IMO 2026 · Problem 1

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_Problem1.pdfDownload PDF

created 2026-07-16level: L4-candidate (human-checkable write-up)audit: AUDIT-PENDINGsha256:3fa9e154caa6…

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

Official problem

There are $2026$ integers greater than $1$ written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m>1$ and $n>1$ from different places on the blackboard and replaces these two integers with $\gcd(m,n) \qquad \text{and} \qquad \frac{\operatorname{lcm}(m,n)}{\gcd(m,n)}.$ He continues to make moves while it is possible to do so.

(a) Prove that, regardless of the choices of Confucius, after finitely many moves, exactly one integer $M$ on the blackboard is greater than $1$ .

(b) Prove that the value of $M$ does not depend on the choices of Confucius.

(Note that $\gcd(x,y)$ denotes the greatest common divisor of positive integers $x$ and $y$ , and $\operatorname{lcm}(x,y)$ denotes the least common multiple of $x$ and $y$ .)

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

Vietnamese translation

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

Có $2026$ số nguyên lớn hơn $1$ được viết trên một tấm bảng, không nhất thiết khác nhau. Trong một nước đi, Confucius chọn hai số nguyên $m>1$ và $n>1$ ở hai vị trí khác nhau trên bảng và thay hai số này bằng $\gcd(m,n) \qquad \text{và} \qquad \frac{\operatorname{lcm}(m,n)}{\gcd(m,n)}.$ Ông ấy tiếp tục thực hiện các nước đi khi còn có thể.

(a) Chứng minh rằng, bất kể Confucius chọn thế nào, sau hữu hạn nước đi, có đúng một số nguyên $M$ trên bảng lớn hơn $1$ .

(b) Chứng minh rằng giá trị của $M$ không phụ thuộc vào cách chọn của Confucius.

(Ở đây $\gcd(x,y)$ ký hiệu ước chung lớn nhất của các số nguyên dương $x$ và $y$ , và $\operatorname{lcm}(x,y)$ ký hiệu bội chung nhỏ nhất của $x$ và $y$ .)

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:c6d3faaab3d1…

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.mdsha256:61afd1f532…ed4d96 problem_en.normalized.jsonsha256:83cd21b0fc…7a84f3 problem_vi.reviewed.mdsha256:dedcfb4c38…6a9a78 source_record.jsonsha256:319fbb22eb…cb35d9 contamination_ledger.jsonsha256:5f50d50b4d…70def8
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.mdsha256:61afd1f532…ed4d96 problem_en.normalized.jsonsha256:83cd21b0fc…7a84f3 problem_vi.reviewed.mdsha256:dedcfb4c38…6a9a78 source_record.jsonsha256:319fbb22eb…cb35d9 contamination_ledger.jsonsha256:5f50d50b4d…70def8

MANIFEST sha256: 34b34081d3218512555cdb9dfd33dc2c68f67cc829e076ab9a3e0277133ecadb

Revision history

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