## wp18268

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### The International Mathematics Olympiads (IMO): structure and notable facts
- Annual competition since 1959.
- Participants:
  - High school students younger than 20 years and not enrolled at a tertiary education institution.
  - National teams up to six participants.
  - Some repeat participants; majority compete once.
- Contest format:
  - Six problems from geometry, number theory, algebra and combinatorics.
  - Each problem worth seven points; maximum score 42 points.
  - Medals awarded on sum of points; slightly fewer than half receive medals (gold, silver, bronze).
  - “Honourable mention” (perfect solution to one problem for non-medalists) awarded since 1987.
- Participation history and notable achievements:
  - Participation expanded to over 100 countries; U.K. and France joined in 1967; U.S. in 1974; China in 1985.
  - Of the 26 Fields medals awarded between 1994 and 2018, 14 went to former IMO medalists.
  - Examples: Maryam Mirzakhani (IMO gold, perfect score; first woman to win the Fields medal), Terence Tao (gold medal at 29th IMO; Fields medalist), Grigori Perelman (IMO gold; solved Poincaré conjecture).

### Data: construction, coverage, and key sample statistics
- Primary construction:
  - Official IMO website: names, country, year, points by problem, medal type.
  - Selected participants from 1981–2000 inclusive; for repeat participants only last participating year kept.
  - Final IMO participant list: 4,710 individuals.
- Long-term outcomes and ancillary data:
  - PhD data from Mathematics Genealogy Project; bibliometrics from MathSciNet (manual author disambiguation).
  - IMC invited speakers and Fields medalists tagged.
  - Manual online employment search for IMO medalists (n = 2,272) into five categories: (1) mathematics academia, (2) non-mathematics academia, (3) IT industry, (4) finance industry, (5) another industry.
  - Ancillary dataset: PhD graduates in Math Genealogy Project graduating 1990–2010 (n = 89,086).
- Final database fields for IMO participants (1981–2000; n = 4,710):
  - IMO year, country, points, medal type.
  - Indicator for PhD in mathematics; PhD year and school.
  - Publications and cites counts until 2015.
  - Indicators: IMC speaker, Fields medalist.
  - For math PhDs: indicator for graduating from a top ten school (proxied by Shanghai 2010 mathematics rankings).
- Descriptive statistics (exact figures reported):
  - Medal shares: around 8% gold, 16% silver, 24% bronze; 10% honourable mention.
  - Around 22% hold a PhD in mathematics; of those around a third have PhD from a top 10 school.
  - 1% became IMC speakers.
  - 0.2% became Fields medalists.
  - Collective output: more than 15,000 publications and more than 160,000 cites.
  - Country of origin: around half high-income (2000 World Bank), 23% upper middle income, 16% lower middle-income, 11% low-income.
  - Historical gold medal counts in sample: China 54, USSR/Russia 43, U.S.A. 27, Romania 26; Germany, Bulgaria, Iran, Vietnam, the U.K., Hungary and France also >10 gold medalists.

### How much does teenage talent affect long-term performance?
- Research question: whether teenage talent (IMO scores) correlates with long-term mathematics outcomes (PhD attainment, publications, cites, IMC invitations, Fields medals).
- Interpretation considerations:
  - IMO problems differ from research problems: known solutions, short time frame, no literature or collaboration.
  - Measurement error possible (luck, extraneous factors).
  - Possibility of causal effects of IMO performance via confidence, access, or signaling.

#### Graphical and regression evidence (Section 4.1)
- Graphical patterns (Figure 3): higher IMO scores associated with higher likelihoods of PhD, PhD from top school, and higher publications and cites; noisier but positive patterns for IMC speaker and Fields medal.
- Individual-level OLS regression specification: Y_it = β IMOscore_it + δ X_it + ε_it, with cohort and country fixed effects.
- Regression associations (Table 4; exact associations):
  - Each additional IMO point (out of 42) associated with:
    - a 1 percent point increase in likelihood of obtaining a Ph.D.;
    - a 2.6 percent increase in publications;
    - a 4.5 percent increase in citations;
    - a 0.1 percent point increase in likelihood of becoming an IMC speaker;
    - a 0.03 percent point increase in likelihood of becoming a Fields medalist.
  - IMO scores alone explain around 8% of variation in math PhD, publications and citations.
- Problem difficulty: scores on more difficult problems (3 and 6) have larger coefficients than on less difficult problems (1, 2, 4, 5) (Appendix Table A2).

#### Intensive margin: link conditional on obtaining a PhD (Section 4.2)
- Sample restricted to IMO participants with a mathematics PhD: n = 1,023.
- Table 5, panel A (controls: country FE, olympiad year FE):
  - Conditional on a PhD, each additional IMO point associated with:
    - a 2% increase in publications;
    - a 4% increase in cites;
    - a 0.2 percentage point increase in propensity to become an IMC speaker;
    - a 0.07 percentage point increase in propensity to become a Fields medalist.
  - Estimates hardly lower than unconditional Table 4.
- Table 5, panel B (adds graduate school by olympiad year FE):
  - Positive correlation remains; point estimates somewhat lower for publications and cites, similar for IMC speaker and Fields medal.

#### Investigating causal effects via medals and regression discontinuity (Section 4.3)
- Motivation: medals awarded by explicit score cutoffs and act as public signals.
- RD design:
  - y_it = α + β AboveThreshold_it + δ1(IMOScore_it − Threshold_t) + δ2 AboveThreshold_it*(IMOScore_it − Threshold_t) + λ X_it + ε_it.
  - AboveThreshold_it indicates at-or-above medal threshold; β is causal parameter of interest.
  - Pool across gold, silver, bronze thresholds by creating three copies per observation; use optimal bandwidth selector (Calonico, Cattaneo & Titiunik 2014).
- Manipulation checks:
  - Running variable not manipulable; medal thresholds unknown during solving.
  - Frandsen (2017) test p-value = 0.974 (k = 0.02) — no manipulation detected.
- RD results (Table 6 summary):
  - Above (better) medal threshold estimates imprecisely estimated and not statistically significant:
    - Math PhD 0.0138 (0.0250)
    - Math PhD (top 10) 0.0147 (0.0142)
    - Pubs (log) 0.0117 (0.0719)
    - Cites (log) 0.0114 (0.1043)
  - Distance from threshold coefficients significant and positive:
    - Math PhD 0.0098 (0.0032) ∗∗∗
    - Math PhD (top 10) 0.0038 (0.0015) ∗∗
    - Pubs (log) 0.0202 (0.0104) ∗
    - Cites (log) 0.0385 (0.0133) ∗∗∗
- Interpretation: point estimates for causal effect of receiving a better medal are imprecisely estimated; evidence points to underlying talent (points scored) rather than medal per se driving long-term outcomes.

