 3.458015267 | 1.886202 | 150896.16 | 0.316789192 |
| Hong Kong | 2.651515152 | 7.5007 | 600056 | 0.312235473 |
| Belarus | 2.554945055 | 9.398861 | 751908.88 | 0.303046414 |
| South Korea | 1.751173709 | 51.709098 | 4136727.84 | 0.293032751 |
| Slovakia | 2.902298851 | 5.459642 | 436771.36 | 0.287552683 |
| Israel | 2.595454545 | 9.2169 | 737352 | 0.287446069 |
| Slovenia | 3.519685039 | 2.100126 | 168010.08 | 0.271654812 |
| Georgia | 3.126506024 | 3.982 | 318560 | 0.269197197 |
| Lithuania | 3.388059701 | 2.722289 | 217783.12 | 0.261560773 |
| Russia | 1.477777778 | 145.912025 | 11672962 | 0.23287729 |
| Norway | 3.280701754 | 5.378857 | 430308.56 | 0.15045889 |
| Austria | 2.991266376 | 9.0064 | 720512 | 0.145302822 |
| New Zealand | 3.328767123 | 5.097164 | 407773.12 | 0.144779238 |
| Kosovo | 3.9375 | 1.877 | 150160 | 0.144581013 |
| Ukraine | 2.188172043 | 44.134693 | 3530775.44 | 0.130196775 |
| Czech Republic | 2.953216374 | 10.708981 | 856718.48 | 0.124564205 |
| Australia | 2.5 | 25.687041 | 2054963.28 | 0.115958442 |
| Albania | 3.767857143 | 2.837743 | 227019.44 | 0.112134124 |
| Kazakhstan | 2.704142012 | 18.776707 | 1502136.56 | 0.101484742 |
| Finland | 3.420689655 | 5.540718 | 443257.44 | 0.094639919 |
| Switzerland | 3.228346457 | 8.635 | 690800 | 0.071226202 |
| Canada | 2.41991342 | 38.005238 | 3040419.04 | 0.066123544 |
| Iran | 2.04109589 | 83.992953 | 6719436.24 | 0.063837594 |
| Denmark | 3.477477477 | 5.792202 | 463376.16 | 0.062798204 |
| Poland | 2.546052632 | 38.386 | 3070880 | 0.017009841 |
| Netherlands | 2.974093264 | 17.470267 | 1397621.36 | 0.012916275 |
| United States of America | 1.605960265 | 331.893745 | 26551499.6 | 0.007752475 |
| China | 1.214912281 | 1444.216107 | 115537288.6 | 0.002513635 |
| Portugal | 3.349514563 | 10.276617 | 822129.36 | -0.012459756 |
| United Kingdom | 2.337386018 | 67.886011 | 5430880.88 | -0.017210821 |
| Ireland | 3.797297297 | 4.994724 | 399577.92 | -0.019261533 |
| Germany | 2.250909091 | 83.783945 | 6702715.6 | -0.023110883 |
| Belgium | 3.31547619 | 11.589623 | 927169.84 | -0.023385945 |
| Japan | 2.078431373 | 125.836021 | 10066881.68 | -0.027486014 |
| Thailand | 2.387978142 | 69.799978 | 5583998.24 | -0.043070586 |
| Azerbaijan | 3.467213115 | 10.139177 | 811134.16 | -0.048722661 |
| Paraguay | 3.675 | 7.13253 | 570602.4 | -0.048792133 |
| Peru | 2.863013699 | 33.050325 | 2644026 | -0.073561987 |
| El Salvador | 3.80952381 | 6.486201 | 518896.08 | -0.081244924 |
| France | 2.622568093 | 65.273504 | 5221880.32 | -0.116591956 |
| Italy | 2.670157068 | 59.007743 | 4720619.44 | -0.117279466 |
| Turkey | 2.5 | 85.342242 | 6827379.36 | -0.12165023 |
| Malaysia | 3.12745098 | 32.365999 | 2589279.92 | -0.164060373 |
| Saudi Arabia | 3.092105263 | 34.813871 | 2785109.68 | -0.16868207 |
| Honduras | 3.916666667 | 9.904608 | 792368.64 | -0.208058707 |
| Argentina | 3.070063694 | 45.195777 | 3615662.16 | -0.21278441 |
| Chile | 3.7 | 19.107216 | 1528577.28 | -0.265194105 |
| Sri Lanka | 3.720430108 | 21.919 | 1753520 | -0.301474676 |
| Spain | 3.410071942 | 46.754778 | 3740382.24 | -0.340068017 |
| Ghana | 3.666666667 | 32.98 | 2638400 | -0.362634613 |
| Iraq | 3.571428571 | 40.220503 | 3217640.24 | -0.368731771 |
| Philippines | 3.04040404 | 109.581085 | 8766486.8 | -0.375500879 |
| Mexico | 2.962962963 | 128.932753 | 10314620.24 | -0.376269896 |
| Morocco | 3.642276423 | 36.91056 | 2952844.8 | -0.376891023 |
| Brazil | 2.733668342 | 212.559417 | 17004753.36 | -0.383948697 |
| Algeria | 3.571428571 | 43.851044 | 3508083.52 | -0.386047524 |
| Indonesia | 2.962686567 | 276.361783 | 22108942.64 | -0.5205246 |
| Bangladesh | 3.393258427 | 164.689383 | 13175150.64 | -0.576395871 |
| India | 2.579207921 | 1393.409525 | 111472762 | -0.670558903 |
| Pakistan | 3.628571429 | 225.199937 | 18015994.96 | -0.717266842 |
| Nigeria | 3.761904762 | 211.400708 | 16912056.64 | -0.754117741 |
Higher the inferredMu, better the country's "Mathematical ability", so this is the ranking from the best to worst.
