IMO 2026 · Problem 3

Frozen · pending review·Day 1·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_Day1_Problem3.pdfDownload PDF

created 2026-07-16level: L4-candidate (human-checkable write-up)audit: AUDIT-PENDINGsha256:74bfa6311fca…

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

Official problem

Let $n$ be a positive integer. Liu Bang and Xiang Yu have a stick of length $1$ and want to divide it between themselves. Liu marks at most $n$ points on the stick, and then Xiang marks at most $n$ points on the stick. The marked points are distinct. Then, the stick is cut at all marked points, creating a number of pieces. Afterwards, they take turns claiming any unclaimed piece of the stick, with Liu going first. Each player's goal is to maximise the total length of their own pieces.

For each $n$ , determine the largest value $c$ such that Liu may guarantee a total length of at least $c$ , regardless of Xiang's play.

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

Vietnamese translation

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

Cho $n$ là một số nguyên dương. Liu Bang và Xiang Yu có một chiếc gậy dài $1$ và muốn chia nó cho nhau. Liu đánh dấu nhiều nhất $n$ điểm trên gậy, sau đó Xiang đánh dấu nhiều nhất $n$ điểm trên gậy. Các điểm được đánh dấu là phân biệt. Sau đó, chiếc gậy được cắt tại tất cả các điểm đã đánh dấu, tạo thành một số mảnh. Tiếp theo, họ lần lượt nhận bất kỳ mảnh nào chưa được nhận, Liu đi trước. Mục tiêu của mỗi người chơi là tối đa hóa tổng độ dài các mảnh của mình.

Với mỗi $n$ , hãy xác định giá trị lớn nhất $c$ sao cho Liu có thể bảo đảm tổng độ dài ít nhất bằng $c$ , bất kể Xiang 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: 6 (4 load-bearing)audited: 0/6

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:ca114fdb6769…

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:a0cd720e8c…03e3d7 problem_en.normalized.jsonsha256:0b08ff64eb…60ae30 problem_vi.reviewed.mdsha256:a3088291bd…4e33f1 source_record.jsonsha256:dac43bcb6e…3dcc8c contamination_ledger.jsonsha256:8b176ec5e5…4c102b
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:a0cd720e8c…03e3d7 problem_en.normalized.jsonsha256:0b08ff64eb…60ae30 problem_vi.reviewed.mdsha256:a3088291bd…4e33f1 source_record.jsonsha256:dac43bcb6e…3dcc8c contamination_ledger.jsonsha256:8b176ec5e5…4c102b

MANIFEST sha256: 3a2d1b5820695723524934ec064311d5598bd3de183e8c108216a6ea4993461b

Revision history

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