### Effect of receiving a better medal versus underlying talent
- Controlling for score, being awarded a better medal appears to have no additional impact on becoming a professional mathematician or future knowledge production.
- Complementary evidence:
  - Honourable mentions do not have a causal effect on long-term performance (Appendix Table A1).
  - Positive gradients between points scored and long-term performance within medal bins:
    - Gold medalists: points positively correlated with outcomes (Table A4 panel A).
    - Bronze medalists: same positive correlation.
    - Silver medalists: significant correlation for two of four outcomes.
- Interpretation: link reflects differences in underlying talent rather than causal effect of medal.

### Link between IMO score and long-term performance by country income group
- Countries grouped by 2000 World Bank classification; about half sample from low- and middle-income countries.
- Participants from developing countries do not score lower at IMO than from developed countries (Table A5 shows Low-income coefficient 3.206*** (0.583) on IMO Score).
- Graphical pattern (Figure 6): for a given number of points, share obtaining a PhD in math is typically highest for high-income, then upper middle-income, then lower middle-income, then low-income countries.
- Regression specification: Y_it = β1 IMOscore_it + β2 CountryIncomeGroup_i + η_t + ε_it (country income indicators: low-income, lower middle-income, upper middle-income; high-income omitted).
- Main quantitative findings (Table 7; cohort FE; N = 4,710):
  - Low-income:
    - Math PhD −0.152 (0.017) ∗∗∗
    - Math PhD (top 10) −0.031 (0.012) ∗∗∗
    - Pubs (log) −0.337 (0.041) ∗∗∗
    - Cites (log) −0.560 (0.067) ∗∗∗
  - Lower middle-income:
    - Math PhD −0.101 (0.014) ∗∗∗
    - Math PhD (top 10) −0.022 (0.009) ∗∗
    - Pubs −0.194 (0.034) ∗∗∗
    - Cites −0.321 (0.057) ∗∗∗
  - Upper middle-income:
    - Math PhD −0.040 (0.017) ∗∗
    - Math PhD (top 10) −0.025 (0.010) ∗∗
    - Pubs −0.083 (0.041) ∗∗
    - Cites −0.171 (0.066) ∗∗∗
  - IMO Score main effects remain positive and significant across outcomes: 0.011 (0.001) ∗∗∗; 0.005 (0.000) ∗∗∗; 0.027 (0.001) ∗∗∗; 0.045 (0.002) ∗∗∗.
- Robustness:
  - Similar results when replacing country income groups with GDP per capita (Table A6) or decile indicators (Table A7).
  - Interaction (Table 9): low-income × IMO Score negative, indicating larger penalty for higher-scoring individuals from low-income countries (e.g., Low-income × IMO Score −0.003 (0.001) ∗∗).
- Productivity conditional on having a PhD (Table 10):
  - Point estimates for income groups negative but not significant (or marginally significant for low-income).
  - With IMO year × graduate school FE (panel B) estimates positive though not significant (except upper middle-income in cites).
  - Interpretation: extensive margin (getting a PhD) may matter more than intensive margin (productivity conditional on PhD) for country-origin differences.

### Cross-country differences in IMO training and migration
- Training proxy — repeat participation:
  - Participants from low- and middle-income countries are less likely to have multiple IMO participations than those from developed countries (Table 8 column 1).
- Migration and PhD location (Table 11):
  - Difference in propensity to do a PhD between high and low/middle-income countries driven entirely by propensity to do a PhD at home.
  - Low-/middle-income participants significantly less likely to do a PhD in their home country; no corresponding increase in PhD abroad.
- Controls for home country capacity:
  - Indicator: home country has at least one university ranked among the 100 best in mathematics (Shanghai 2009).
  - Indicator: production of mathematics articles in home country (log).
  - Including these reduces coefficients for low- and middle-income countries.
- Interpretation: weak home-country research and training capacity helps explain lower conversion of developing-country medalists into knowledge producers.

### Time trends: is the country-of-origin penalty decreasing?
- Specification includes interaction of country income group with ‘late’ cohorts (1991–2000 vs 1981–1990).
- Results (Table 12):
  - Interaction late × low-income positive and significant for three of four outcomes: penalty for low-income countries has decreased over time.
  - Interaction late × upper middle-income negative: gap between upper middle-income and high-income countries may have increased.
- Graphical evidence (Figure 7):
  - Difference in share getting a PhD between high- and low-income countries by five-year bands and medal type always positive.
  - For bronze and silver medalists difference has considerably diminished over time.
  - For gold medalists there is no decrease.
- Overall: coming from a low-income country is less detrimental than before for becoming a professional mathematician and producing mathematical knowledge, but the top (gold medalists) gap persists.

### Quantifying the size of lost knowledge production
- Aggregate counterfactual (Table 13): multiply country income group coefficients by group shares and aggregate.
  - Low-income share = 0.11; Coeff (pubs) = −0.337; Coeff (cites) = −0.561; Loss (pubs) = −0.037; Loss (cites) = −0.062.
  - Lower middle-income share = 0.23; Loss (pubs) = −0.045; Loss (cites) = −0.075.
  - Upper middle-income share = 0.23; Loss (pubs) = −0.019; Loss (cites) = −0.039.
  - Total weighted loss: pubs −0.101; cites −0.176.
- Interpretation: knowledge production could be 10% higher (publications) and 17% higher (cites) if developing-country participants produced at same rate as high-income counterparts.
- Caveats:
  - Costs to enable such increases unquantified and potentially substantial.
  - Potential crowding-out or displacement effects.
  - Induced shifts away from other valuable occupations could mitigate welfare costs.

### Comparing IMO participants with other mathematicians
- Comparison samples constructed:
  - All PhD students obtaining PhD in mathematics between 1990 and 2010: n = 89,068.
  - PhD graduates from top 10 schools: n = 9,049.
  - IMO bronze/silver medalists with math PhD: n = 520.
  - IMO gold medalists with math PhD: n = 145.
- Outcomes (Figure 8 and Table 14):
  - Medalists, especially gold medalists, outperform other PhD graduates and top-10 graduates in publications, citations, IMC speaking invitations, and Fields medals.
  - Differences larger for exceptional achievements (IMC, Fields) than routine outputs.
  - Regression results (Table 14; N = 89,068):
    - Gold medalist coefficients sizeable: Pubs (log) 1.2597 (0.1087) ∗∗∗; Cites (log) 2.1425 (0.1764) ∗∗∗; IMC speaker 0.0799 (0.0231) ∗∗∗; Fields medalist 0.0333 (0.0147) ∗∗.
    - Conditional probability: IMO gold medalist fifty times more likely to become a Fields medalist than a PhD graduate from a top 10 program (interpretation from paper’s text).
- Sorting across graduate schools (Figure 9):
  - Medalists cluster in top schools; 36% of IMO medalists who get a PhD graduate from top 10 schools.