I did not include countries with almost no medals, because most of them had either small populations, and also it is bit unreliable to infer actual medal distribution.
But now let's see how this data correlates with Lynn's IQ measurements [2].
r = 0.61 has R^2 of 0.37, so national IQs explain roughly 37% variance between nations being successful at the IMO!
Discussion:
My main point in this analysis was to see how well Lynn's measures predict success at arguably the most cognitively demanding Olympiad. While there is undeniably a significant correlation, one might naturally expect an even higher correlation. I was somewhat skeptical of this. Not because IMO performance is unrelated to IQ (g-factor), quite the opposite, but because I am a bit skeptical of Lynn's measures [2]. I believe that if Lynn’s measures were more accurate, they would have captured the correlation more strongly, rather than explaining only 37% of the variance.
My issue with Lynn’s studies is that he uses PISA scores as a strong proxy for IQ, which I don’t fully accept. While I do think there is a significant correlation between PISA and IQ (I’d estimate roughly r = 0.7), Lynn claims the correlation is r = 0.9 [3]. To me, that seems bit absurd, especially considering that even the WAIS’s own Cronbach’s alpha is roughly in that range. Additionally, there are other factors that may influence PISA scores. In my opinion, IMO performance is a more meaningful proxy than PISA for assessing the talent of a nation. While it may not reflect the average population’s ability as accurately, it does reflect a nation’s ability to produce top-level talent.
Let’s also discuss some potential limitations of my analysis.
1) Eligible population may be smaller than estimated.
We could argue that the eligible population is even smaller than just taking, say, 8% of the total population (essentially a demographic slice). The main argument here is that most IMO medalists come from urbanized cities, which provide better exposure, training, and opportunities compared to rural areas. So the actual demographic pool may be smaller. However, I don’t think this is a major concern, as the model’s goal is to offer a comparative analysis between countries rather than a detailed simulation controlling for every possible covariate.
2) How fair is the model? Are our assumptions flawed?
The model is not 100% precise—it’s essentially a toy model. For example, it may seem as though 8% of the population is actively competing regionally for the IMO, which would imply extremely high Z-scores. But obviously, that’s not the case. Most people are not interested in mathematics, do not possess extensive math knowledge, or have never even heard of the IMO.
That said, the model’s assumption of math ability implicitly includes all of this. In fact, I’m being generous—because if someone has actual innate aptitude for mathematics, they are more likely to be interested in math, more likely to have heard about the IMO, and much more likely (at the extreme) to qualify for the IMO. I think these are fair assumptions. Sometimes I like to draw an analogy: 17% of people taller than 7 feet in the U.S. are or were in the NBA (although from what i've heard this is bit of a myth). I strongly suspect the proportion is even higher when it comes to innate mathematical aptitude and the IMO, compared to height and basketball skill.
Also, because this model is comparative, I am being more liberal with the number of assumptions and less strict about controlling for covariates. The goal was to reasonably predict differences between countries—not to construct a detailed, high-fidelity simulation.
3) Repeat participation and demographic window.
Some participants qualify for the IMO multiple times. Doesn’t this skew the medal distribution for a country? Also, the eligible population is essentially a "sliding window" of the demographic.
Yes, repeat participation probably does skew results somewhat. For example, China almost never sends the same students twice, which might partly explain why China performs slightly lower than expected. But again, these are idealized measures intended for cross-country comparison, not a comprehensive academic study.
In conclusion, it is actually quite difficult to measure "IMO ability" due to the structure of the Olympiad itself. I’m not certain if there’s a more straightforward way to make these comparisons—maybe there is, and I haven’t thought of it yet. So we shouldn’t take this too seriously. While I believe these are rough but meaningful estimates, a much more meticulous analysis could certainly be conducted—controlling for various factors, refining assumptions, and using richer data.
References:
[1] https://www.imo-official.org/
[2] https://www.datapandas.org/ranking/average-iq-by-country
[3] Lynn, R., & Mikk, J. (2009). National IQs predict educational attainment in math, reading and science across 56 nations. Intelligence, 37(3), 305–310. https://doi.org/10.1016/j.intell.2009.01.002
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