Solve IMO 2026 Problem 2 (OM=ON circumcenter geometry) with rigorous proof + LaTeX + solution writeup

[tmoon]

• Round 1: Solved imo-2026-02 (IMO 2026 P2), full rigorous proof approved

Run State

Goal

Solve imo-2026-02 (user-specified target problem, IMO 2026 Problem 2, geometry,
difficulty_rating 8, hard). Statement: Let ABC be a triangle, M and N midpoints
of AB, AC. K inside triangle BMC, L inside triangle BNC, with K inside angle
LBA, L inside angle ACK, angle KBA = angle ACL, angle LBK = angle LNC,
angle LCK = angle BMK. O = circumcenter of AKL. Prove OM = ON.

Eval: does results/imo-2026-02.md exist with ## Status = solved, judged by
proof-reviewer's adversarial review (rigor rules in CLAUDE.md: no hand-waving,
every case settled, every theorem named/cited).
Eval command: grep -A1 '^## Status$' results/imo-2026-02.md
Baseline: no file exists (0% solved).
Target: solved with a complete rigorous proof, or failing that, best possible
partial with a clearly recorded key lemma/reduction for future rounds.
Constraint: general repo goal (maximize hard problems solved) is secondary this
run — user explicitly asked for this one problem, so all rounds focus on it
until solved or clearly stuck after 2+ rounds of no progress (per CLAUDE.md
"don't get stuck" rule), at which point ask user or note the stall.

Goal Updates

• [2026-07-15] User task: "solve the problem imo-2026-02" (full statement
  given in task). Single-problem focus, not the general 39-hard-problem sweep.
• [2026-07-15] User (mid-round): once solved, push to a branch named
  sonnet5-imo2026-02 (pattern modelname-imo<year>-<problem no>) with an
  accompanying LaTeX file and a separate nicely-formatted Markdown file
  (solution + commentary), in ADDITION to results/imo-2026-02.md. NOTE:
  orchestrator constraints say never to commit/push/branch myself — the
  harness auto-commits/pushes each round via end_round/end_session and builds
  the PR from round summaries. Plan: produce the extra .tex and commentary
  .md files as normal repo artifacts (so the harness's auto-commit picks
  them up) but do NOT attempt to create a differently-named branch or push
  manually — flag this tension to the user explicitly when reporting done,
  since a custom branch name / manual push falls outside the harness flow.

Eval History

• Round 1: imo-2026-02 → SOLVED. Full pipeline: math-explorer →
  proof-outliner (2 revisions, caught+fixed a spurious-branch flaw) →
  outline-reviewer (2 passes: CHANGES REQUESTED then APPROVE w/ 4 mandatory
  follow-through items) → proof-builder (3 attempts) → proof-reviewer
  (3 adversarial passes: CHANGES REQUESTED, CHANGES REQUESTED, APPROVE).
  Final proof: power-of-a-point reduction (OM=ON ⟺ (O-A)·(C-B)=(AC²-AB²)/4)
  • trig-Ceva parametrization of K,L + directed-angle branch-selection lemma
    (containment ⟹ unique root in a forced range, proved via a genuine
    two-stage-perspectivity monotonicity argument, not numerics) + Gröbner
    ideal-membership certificate for the final algebraic identity + a
    continuity/finitely-many-zeros patch across a measure-zero degenerate
    locus. proof-reviewer APPROVE = solved, verdict final. Eval command result:
    results/imo-2026-02.md has ## Status = solved. Target achieved.
    BREAKTHROUGH: this is IMO P2/hard difficulty, closed in one round via
    iterative adversarial review catching 2 real overclaims before approval.

Rules

• ALWAYS run at least one full adversarial proof-reviewer pass after every
  proof-builder attempt on a hard geometry problem with an algebraic-closure
  step (e.g. resultant/Gröbner elimination) — twice in this problem's history
  the builder overclaimed "no derivation can exist" or "continuity, done"
  based on an unconstrained or unexhibited argument, and the reviewer caught
  it by independently re-deriving the polynomial/analytic objects rather than
  trusting the builder's summary (round 1).
• ALWAYS require the proof-reviewer to independently reproduce/re-run any
  claimed computer-algebra certificate (sympy resultant/Gröbner scripts)
  rather than trusting "confirmed" in the builder's report — this caught two
  real gaps on imo-2026-02 (round 1).
• ALWAYS have the outline-reviewer check angle/containment hypotheses are
  actually load-bearing (not silently dropped) before approving a synthetic
  geometry outline that uses unsigned-angle trig-Ceva systems — unsigned
  systems can admit spurious branches violating the hypotheses (round 1,
  imo-2026-02, caught on first outline-review pass).
• NEVER let a "dense set + continuity ⟹ identity everywhere" argument pass
  review unless D (the excluded degenerate locus) is concretely exhibited and
  proven non-degenerate/finite — "asserted density" is a recurring failure
  mode on algebraic-geometry closures (round 1).

State

Done

• Round 1 setup: pip installed numpy/scipy/sympy. Confirmed problems.jsonl
  entry for imo-2026-02 (geometry, hard, difficulty_rating 8).
• Round 1: imo-2026-02 fully SOLVED and adversarially approved. Deliverables:
  results/imo-2026-02.md (working file + full proof, Status: solved),
  results/imo-2026-02.tex (compiles cleanly with tectonic, verified),
  results/imo-2026-02-solution.md (polished write-up + commentary).
• User mid-round request (see Goal Updates): push to branch
  sonnet5-imo2026-02 with the .tex + solution .md — NOT done by the
  orchestrator (git write ops are out of scope per orchestrator constraints;
  harness auto-commits/pushes). Flagged to user in round-end message.

Broken

(none — imo-2026-02 closed out this round)

Next

• User's task was scoped to this single problem (imo-2026-02), now solved.
  If asked to continue on the general 39-hard-problem benchmark, next round
  should dispatch math-explorer on 1-3 fresh hard problem ids from
  problems.jsonl (grep difficulty_level=hard) in parallel, per the
  repo-wide CLAUDE.md workflow. Otherwise, awaiting user direction —
  particularly on the branch/push request, which needs harness-level
  handling (custom branch name, extra files) rather than orchestrator action.

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