### Careers of IMO participants outside mathematics (Section 6.3)
- Occupation coding for medalists (n = 2,272): mathematics academia, academia outside mathematics, finance industry, IT industry, other industry, no online profile.
- Regression results on occupations (Table 15):
  - Low- and middle-income medalists significantly less likely to be in mathematics academia.
  - Low-income medalist effects:
    - Academia (math) −0.111 (0.026) ∗∗∗
    - Finance industry 0.031 (0.017) ∗
    - No online profile 0.087 (0.033) ∗∗∗
  - IMO Score associations with current occupation:
    - Academia (math) 0.007 (0.001) ∗∗∗
    - Academia (not math) 0.002 (0.001) ∗∗∗
    - No online profile −0.009 (0.001) ∗∗∗
- Interpretation:
  - Many developing-country medalists are less likely to be in math academia and more likely to have no online presence.
  - The paper frames allocation in terms of comparative advantage and leans toward under-utilization of developing-country talent in mathematics being likely inefficient, while acknowledging countervailing displacement or spillover effects.

### Main conclusions and policy-relevant implications
- Two central findings:
  - Strong, consistent link between IMO scores and later achievements: PhD attainment, publications, cites, IMC invitations, Fields medals.
  - IMO participants from low- and middle-income countries produce less mathematical knowledge than equally talented peers from high-income countries.
- Magnitude: within extreme-right tail, small differences in IMO performance translate into sizeable differences in long-term achievements.
- Aggregate impact: substantial lost knowledge production attributable to cross-country productivity differences among IMO participants (approximate totals: pubs −0.101; cites −0.176 in Table 13 aggregation).
- Suggested supply-side policies (push programs):
  - Fellowships for high-end talent to study mathematics at undergraduate and/or graduate levels.
  - Top schools should encourage applications from developing countries.
  - Strengthening mathematics research and training capacity in developing countries.
- Gaps and further research:
  - Paper does not precisely identify why developing-country participants are less likely to become professional mathematicians; home-country research capacity, preferences, private incentives, and relative pay may matter.
  - Similar under-utilization may exist in other disciplines (biomedicine, computer science) but warrants further investigation.

_Italic: Source — wp18268 (IMF working paper content unit as provided)._

### Section 3 presents the data.  The results on the link between IMO success and long-term

### wp18268 - Section 3 presents the data.  The results on the link between IMO success and long-term

### The International Mathematics Olympiads (IMO): structure and notable facts
- The IMO is a competition held annually since 1959.
- Participants:
  - Are high school students younger than 20 years of age and not enrolled at a tertiary education institution.
  - Travel as part of a national team; national federations select up to six participants per country.
  - Some participants compete in multiple successive years, but the majority compete once.
- Contest format:
  - Six problems drawn from geometry, number theory, algebra and combinatorics.
  - Each problem is worth seven points; maximum score is 42 points.
  - Medals are awarded solely on the sum of points. Slightly fewer than half of participants receive medals (gold, silver, bronze).
  - An “honourable mention” recognizing a perfect solution to one problem has been awarded to non-medalists since 1987.
- Participation history and notable achievements:
  - Participation expanded over time to include over 100 countries.
  - The United Kingdom and France joined in 1967; the U.S. joined in 1974; China joined in 1985.
  - Of the 26 Fields medals awarded between 1994 and 2018, 14 went to former IMO medalists.
  - Examples of prominent former IMO participants: Maryam Mirzakhani (IMO gold, perfect score; first woman to win the Fields medal), Terence Tao (gold medal at 29th IMO; Fields medalist), Gregori Perelman (IMO gold; solved Poincaré conjecture; declined Fields medal and Millennium Prize).

### Data: construction, coverage, and key sample statistics
- Primary sources and construction:
  - Started with official IMO website: http://www.imo-official.org (names, country represented, year, points by problem, medal type).
  - Selected participants who participated between 1981 and 2000, inclusive.
    - For participants competing multiple years, only the last participating year was kept.
    - Final IMO participant list: 4,710 individuals.
  - Long-term outcomes:
    - PhD information from the Mathematics Genealogy Project (coverage >200,000 mathematicians).
    - Bibliometric data from MathSciNet: total publications and cites by author (manual author disambiguation by Mathematical Reviews).
    - IMC (International Mathematics Congress) invited speakers list and Fields medalists were tagged.
  - Manual online search for current employment measures was performed only for IMO medalists (2,272 people), coded into employment categories: (1) mathematics academia, (2) non-mathematics academia, (3) the IT industry, (4) the finance industry, (5) another industry.
  - Ancillary dataset: all PhD graduates in the Math Genealogy Project graduating between 1990 and 2010 (n=89,086) with school, year, publications, cites, IMC speaker, and Fields medalist indicators.
- Final database content for IMO participants (1981–2000; n=4,710):
  - IMO participation details: year, country, points scored, medal type.
  - Whether the person holds a PhD in mathematics, PhD year and school (if applicable).
  - Mathematics publications and cites counts until 2015.
  - Indicators for being an IMC speaker and being a Fields medalist.
  - For those with a mathematics PhD, an indicator for graduating from a top ten school (proxied by graduating from one of the top ten schools in the Shanghai 2010 mathematics rankings).
- Descriptive statistics (exact figures as reported):
  - Medal shares: around 8% gold, 16% silver, 24% bronze; a further 10% have an honourable mention.
  - Around 22% of IMO participants hold a PhD in mathematics; of those around a third have a PhD in mathematics from a top 10 school.
  - 1% of IMO participants became IMC speakers.
  - 0.2% became Fields medalists.
  - Collective output of the sample: more than 15,000 publications and more than 160,000 cites.
  - Country of origin (proxied by country represented at the IMO): around half from high-income countries (2000 World Bank classification), 23% from upper middle income, 16% from lower middle-income, and 11% from low-income countries.
  - Historically most successful countries in sample (gold medal counts): China 54, USSR/Russia 43, U.S.A. 27, Romania 26. Germany, Bulgaria, Iran, Vietnam, the U.K., Hungary and France also have more than 10 gold medalists in the sample.

### How much does teenage talent affect long-term performance?
- Research question: whether teenage talent (proxied by IMO scores) correlates with long-term performance in mathematics (PhD attainment, publications, cites, IMC invitations, Fields medals).
- Considerations on interpretation:
  - IMO problems differ from research problems (solutions are known; short time frame; no literature or collaboration).
  - Potential measurement error if luck or extraneous factors affect IMO scores.
  - Possibility that IMO performance causally affects later outcomes (through confidence, access to schools, signaling via medals).

#### 4.1 Link between IMO score and long-term performance (graphical and regression evidence)
- Graphical evidence (Figure 3): mean achievements by IMO points (linear fit) for six outcomes—PhD in mathematics; PhD from a top 10 school; publications (logs); cites (logs); IMC speaker; Fields medal.
  - First two rows: clear positive gradient—higher IMO scores associated with higher likelihood of PhD and PhD from a top school, and higher publications and cites.
  - Bottom rows (rare outcomes): noisier but broadly similar positive pattern.
- Individual-level regression specification:
  - Y_it = β IMOscore_it + δ X_it + ε_it, where X_it includes cohort (Olympiad year) fixed effects and country of origin fixed effects.
- Regression associations (Table 4; exact reported associations):
  - Each additional point scored at the IMO (out of 42) is associated with:
    - a 1 percent point increase in likelihood of obtaining a Ph.D.;
    - a 2.6 percent increase in publications;
    - a 4.5 percent increase in citations;
    - a 0.1 percent point increase in the likelihood of becoming an IMC speaker;
    - a 0.03 percent point increase in the likelihood of becoming a Fields medalist.
  - IMO scores alone explain around 8% of the variation in math PhD, publications and citations (not shown on the table).
- Additional evidence on problem difficulty (appendix Table A2): scores on more difficult problems (problems 3 and 6) have consistently larger coefficients than scores on less difficult problems (problems 1, 2, 4, 5), though the latter also tend to be significant.

