IMO-Steps Dataset

This dataset is a benchmark that consists of building blocks for 13 IMO problems, and also the complete formal proofs for 20 IMO problems. The topics cover a variety of concepts ranging from divisibility to finite sets and functions. All proof steps are written in Lean 4.

All files compile with no error in Lean v4.17.0.

The purpose of the dataset is to expose current theorem provers' ability in solving IMO problems and highlight their strengths and weaknesses.

Paper is available at arxiv.org/abs/2411.18872. It is also published at TMLR: https://openreview.net/forum?id=CrKMqRAhBo.

The dataset is released under MIT license. Contributions are welcome through pull requests.

IMO problems formalized in this paper are as follows:

#	Year	Problem	Topic	in miniF2F	Lean proof publicly available	# of lines of Lean4 code
1	1959	P1	number theory	Yes	Yes	9
2	1960	P2	algebra	Yes	Yes	40
3	1962	P2	algebra	Yes	No	60
4	1963	P5	algebra	Yes	No	50
5	1964	P2	algebra	Yes	Yes	50
6	1965	P2	algebra	Yes	No	210
7	1968	P5	algebra	Yes	No	30
8	1969	P2	algebra	Yes	No	150
9	1974	P3	number theory	Yes	No	510
10	1981	P6	algebra	Yes	No	40
11	1982	P1	algebra	Yes	No	75
12	1983	P6	algebra	Yes	No	180
13	1984	P6	number theory	Yes	No	380
14	1985	P6	algebra	Yes	No	1,310
15	1992	P1	number theory	Yes	No	480
16	1997	P5	number theory	Yes	No	390
17	2007	P6	algebra	Yes	No	570
18	2022	P2	algebra	No	No	260
19	2022	P5	number theory	No	No	640
20	2023	P4	number theory	No	No	450
total						5,884

Performance of SOTA LLMs on the Lemmas Dataset

Problem	# of lemmas	DeepSeek Prover-v1.5-RL (@32)	Goedel-Prover (@32)	ReProver retrieval ✗	ReProver retrieval ✓	o3-mini (with 10 e.f.)
1959-p1	4	3 (75.0%)	2 (50.0%)	2 (50.0%)	2 (50.0%)	2 (50.0%)
1960-p2	9	7 (77.8%)	6 (66.7%)	3 (33.3%)	4 (44.4%)	1 (11.1%)
1962-p2	14	13 (92.9%)	12 (85.7%)	7 (50.0%)	8 (57.1%)	6 (42.9%)
1964-p2	9	5 (55.6%)	5 (55.6%)	5 (55.6%)	5 (55.6%)	3 (33.3%)
1965-p2	73	48 (65.8%)	47 (64.4%)	47 (64.4%)	46 (63.0%)	12 (16.4%)
1983-p6	53	25 (47.2%)	32 (60.4%)	28 (52.8%)	29 (54.7%)	18 (34.0%)
1984-p6	64	31 (50.0%)	33 (51.6%)	25 (39.1%)	24 (37.5%)	13 (20.3%)
1985-p6	427	116 (27.2%)	116 (27.2%)	89 (20.8%)	89 (20.8%)	84 (19.7%)
1992-p1	91	48 (52.7%)	54 (59.3%)	35 (38.5%)	34 (37.4%)	25 (27.5%)
1997-p5	122	51 (41.8%)	49 (40.2%)	48 (39.3%)	51 (41.8%)	30 (24.6%)
2022-p2	61	34 (55.7%)	30 (49.2%)	24 (39.3%)	25 (41.0%)	25 (41.0%)
2022-p5	265	89 (33.6%)	76 (28.7%)	80 (30.2%)	77 (29.1%)	60 (22.6%)
2023-p4	137	52 (38.0%)	41 (29.9%)	43 (31.4%)	45 (32.8%)	37 (27.0%)
Total	1,329	522 (39.3%)	504 (37.9%)	436 (32.8%)	439 (33.0%)	316 (23.8%)

Distribution of Proof Length of LLM-Generated Proofs

The figure demonstrates that the SOTA LLMs tend to write shorter proofs, which is a potential challenge for lengthier proof procedures that require long-term planning.

Citation

The dataset, motivations and additional results are available here

@article{
yousefzadeh2025a,
title={A Lean Dataset for International Math Olympiad: Small Steps towards Writing Math Proofs for Hard Problems},
author={Roozbeh Yousefzadeh and Xuenan Cao and Azim Ospanov},
journal={Transactions on Machine Learning Research},
year={2025},
url={https://openreview.net/forum?id=CrKMqRAhBo}
}

Versioning

The Lemmas Dataset is explicitly written for the latest available Lean4 version at the moment - v4.17.0. Any subsequent release can impact the released lemmas and some proofs might need future revision.

Formalised IMO Problems Statistics

#	Year	Problem	Topic	in miniF2F	Lean proof publicly available	# of lemmas	# of lines of Lean4 proof
1	1959	p1	number theory	Yes	Yes	4	9
2	1960	p2	algebra	Yes	Yes	9	40
3	1962	p2	algebra	Yes	No	14	60
4	1964	p2	algebra	Yes	Yes	9	50
5	1965	p2	algebra	Yes	No	73	210
6	1983	p6	algebra	Yes	No	53	180
7	1984	p6	number theory	Yes	No	64	380
8	1985	p6	number theory	Yes	No	427	1,310
9	1992	p1	number theory	Yes	No	91	480
10	1997	p5	number theory	Yes	No	122	390
11	2022	p2	algebra	No	No	61	260
12	2022	p5	number theory	No	No	265	640
13	2023	p4	number theory	No	No	137	450
	total					1,329	4,459

Additional Analysis of o3-mini Generated Proofs

#	Problem	# of lemmas	Correct proof in NL	Correct proof in Lean	Match between NL and Lean
1	1959-p1	4	100%	50.0%	100%
2	1960-p2	9	55.6%	11.1%	100%
3	1962-p2	14	92.9%	42.9%	100%
4	1964-p2	9	77.8%	33.3%	100%
5	1965-p2	73	97.3%	16.4%	100%
6	1983-p6	53	64.2%	34.0%	100%
7	1984-p6	64	73.4%	20.3%	100%
8	1985-p6	427	75.2%	19.7%	95.6%
9	1992-p1	91	87.6%	27.5%	100%
10	1997-p5	122	69.7%	24.6%	100%
11	2022-p2	61	77.0%	41.0%	100%
12	2022-p5	265	63.4%	22.6%	92.8%
13	2023-p4	137	88.3%	27.0%	92.7%
	total	1,329	75.5%	23.8%	96.4%