IMO 2026 · Problem 4

Frozen · pending review·Day 2·domain pending·Evidence I0 - Unscoped

Official-solution dependence: None·Official solution viewed: NO

Proof audit: NOT-STARTED·Internal prior-solution exposure: YES

Legacy solution manuscript

A complete solution write-up produced earlier in the project. Preserved here in full; not an official IMO solution and not yet independently audited.

IMO2026_Day2_Problem4.pdfDownload PDF

created 2026-07-16level: L4-candidate (human-checkable write-up)audit: AUDIT-PENDINGsha256:2f78c72a43e9…

Independent HIVE-IMO X manuscript. Not an official IMO solution, and not independently verified under the new pipeline.

Official problem

Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^\circ<\theta<180^\circ$ known to both players. Initially, Shan-Yu makes a paper triangle $\mathcal{T}$ with measurements of his choice. Then, they repeatedly perform the following steps:

If $\mathcal{T}$ has at least one angle measuring exactly $\theta$ , then the game stops and Mulan wins.
Otherwise, Mulan chooses a point $P$ on the perimeter of $\mathcal{T}$ , different from its three vertices. She then makes a straight cut from $P$ to the opposite vertex of $\mathcal{T}$ , splitting it into two triangles.
Shan-Yu discards one of the two triangles. The remaining triangle becomes the new $\mathcal{T}$ .

For which real values of $\theta$ can Mulan guarantee her victory in finitely many steps, no matter how Shan-Yu plays?

official source ↗frozen 2026-07-21sha256:cbe3015874…

Vietnamese translation

Faithful literal draft by Fable, pending Human review (proper names kept in original form).

Shan-Yu và Mulan chơi một trò chơi. Cho $\theta$ là một góc với $0^\circ<\theta<180^\circ$ mà cả hai người chơi đều biết. Ban đầu, Shan-Yu làm một tam giác bằng giấy $\mathcal{T}$ với các số đo tùy ý theo lựa chọn của mình. Sau đó, họ lặp lại các bước sau:

Nếu $\mathcal{T}$ có ít nhất một góc bằng đúng $\theta$ , thì trò chơi dừng lại và Mulan thắng.
Ngược lại, Mulan chọn một điểm $P$ trên chu vi của $\mathcal{T}$ , khác ba đỉnh của nó. Sau đó cô cắt thẳng từ $P$ đến đỉnh đối diện của $\mathcal{T}$ , chia nó thành hai tam giác.
Shan-Yu bỏ đi một trong hai tam giác. Tam giác còn lại trở thành $\mathcal{T}$ mới.

Với những giá trị thực nào của $\theta$ thì Mulan có thể bảo đảm chiến thắng của mình sau hữu hạn bước, bất kể Shan-Yu chơi thế nào?

What the problem asks

A short plain-language explanation is written during S1–S3, after independent human and AI reads. S0 freezes the statement only - no interpretation yet.

Human × AI contribution matrix

Attribution per stage. Final approval is always human-led and verified independently.

Contribution	Human	Sol	Fable	Independent verifier
Problem interpretation	-	-	-	-
Core idea	-	-	-	-
Lemma discovery	-	-	-	-
Counterexample search	-	-	-	-
Proof writing	-	-	-	-
Formal checking	-	-	-	-
Final approval	-	-	-	-

Contribution attribution is recorded stage by stage; no roles are assigned until work begins.

Swipe horizontally to see all columns →

Audit preparation (S0-R)

The manuscript has been reconciled into a verifier-ready dossier: statement binding, section map, and a proof-obligation skeleton. No obligation has been verified.

state: RECONCILEDstatement: MATCHproof obligations: 5 (3 load-bearing)audited: 0/5

Every obligation is marked CLAIMED (asserted by the manuscript), never VERIFIED. Independent audit by an unexposed verifier has not started.

proof_obligations.json ↗verifier_input_manifest.json ↗sha256:d2ab2d1be9b9…

Timeline · S0–S14

S0Official problem freeze
HUMAN · Pending: Human V3 translation review + contamination attestation.AI · Fable froze the official statement, produced normalized EN + VI draft, source record, contamination ledger and SHA-256 manifest. Sol V2 semantic review pending. No solving.
problem_en.mdsha256:cdbe197eb5…b36896 problem_en.normalized.jsonsha256:b20d4f1563…874bf7 problem_vi.reviewed.mdsha256:ae1719845f…4299c5 source_record.jsonsha256:bfb81a365f…be6462 contamination_ledger.jsonsha256:c2855af94e…1ffe71
S1Human blind read
Not started.
S2AI independent read
Not started.
S3Contrastive merge
Not started.
S4Formal decomposition
Not started.
S5Small-domain reconnaissance
Not started.
S6Route tournament
Not started.
S7Lemma forge
Not started.
S8Candidate proof assembly
Not started.
S9Adversarial review
Not started.
S10Independent reconstruction
Not started.
S11Human proof rewrite
Not started.
S12Independent / formal verification
Not started.
S13Official-solution comparison
Not started.
S14Publication and postmortem
Not started.

Route map

Up to three main routes survive the tournament at S6.

No routes proposed yet.

Failed attempts

Failed routes and rejected proofs are kept permanently, never deleted.

None recorded yet.

Key lemmas

Lemmas appear here once forged at S7, each with an exact statement and an evidence class.

Proof - compact

Written at S11 in natural olympiad style, scorable 0–7 per step. Not yet available.

Proof - annotated

The annotated proof with justifications and dependencies is not yet available.

Independent verification

Required before the status VERIFIED-INDEPENDENT may be shown.

No independent verification has been performed. Status will not reach VERIFIED-INDEPENDENT until an olympiad expert, an independent model family, or a formal proof (Lean/Isabelle/Coq) confirms the proof.

Official-solution comparison

Official solution viewed after proof freeze: NO
Official solution dependence: NONE

A full comparison (core idea, length, naturalness, differences) runs at S13, only after the proof is frozen.

Artifact downloads

Every file is published with a SHA-256 checksum.

problem_en.mdsha256:cdbe197eb5…b36896 problem_en.normalized.jsonsha256:b20d4f1563…874bf7 problem_vi.reviewed.mdsha256:ae1719845f…4299c5 source_record.jsonsha256:bfb81a365f…be6462 contamination_ledger.jsonsha256:c2855af94e…1ffe71

MANIFEST sha256: 12e60de0a27214abe20e88ca3944771ab9617b992f490901e73fbb838dae9c2d

Revision history

21/07/2026Record initialized
Ledger created at status FROZEN-PENDING-REVIEW. No mathematical progress recorded yet.