#### 4.2 Link conditional on obtaining a PhD (intensive margin)
- Motivation: to distinguish the extensive margin (higher-scorers more likely to pursue a PhD) from the intensive margin (higher-scorers produce more research conditional on a PhD).
- Sample restricted to IMO participants who have a PhD in mathematics: n = 1,023.
- Table 5, panel A: regressions of publications (logs), cites (logs), IMC speaker, and Fields medal on IMO scores controlling for cohort and Olympiad fixed effects.
  - Conditional on a PhD, an additional IMO point is associated with:
    - a 2% increase in publications;
    - a 4% increase in cites;
    - a 0.2 percentage point increase in propensity to become an IMC speaker;
    - a 0.07 percentage point increase in propensity to become a Fields medalist.
  - These estimates are hardly lower than those in Table 4 panel A.
- Table 5, panel B: regressions adding graduate school by Olympiad year fixed effects (comparing IMO participants who competed in the same Olympiad and completed a mathematics PhD in the same school).
  - Even in this demanding specification, a positive correlation remains between IMO scores and long-term performance.
  - Point estimates are somewhat lower for publications and cites, but similar for becoming an IMC speaker and receiving the Fields medal.

#### 4.3 Investigating causal effects via medals and regression discontinuity (RD) design
- Rationale:
  - Medals are awarded based on explicit cutoffs in IMO scores; medals are commonly used as public signals (CVs, LinkedIn).
  - Example (year 2000): participants scoring 30 and above received a gold medal; scores 21–29 received silver; scores 11–20 received bronze.
  - If medals causally affect later outcomes (via signaling or confidence), RD around medal cutoffs can provide evidence.
- RD design setup:
  - Model estimated: y_it = α + β AboveThreshold_it + δ1(IMOScore_it − Threshold_t) + δ2 AboveThreshold_it*(IMOScore_it − Threshold_t) + λ X_it + ε_it.
  - Outcomes y_it: obtaining a PhD in mathematics, PhD from a top 10 school, publications (logs), cites (logs).
  - AboveThreshold_it: indicator for being at or above the medal threshold (key parameter β).
  - Controls: linear distance to cutoff on each side, cohort fixed effects, country fixed effects.
  - Pool across three time-varying thresholds (gold, silver, bronze) to maximize power by creating three copies of each observation (distance to each threshold).
  - Optimal bandwidth selector of Calonico, Cattaneo & Titiunik (2014) used to select narrowly adjacent observations to the cutoff.
- Assumptions and manipulation checks:
  - Running variable (points scored) cannot be precisely manipulated by participants; medal thresholds unknown during solving.
  - Figure A1 shows the distribution of IMO scores by distance to medal threshold.
  - Frandsen (2017) test for manipulation when running variable is discrete: does not reject null of no manipulation (p-value=0.974 for k=0.02).
- Note: Results of the RD analysis are presented in Table 6 and Figure 4 (not reproduced here).

*Source: wp18268 - Section 3 presents the data.  The results on the link between IMO success and long-term*

### 4.   The  point  estimates  for  the  effect  of  a  (better)  medal  are  imprecisely  estimated  but

### 4.   The  point  estimates  for  the  effect  of  a  (better)  medal  are  imprecisely  estimated  but

### Effect of receiving a better medal versus underlying talent
- Controlling for score, being awarded a better medal appears to have no additional impact on:
  - becoming a professional mathematician
  - future knowledge production
- Two complementary pieces of evidence:
  - Receiving an honorable mention does not appear to have a causal effect on long-term performance.
  - Positive gradient between points scored and long-term performance even within medal bins:
    - Gold medalists: number of IMO points scored is positively correlated with each of the four outcomes (Table A4 panel A).
    - Bronze medalists: same positive correlation (Table A4 panel A).
    - Silver medalists: number of points scored is significantly correlated with two of the four outcomes.
- Interpretation: link between IMO scores and long-term performance reflects differences in underlying talent of medalists rather than a causal effect of IMO success.

### Link between IMO score and long-term performance by country income group
- Countries grouped by 2000 World Bank classification; about half of sample from low- and middle-income countries.
- Observation: participants from developing countries do not score lower at the IMO than participants from developed countries.
- Graphical pattern (Figure 6): for a given number of points, share obtaining a PhD in math is typically:
  - highest for high-income countries
  - followed by upper middle-income
  - then lower middle-income
  - with low-income countries having the lowest share
- Regression specification:
  - Y_it = β1 IMOscore_it + β2 CountryIncomeGroup_i + η_t + ε_it
  - Y_it = indicator for: getting a PhD in mathematics; getting a PhD in mathematics from a top school; publications in logs; cites in logs
  - Country income indicators: low-income, lower middle-income, upper middle-income (high-income omitted)
  - Controls: number of points scored at the IMO; Olympiad year fixed effects (η_t)
- Main quantitative findings (Table 7):
  - IMO participants from low-income countries are:
    - 16 percentage points less likely to do a PhD
    - 3.4 percentage points less likely to do a PhD in a top school
    - producing 35% fewer publications
    - producing 57% fewer cites
  - The low-income penalty for getting a math PhD is equivalent to scoring 15 fewer points at the IMO.
  - Similar, though less pronounced, pattern for middle-income countries.
- Robustness:
  - Similar results when replacing country income groups with linear income per capita (Table A6) or decile indicators (Table A7).
  - Interaction of points scored with country income group (Table 9): low-income penalty larger for individuals who score more points.
- Productivity conditional on having a PhD (Table 10):
  - Point estimates for low-, lower middle-, upper middle-income are negative but not significant (or only marginally significant for low-income).
  - With IMO year by graduate school fixed effects (panel B), point estimates are positive though not significant (except upper middle-income in cites).
  - Interpretation: extensive margin (getting a math PhD) may be more important than intensive margin (productivity conditional on having a math PhD) in explaining the country-origin relationship.

### Cross-country differences in IMO training
- Hypothesis: systematic differences in intensity/quality of training across countries could explain lower PhD and productivity rates for developing-country participants.
- Although regressions control for number of points scored, participants from developing countries might train more extensively to reach same scores.
- Observable proxy: repeat participation at IMO (competing multiple times implies more training).
- Finding (Table 8 column 1): participants from low- and middle-income countries are less likely to have participated at the IMO multiple times than those from developed countries.
- Interpretation: on this observable margin of training quantity, developing country participants receive generally less training than high-income counterparts.

