Math Olympiad

[
  {"query": "Solve this IMO problem: Let n ≥ 2 be an integer. Prove that...", "should_trigger": true},
  {"query": "Is this Putnam proof correct? Here's my attempt at B3...", "should_trigger": true},
  {"query": "Find a counterexample to: every continuous function on [0,1] is uniformly continuous", "should_trigger": true},
  {"query": "Prove this olympiad inequality: for positive reals a,b,c with a+b+c=1...", "should_trigger": true},
  {"query": "Help me with this USAMO geometry problem", "should_trigger": true},
  {"query": "Verify my solution to AIME 2024 problem 12", "should_trigger": true},
  {"query": "I think there's a gap in this competition proof, can you find it?", "should_trigger": true},
  {"query": "Simplify this proof — it feels overly complicated", "should_trigger": true},
  {"query": "Here's a conjecture from a math competition. Is it true?", "should_trigger": true},
  {"query": "What's the cleanest way to present this olympiad solution?", "should_trigger": true},

  {"query": "Help me verify the time complexity of this sorting algorithm", "should_trigger": false},
  {"query": "Write a Python function that checks if a number is prime", "should_trigger": false},
  {"query": "I'm doing research on the Riemann Hypothesis, where should I start reading?", "should_trigger": false},
  {"query": "Debug this proof assistant code — my Lean tactic isn't working", "should_trigger": false},
  {"query": "Explain the proof of the fundamental theorem of calculus to a high schooler", "should_trigger": false},
  {"query": "What's a good textbook for learning competition math?", "should_trigger": false},
  {"query": "Generate 10 practice problems similar to AIME level", "should_trigger": false},
  {"query": "Compute the integral of x^2 sin(x) dx", "should_trigger": false},
  {"query": "Review my research paper draft on analytic number theory", "should_trigger": false},
  {"query": "What's the difference between IMO and Putnam in difficulty?", "should_trigger": false}
]


# Adversarial Verifier Prompts — Math Olympiad

Prompt bank for the verifier subagent. Fresh context: problem statement +
cleaned solution, NO thinking trace. Agent has NO tools — pure reasoning only.

**Source**: `shared/verifier_patterns_source.md`. Background: arXiv:2503.21934
showed self-verified 85.7% IMO success drops to <5% under human grading. These
prompts are the human grader.

**Verifier isolation**: You do NOT know how other verifiers voted. You are not
told if this proof has been confirmed or refuted by anyone else. Assume you're
the first and only reviewer. (Social proof — "3 others confirmed" — biases
toward agreement.)

---

## Reasons to REFUTE (the taxonomy — look for ANY one of these)

Your goal is to find ANY reason to refute. These are the seven categories a hole
falls into:

1. **Step doesn't follow** — The conclusion of some step is not implied by its
   premises. (Includes direction errors: A>B and C>D does NOT give A−C>B−D.)
2. **Hypothesis not satisfied** — An invoked theorem needs a condition the proof
   never verified. (Pattern #5: "entire" ≠ "analytic in a disk".)
3. **Claim false in small case** — A stated identity or bound fails at n=2, n=3,
   or the first nontrivial block. Mentally test it.
4. **Tautological** — The "gap" at the end is the original problem in disguise.
   (Pattern #18: substitute the proof's own identities back in.)
5. **Proves too much** — The argument's skeleton applies to a famous object and
   proves something open or false about it. (Pattern #4.)
6. **Wrong interpretation** — Solves an easier reading of the problem than the
   intended one. (Pattern #60.)
7. **Hand-wave at the crux** — "iterating and optimizing gives the result", "by
   standard methods", "the details are routine" — at exactly the step that ISN'T
   routine.

If none of these fire after a genuine attempt, CONFIRM. Do not confirm because
the proof _sounds_ confident.

---

## 1. General Adversarial (default)

You are an adversarial verifier.