### Migration and home country environment
- Distinguish doing a PhD abroad versus in the home country (Table 11 panel A).
  - Difference in propensity to do a math PhD between high and low/middle income countries is driven entirely by propensity to do a PhD at home.
  - IMO participants from developing countries are significantly less likely to do a PhD in their home country, with no corresponding increase in propensity to do a PhD abroad.
- Controls for home country capacity:
  - (1) whether home country has at least one university ranked among the 100 best in the world in mathematics (2009 Shanghai mathematics rankings)
  - (2) production of mathematics articles in the home country in log (panel C)
  - Either indicator is strongly correlated with doing a PhD at home and including them reduces coefficients for low- and middle-income countries.
- Interpretation: weak mathematics research and training capacity in home countries could help explain why developing-country medalists are less likely to become knowledge producers.

### Time trends: is the country-of-origin penalty decreasing?
- Specification includes interaction of country income group with indicator for ‘late’ cohorts (IMO participants who competed between 1991 and 2000; 1981–1990 omitted).
- Results (Table 12):
  - Interaction late × low-income is positive (significant for three of four outcomes): penalty for low-income countries has decreased over time.
  - Interaction late × upper middle-income is negative: suggesting an increasing gap over time between upper middle-income and high-income countries.
- Graphical evidence (Figure 7):
  - Plot difference in share of medalists getting a PhD in math between high- and low-income countries by five-year bands and medal type.
  - Difference always positive.
  - For bronze and silver medalists the difference has considerably diminished over time.
  - For gold medalists there is no decrease.
- Overall interpretation: coming from a low-income country is less detrimental than it used to be for becoming a professional mathematician and producing mathematical knowledge, but the gap at the top of the talent distribution (gold medalists) has not narrowed.

### Quantifying the size of lost knowledge production
- Approach: multiply coefficients on country income groups by share of IMO participants in each group and aggregate (cf Table 13).
- Context: around half of IMO participants and medalists are from low- and middle-income countries (cf descriptive statistics Table 2).
- Aggregate counterfactual:
  - Knowledge production could be 10% higher in terms of publications if IMO participants from low-income countries produced at same rate as those from high-income countries.
  - Knowledge production could be 17% higher in terms of cites under same assumption.
- Caveats:
  - Costs of enabling such increases are unquantified and may be substantial.
  - Potential crowding-out effects: enabling developing-country individuals to produce more may reduce knowledge produced by developed-country individuals (e.g., fixed important problems or competition for research positions).
  - Inducing individuals into mathematical careers may reduce their distinctive contributions outside mathematics.

### Comparing IMO participants with other mathematicians
- Evidence that talent matters for exceptional discoveries:
  - More than half of Fields medalists were IMO medalists; all but two were gold medalists (cf Table 1).
  - Example: Grigori Perelman won a gold medal with a perfect score at the 1982 IMO and solved the Poincaré conjecture (only ‘Millenium Prize Problem’ solved as of 2018).
- Constructed comparison samples:
  - All PhD students obtaining a PhD in mathematics between 1990 and 2010: n = 89,068.
  - PhD graduates from top 10 schools: n = 9,049.
  - IMO bronze and silver medalists with a mathematics PhD: n = 520.
  - IMO gold medalists with a mathematics PhD: n = 145.
- Outcomes plotted (Figure 8): average publications, average citations, share becoming IMC speakers, share becoming Fields medalists.
  - Medalists, especially gold medalists, outperform both other PhD graduates and PhD graduates from top schools on each outcome.
  - Differences larger for exceptional achievements (IMC speaking invitations, Fields medals) than for routine outputs (papers, citations).
- Regression comparisons (sample of 89,068 math PhD graduates; Table 14):
  - Regress outcomes on indicators for bronze/silver medalist and gold medalist.
  - Panel A: no other controls; Panel B: include PhD graduation year fixed effects and PhD graduate school fixed effects.
  - Findings:
    - Positive and significant coefficients for both bronze/silver and gold medalists across outcomes.
    - Magnitude sizeable for papers and citations; considerably larger for exceptional achievements:
      - For propensity to become an IMC speaker: coefficient for IMO gold medalist is an order of magnitude larger than mean propensity.
      - For propensity to become a Fields medalist: coefficient for IMO gold medalist is two orders of magnitude larger than mean propensity.
      - Conditional probability that an IMO gold medalist will become a Fields medalist is fifty times larger than corresponding probability for a PhD graduate from a top 10 mathematics program.
    - Coefficients similar when controlling for PhD graduate school fixed effects.
  - Interpretation: IMO medalists who get a PhD tend to outperform other mathematics PhD graduates and their classmates from the same school, especially on exceptional research achievements.
- Sorting across graduate schools (Figure 9):
  - Compute number of IMO medalists graduating with PhD from each school and plot against Shanghai mathematics ranking.
  - Medalists cluster in the very best schools.
  - 36% of the IMO medalists who get a PhD in math graduate from the top 10 schools.
  - Interpretation: highly talented individuals (as proxied by IMO medals) are scarce and concentrate at top institutions.

*Source: wp18268 - 4.   The  point  estimates  for  the  effect  of  a  (better)  medal  are  imprecisely  estimated  but*

### 6.3    Careers of IMO participants outside mathematics

### 6.3    Careers of IMO participants outside mathematics

### Occupational outcomes of IMO medalists
- Occupations were manually classified into five broad categories: mathematics academia, academia outside mathematics, occupations in the finance industry, occupations in the IT industry, other occupations (mostly other industries), and ‘no online profile’.
- Regressions of indicator variables for these occupational categories on country income group indicators, points scored and cohort dummies were conducted (cf Table 15).
- Low- and middle-income country medalists are significantly less likely to be employed in mathematics academia.
- There are some differences across country income groups in the propensity to be employed in non-math academia or in finance, but the paper finds no systematic pattern.
- Low-income and lower-middle income medalists are more likely to have no online presence; the coefficients are quantitatively large (around 9 percentage points) and highly significant.
- Medalists with no online presence do not produce either mathematical or non-mathematical knowledge (at least in academia) nor are they otherwise visible online.

### Interpretation and implications for talent allocation
- The paper frames the allocation question in terms of comparative advantage: IMO participants are viewed as having a strong natural comparative advantage in mathematics given the high specificity of the discipline.
- Less engagement of developing-country IMO talent in mathematics does not by itself imply inefficiency; efficiency depends on the social value of mathematical knowledge relative to other goods and services and on individuals’ comparative advantages.
- The authors lean toward the view that under-utilization of developing-country talent in mathematics is likely inefficient because IMO participants tend to have a specific comparative advantage in mathematics.
- There are potential countervailing effects on developed-country mathematicians: increased participation by developing-country talent may generate learning and spillovers (Azoulay et al. 2010) but may also induce displacement and crowding out if graduate school and faculty slots are limited (Borjas & Doran 2012).

### Main findings summarized in the paper’s conclusion
- The paper addresses two questions: (1) how lifetime knowledge production depends on talent displayed in teen years, and (2) conditional on teenage talent, how country of birth affects knowledge produced.
- Using the IMO to observe extreme-right-tail talent avoids selection on eventual lifetime success and enables comparable measurement across countries.
- There is a strong and consistent link between IMO scores and later achievements in mathematics: obtaining a PhD, mathematics publications and cites, and being awarded a Fields medal.
- Even within the extreme-right tail, small differences in IMO performance translate into sizeable differences in long-term achievements.
- IMO participants from low- and middle-income countries produce consistently less mathematical knowledge than equally talented participants from high-income countries.
- The quantity of lost knowledge production arising from cross-country differences in the productivity of IMO participants is sizeable, and this lost knowledge production is not easily replaceable by that of other mathematicians.
- The authors caution that IMO performance is not a necessary condition to become a successful mathematician; IMO medals are used as a tool to observe part of the extreme right tail of the ability distribution.
- The paper notes anecdotal evidence that mathematical discoveries have direct or indirect practical applications (examples given: weather simulations, cryptography and telecommunications), and cites a Deloitte report that quantified the benefits of mathematical research to the UK economy as above 200 billion pounds.
- If developing-country talent is instead used in valuable non-mathematical occupations, the welfare cost of lost mathematical production may be mitigated; the paper cannot rule out this possibility.

### Gaps identified and directions for policy
- The paper does not identify precisely why developing-country participants are less likely to become professional mathematicians; research and training capacity in the home country appears to play a role, but other factors (preferences, private incentives, relative pay outside mathematics) may also matter.
- Suggested supply-side policies (push programs) consistent with the paper’s findings:
  - Fellowships for high-end talent to study mathematics at undergraduate and/or graduate levels to alleviate resource constraints and make mathematics careers more attractive.
  - Top schools should encourage applications from developing countries; recruiting elite talent to student programs is likely in their interest.
  - Strengthening mathematics research and training capacity in developing countries to improve training for those who prefer to stay and to make mathematics research careers more attractive.
- The authors note similar under-utilization of developing-country talent might exist in other disciplines such as biomedicine and computer science, though the importance of talent in those fields relative to mathematics is less clear and warrants further research.

*Source: 6.3 Careers of IMO participants outside mathematics (wp18268).*

### References

### wp18268 - References

### Major bibliographic sources
- Citations include working papers and journal articles on innovation, allocation of talent, incentives in R&D, and the economics of science; examples (authors and years as listed): Aghion et al. (2018); Akcigit et al. (2017); Agrawal, Goldfarb & Teodoridis (2016); Azoulay et al. (2010, 2011); Bell et al. (forthcoming); Baumol (1990); Bloom et al. (2017); Borjas & Doran (2012, 2015a, 2015b); Calonico, Cattaneo & Titiunik (2014); Hsieh et al. (2013); Iaria, Schwarz & Waldinger (2017); Waldinger (2010, 2011); Williams (2012).
- Methodological and empirical references include regression-discontinuity methods (Calonico et al. 2014; Frandsen 2017), nonparametric inference, and standard econometric approaches used throughout the paper.

### Key datasets and descriptive statistics (from Tables)
- Sample of IMO participants (1981-2000): N = 4,710 (Table 2).
  - IMO Score: Mean = 16.01, Std. Dev. = 11.30, Min = 0, Max = 42.
  - Gold Medal mean indicator = 0.08, Silver = 0.16, Bronze = 0.24, Honourable Mention = 0.10.
  - Math PhD indicator = 0.22; Math PhD (top 10) = 0.07.
  - Pubs: Mean = 3.31, Std. Dev. = 11.50, Max = 264.
  - Cites: Mean = 34.62, Std. Dev. = 221.10, Max = 11,062.
  - Country income group shares: High-income = 0.50, Upper middle-income = 0.23, Lower middle-income = 0.16, Low-income = 0.11.

- Sample of IMO medalists: N = 2,272 (Table 3).
  - IMO Score: Mean = 25.4, Std. Dev. = 8.3, Min = 1, Max = 42.
  - Math PhD indicator = 0.32; Math PhD (top 10) = 0.11.
  - Pubs: Mean = 5.3, Std. Dev. = 14.8, Max = 264.
  - Cites: Mean = 59.3, Std. Dev. = 305.7, Max = 11,062.
  - Current employment (manually constructed): Academia (math) = 0.24, Academia (not math) = 0.13, Finance industry = 0.05, IT industry = 0.09, Other industry = 0.04, No online profile = 0.46.

### Regression and causal-estimate highlights
- Table 4 (OLS, all IMO participants, N = 4,710): IMO Score (coefficient estimates)
  - Math PhD: 0.0101 (standard error 0.0008) ∗∗∗
  - Math PhD (top 10): 0.0054 (0.0006) ∗∗∗
  - Pubs (log+1): 0.0261 (0.0021) ∗∗∗
  - Cites (log+1): 0.0434 (0.0035) ∗∗∗
  - IMC speaker: 0.0012 (0.0003) ∗∗∗
  - Fields medalist: 0.0003 (0.0001) ∗∗∗

- Table 5 (Conditional on having a PhD, N = 1,032 / 1,023): IMO Score
  - Panel A (controls: country FE, olympiad year FE): Pubs (log) 0.0239 (0.0054) ∗∗∗; Cites (log) 0.0404 (0.0087) ∗∗∗; IMC speaker 0.0026 (0.0009) ∗∗∗; Fields medalist 0.0007 (0.0003) ∗∗.
  - Panel B (adds graduate school by olympiad year FE): Pubs (log) 0.0173 (0.0087) ∗∗; Cites (log) 0.0282 (0.0135) ∗∗; IMC speaker 0.0025 (0.0012) ∗∗; Fields medalist 0.0008 (0.0005).

- Table 6 (Regression discontinuity, pooled medal thresholds):
  - Above (better) medal threshold: Math PhD 0.0138 (0.0250); Math PhD (top 10) 0.0147 (0.0142); Pubs (log) 0.0117 (0.0719); Cites (log) 0.0114 (0.1043).
  - Distance from threshold: Math PhD 0.0098 (0.0032) ∗∗∗; Math PhD (top 10) 0.0038 (0.0015) ∗∗; Pubs (log) 0.0202 (0.0104) ∗; Cites (log) 0.0385 (0.0133) ∗∗∗.
  - Distance × above-threshold interaction coefficients are listed (see table) and country FE, cohort FE, threshold FE are included; bandwidths vary by outcome.

### Cross-country and equity-related findings
- Table 7 (Effect of country income group; cohort FE; N = 4,710)
  - Low-income: Math PhD −0.152 (0.017) ∗∗∗; Math PhD (top 10) −0.031 (0.012) ∗∗∗; Pubs (log) −0.337 (0.041) ∗∗∗; Cites (log) −0.560 (0.067) ∗∗∗.
  - Lower middle-income: Math PhD −0.101 (0.014) ∗∗∗; Math PhD (top 10) −0.022 (0.009) ∗∗; Pubs −0.194 (0.034) ∗∗∗; Cites −0.321 (0.057) ∗∗∗.
  - Upper middle-income: Math PhD −0.040 (0.017) ∗∗; Math PhD (top 10) −0.025 (0.010) ∗∗; Pubs −0.083 (0.041) ∗∗; Cites −0.171 (0.066) ∗∗∗.
  - IMO Score main effects remain positive and significant: 0.011 (0.001) ∗∗∗; 0.005 (0.000) ∗∗∗; 0.027 (0.001) ∗∗∗; 0.045 (0.002) ∗∗∗ across the four outcome columns.

- Table 9 (Interactions of income group with IMO Score; N = 4,710)
  - IMO Score positive main effects: 0.012 (0.001) ∗∗∗; 0.006 (0.000) ∗∗∗; 0.031 (0.004) ∗∗∗; 0.052 (0.006) ∗∗∗.
  - Low-income × IMO Score: −0.003 (0.001) ∗∗; −0.000 (0.001); −0.014 (0.004) ∗∗∗; −0.025 (0.006) ∗∗∗.

- Table 11 (Doing a PhD at home vs abroad; panels A–C; N = 4,710)
  - Low-income main effect on getting a PhD: −0.152 (0.017) ∗∗∗; PhD abroad coefficient in Panel A = −0.004 (0.015); PhD home = −0.148 (0.010) ∗∗∗.
  - IMO Score positive and significant for PhD, PhD abroad, PhD home across panels (e.g., IMO Score 0.011 (0.001) ∗∗∗).

- Table 12 (Time trends; interactions with late cohorts)
  - Low-income: Math PhD −0.237 (0.030) ∗∗∗; Low-income × late cohort 0.106 (0.036) ∗∗ indicating some diminution over time for late cohorts (late cohort = IMO 1991–2000).

- Table 13 (Back-of-the-envelope lost knowledge production)
  - Shares and weighted losses:
    - Low-income share = 0.11; Coeff (pubs) = −0.337; Coeff (cites) = −0.561; Loss (pubs) = −0.037; Loss (cites) = −0.062.
    - Lower middle-income share = 0.23; Coeff (pubs) = −0.194; Coeff (cites) = −0.321; Loss (pubs) = −0.045; Loss (cites) = −0.075.
    - Upper middle-income share = 0.23; Coeff (pubs) = −0.083; Coeff (cites) = −0.171; Loss (pubs) = −0.019; Loss (cites) = −0.039.
    - Total (weighted) loss: pubs −0.101; cites −0.176.

### Comparative and robustness results
- Table 14 (Comparison with non-medalists; ancillary Math Genealogy Project sample N = 89,068 and subsample N = 37,501)
  - Gold medalist effects on PhD-era outcomes (first specification): Pubs (log) 1.2597 (0.1087) ∗∗∗; Cites (log) 2.1425 (0.1764) ∗∗∗; IMC speaker 0.0799 (0.0231) ∗∗∗; Fields medalist 0.0333 (0.0147) ∗∗.
  - Silver/Bronze and top-10 graduate effects are also large and significant (see table for coefficients and standard errors).

- Table 15 (Current occupations of IMO medalists; N = 2,272)
  - Low-income medalist effects on occupation (relative to high-income omitted): Academia (math) −0.111 (0.026) ∗∗∗; Finance industry 0.031 (0.017) ∗; No online profile 0.087 (0.033) ∗∗∗.
  - IMO Score associations with current occupation: Academia (math) 0.007 (0.001) ∗∗∗; Academia (not math) 0.002 (0.001) ∗∗∗; No online profile −0.009 (0.001) ∗∗∗.

### Tables and figures content notes
- Tables include regression specifications with cohort (Olympiad Year) fixed effects, country fixed effects, threshold fixed effects (where applicable), and robust or clustered standard errors. Significance notation: *p <0.1, **p <0.05, ***p <0.01.
- Figures illustrate distributions of medalists by country, relationships between points scored at the IMO and subsequent achievements (PhD, top-10 PhD, publications, citations, IMC speakers, Fields medalists), and regression-discontinuity visualizations of distance to medal thresholds versus long-term outcomes.

_Italic: Source — wp18268 - References (text and tables/figures as provided)._

### 0.01 to 0.07

### wp18268 - 0.01 to 0.07

### Main empirical findings on IMO performance and long-term mathematics outcomes
- Honourable mentions: regressions show no significant causal effect of receiving an honourable mention on long-term outcomes. Table A1 coefficients (non-medal sample, Observations = 2,438):
  - Honourable mention: -0.0550 (0.0359) for Math PhD; -0.0238 (0.0204) for Math PhD (top 10); -0.0443 (0.0882) for Pubs (log); -0.0769 (0.1423) for Cites (log).
  - Perfect score on one problem: 0.0613 (0.0313)* for Math PhD; 0.0320 (0.0188)* for Math PhD (top 10); 0.0678 (0.0814) for Pubs (log); 0.1124 (0.1309) for Cites (log).
  - Adjusted R2: 0.0021, 0.0029, 0.0059, 0.0060 respectively.
- No discontinuity in density around medal thresholds (validating RD): Frandsen (2017) test p-value = 0.974, k = 0.02 (Figure A1 notes).
- Score composition matters: Table A2 (Observations = 4,491) — coefficients for scores on less and more difficult problems across six outcomes:
  - Score on less difficult problems: 0.0087*** (0.0011) for Math PhD; 0.0044*** (0.0007) for Math PhD (top 10); 0.0346*** (0.0044) for Pubs (log); 0.0208*** (0.0027) for Cites (log); 0.0010*** (0.0003) for IMC speaker; 0.0002 (0.0001) for Field medalist.
  - Score on more difficult problems: 0.0129*** (0.0022) for Math PhD; 0.0077*** (0.0015) for Math PhD (top 10); 0.0616*** (0.0095) for Pubs (log); 0.0375*** (0.0059) for Cites (log); 0.0019*** (0.0006) for IMC speaker; 0.0007** (0.0003) for Field medalist.
  - All regressions include Olympiad Year FE and Country FE.
- Regression discontinuity robustness (no country/cohort FE): Table A3 (varied bandwidths, Observations listed per column) — Above (better) medal threshold coefficient estimates:
  - Above threshold: 0.0079 (0.0257) for Math PhD; 0.0154 (0.0147) for Math PhD (top 10); 0.0128 (0.0729) for Pubs (log); -0.0054 (0.1065) for Cites (log).
  - Distance from threshold: 0.0130*** (0.0032); 0.0041*** (0.0016); 0.0240** (0.0104); 0.0504*** (0.0132).
  - Distance X above threshold: -0.0065* (0.0036); 0.0013 (0.0018); 0.0029 (0.0132); -0.0069 (0.0148).
  - Mean of D.V.: 0.2757, 0.0868, 0.5680, 0.9169.
- Within-medal score gradients: Table A4 (gold, silver, bronze subsamples)
  - Gold medalists (Observations = 371): IMO Score coefficients 0.0355*** (0.0105) for Math PhD; 0.0207** (0.0082) for Math PhD (top 10); 0.0933*** (0.0284) for Pubs (log); 0.1446*** (0.0469) for Cites (log).
  - Silver medalists (Observations = 775): IMO Score 0.0017 (0.0067) for Math PhD; -0.0012 (0.0049) for Math PhD (top 10); 0.0323* (0.0185) for Pubs (log); 0.0537* (0.0300) for Cites (log).
  - Bronze medalists (Observations = 1,126): IMO Score 0.0104** (0.0049) for Math PhD; 0.0052* (0.0027) for Math PhD (top 10); 0.0287** (0.0132) for Pubs (log); 0.0495** (0.0211) for Cites (log).
- Country income and scores/outcomes:
  - Table A5 (Observations = 4,710, Mean of D.V. = 16.007): IMO Score coefficients by income group (High-income omitted):
    - Low-income: 3.206*** (0.583)
    - Lower middle-income: 0.623 (0.403)
    - Upper middle-income: 0.030 (0.430)
  - Table A6 (Observations = 4,378, Mean of D.V. = 0.213, 0.069, 0.417, 0.692): joint inclusion of GDP per capita (2000, in thousands USD) and IMO score:
    - GDP per capita (1000 USD): 0.003*** (0.000) for Math PhD; 0.001*** (0.000) for Math PhD (top 10); 0.006*** (0.001) for Pubs (log); 0.011*** (0.001) for Cites (log).
    - IMO score: 0.011*** (0.001); 0.006*** (0.000); 0.028*** (0.002); 0.046*** (0.003) respectively.
  - Table A7 (Observations = 4,710, Mean of D.V. = 0.219, 0.068, 0.431, 0.713): coefficients for income deciles (richest decile omitted) show wide heterogeneity by origin-country income decile; IMO score coefficient across columns: 0.0108*** (0.0005); 0.0054*** (0.0004); 0.0262*** (0.0015); 0.0436*** (0.0025).
- Comparative career outcomes of IMO medalists vs other mathematicians (Figure 8 summary):
  - Outcomes plotted: publications per person, cites per person, share becoming IMC speaker, share getting a Fields medal.
  - Groups compared: All PhD students; Graduates from top 10 schools; IMO bronze or silver medalists; IMO gold medalists.
- Distributional and cohort patterns (figures summary):
  - Figure 6: Share getting a PhD in mathematics plotted by points scored at the IMO and by country income group (High−income, Upper middle income countries, Lower middle income countries, Low−income countries).
  - Figure 7: Difference in share getting a PhD between high- and low-income countries for different cohorts of medalists (by Olympiad year bands 1980, 1985, 1990, 1995) and medal type (Gold, Silver, Bronze); vertical axis ranges from −.2 to .6.
  - Figure 9: Number of IMO medalists graduating from math PhD programs, by graduating school; x-axis shows University rank (Shanghai Math) with values 0 20 40 60 80 100 and counts up to 50.

### Appendix A — IMO problems and examples
- IMO problems are intended to be solvable without knowledge of higher level tertiary mathematics (calculus, analysis) and are typically drawn from geometry, number theory, or algebra.
- Training uses compendia of past problems; problem difficulty is discussed on online forums.
- Example “easy” problem: IMO 1964 Problem 1:
  - (a) Find all natural numbers n such that the number 2^n −1 is divisible by 7.
  - (b) Prove that for all natural numbers n the number 2^n + 1 is not divisible by 7.
  - Suggested solution excerpt: 2^n is equivalent to 2, 4 and 1 (mod 7) for n congruent to 1, 2 and 0 (mod 3) respectively; only n divisible by 3 work for part (a); no n yields 2^n ≡ −1 (mod 7) for part (b).
- Example “difficult” problem: IMO 1988 Problem 6: Let a and b be positive integers such that (1 + ab) | (a^2 + b^2). Show that (a^2 + b^2)/(1 + ab) must be a perfect square.
  - Anecdote: The problem was submitted in 1988 by the FRG, marked superhard, and chosen as the last problem; eleven students gave perfect solutions.

### Appendix B — Identification strategy for honourable mentions
- Identification leverages the introduction of honourable mentions at the 1988 IMO to construct counterfactual awardees (those who would have received the award had it existed earlier).
- Empirical implementation: difference-in-differences comparing those who scored exactly 7 on one problem before vs after 1988, using non-medal participants as control.
- Regression specification:
  - y_it = α + β * Score7_it + δ * Honourable_it + η_t + ε_it
  - Where Score7_it indicates solved one problem perfectly; Honourable_it is the indicator for receiving an honourable mention and equals Score7_it interacted with competing after 1988; η_t are Olympiad year fixed effects.
- Conclusion: no significant causal effect of honourable mentions on obtaining a math PhD, PhD from top 10, publications, or citations; point estimates for honourable mention are negative across four outcomes.

### Appendix C — Validation of RD and distributional checks
- Figure A1 and notes: pooled observations across the three medal thresholds (gold, silver, bronze) with each individual appearing three times; no manipulation detected (Frandsen test p-value = 0.974, k = 0.02).

### Appendix D — Key tables and supplementary statistics
- Table A1: Effect of obtaining an honourable mention (Observations = 2,438; Adjusted R2 values reported).
- Table A2: Score on less vs more difficult problems (Observations = 4,491; Adjusted R2 reported per outcome).
- Table A3: RD estimates robustness to excluding country and cohort fixed effects (Observations and mean of dependent variable reported per column).
- Table A4: Within-medal IMO score gradients by medal type (Gold Observations = 371, Silver Observations = 775, Bronze Observations = 1,126).
- Table A5: IMO score by country income group (Observations = 4,710; Mean of D.V. = 16.007). Low-income coefficient 3.206*** (0.583).
- Table A6: Joint effects of GDP per capita (2000, in thousands USD) and IMO score on outcomes (Observations = 4,378; GDP per capita coefficients 0.003***, 0.001***, 0.006***, 0.011*** across outcomes).
- Table A7: Heterogeneity by country income deciles (Observations = 4,710; decile-specific coefficients reported; IMO score coefficient 0.0108*** (0.0005) for Math PhD).
- Table A8: Top ten schools in mathematics (Shanghai 2010 ARWU subject ranking) listed with ranks 1–10 (Princeton University rank 1; University of California, Berkeley rank 2; Harvard University rank 3; Stanford University rank 4; University of Cambridge rank 5; Pierre and Marie Curie University (Paris 6) rank 6; University of Oxford rank 7; Massachusetts Institute of Technology rank 8; University of Paris Sud (Paris 11) rank 9; University of California, Los Angeles rank 10).
- Table A9: Millennium Prize Problems status as of 2018:
  - Poincaré conjecture — Solved — Grigori Perelman, formerly IMO gold medalist with a perfect score.
  - P versus NP — Unsolved.
  - Hodge conjecture — Unsolved.
  - Riemann hypothesis — Unsolved.
  - Yang-Mills existence and mass gap — Unsolved.
  - Navier-Stokes existence and smoothness — Unsolved.
  - Birch and Swinnerton-Dyer conjecture — Unsolved.
  - Note: As of 2018, only Poincaré conjecture had been solved; solver Grigori Perelman had won a gold medal with a perfect score at the 1982 IMO.

*Source: wp18268 - 0.01 to 0.07 (IMF working paper content unit).*

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