Getting it into your agent
One page per mod, every tool's command on it. A separate URL per tool would split the same page into five that compete with each other.
npx skills add hamzabellouch/agent-skills --skill mathematical-proof-and-competition-methodsgit clone --depth 1 https://github.com/hamzabellouch/agent-skillsWrote this? Show the measurements
A badge with what this costs and how it scanned, read live from this page, so it follows the numbers instead of freezing them. Markdown for a README, HTML for a documentation site or a project page.
[](https://agentmods.dev/skills/hamzabellouch/agent-skills/mathematical-proof-and-competition-methods)<a href="https://agentmods.dev/skills/hamzabellouch/agent-skills/mathematical-proof-and-competition-methods"><img src="https://agentmods.dev/badge/skills/hamzabellouch/agent-skills/mathematical-proof-and-competition-methods/github.svg" alt="Measured on agentmods" height="20"></a>Or the 80×15 button, for a site that already has a row of RSS and ATOM ones. Only the verdict fits; the numbers stay here.
<a href="https://agentmods.dev/skills/hamzabellouch/agent-skills/mathematical-proof-and-competition-methods"><img src="https://agentmods.dev/badge/skills/hamzabellouch/agent-skills/mathematical-proof-and-competition-methods.svg" alt="Reviewed on agentmods" width="80" height="20"></a>What it costs to keep this loaded
Counted locally with the o200k_base tokenizer, which is exact for GPT models; Claude uses its own tokenizer and its counts differ. Treat this as one consistent yardstick across the catalogue rather than a bill. Prices are per million input tokens.
| Model | Per session | Once invoked |
|---|---|---|
| Fable 5.1 | $0.00070 | $0.00532 |
| Opus 5.5 | $0.00028 | $0.00213 |
| Sonnet 5.5 | $0.00014 | $0.00106 |
| Haiku 4.5 | $0.00007 | $0.00053 |
Grade A, and why
mathematical-proof-and-competition-methods scanned grade A with 0 findings against 26 rules in 11 categories — prompt injection, anti-refusal, data exfiltration, privilege escalation, supply chain, agent snooping, system-prompt leakage, SSRF and excessive agency — measured today.
A static scan of the body, not an audit. Every finding is printed with the line that produced it so you can judge whether it matters here. A mod is markdown that instructs an agent; that is exactly why what it instructs is worth reading.
Nothing flagged
None of the 26 patterns this scan looks for appear in this file: no shell pipes, no recursive deletes, no credential paths, no hidden text, no instruction-override or anti-refusal phrasing, no agent-config snooping. That is not a guarantee, it is the absence of the things that are checkable.
What it actually says
Proof & Competition Methods
Polya's four steps
- Understand (restate, small cases, draw). 2. Plan (analogy, auxiliary problem, work backward, special cases, symmetry, generalize). 3. Execute and verify each step. 4. Look back (alternate proof, extensions).
Technique catalog
- Induction (weak, strong, double, Cauchy forward-backward), minimal counterexample / infinite descent.
- Pigeonhole, double counting, bijections, invariants/monovariants, coloring, extremal principle, constructive existence, probabilistic method.
- Inequalities: AM-GM (a+b)/2 >= sqrt(ab); Cauchy-Schwarz (sum a_i b_i)^2 <= (sum a_i^2)(sum b_i^2); Titu/Engel; rearrangement; Jensen; Schur; Muirhead; SOS; substitution a = x/y; tangent-line trick; equality cases always.
- Number theory: mod small numbers (squares mod 4, 8, cubes mod 9), LTE, order arguments, Vieta jumping, factoring (Simon's favorite factoring trick).
- Algebra: symmetric polynomials, Vieta, telescoping, functional equations (substitute x=0,1,-1; injectivity; surjectivity; Cauchy's equation f(x+y) = f(x) + f(y)).
- Geometry: angle chasing, power of a point, spiral similarity, inversion, coordinates/complex/barycentric bash, trigonometric Ceva.
- Combinatorics: generating functions, recursion, states and graphs, games (strategy stealing, symmetry, P/N positions).
Proof writing standards
- State what is assumed and what is to be shown; define notation; every step justified by a definition, theorem, or earlier line.
- Avoid circular reasoning, check division by zero, domain issues, and the quantifier order.
- Cover all cases; conclude explicitly; if you cannot prove something fully, say which gap remains.
Verification habit
Test conjectures on small cases (n = 1..6), look for counterexamples, then prove. If a formula is guessed, prove it by induction.
What ships with it
1 file beside SKILL.md in the same directory: the scripts, references and assets a skill reads on demand. Not counted in the per-session cost; read them before you install if any of them is executable.
What this file has done since we first saw it
Hashed on every crawl. A supply-chain change to an agent config is a question of when, not whether, so the history is kept rather than the latest state alone.
- today First seen · 27 lines · 70 tokens per session scan A 0ebd26d5a500
mathematical-proof-and-competition-methods is a skill published in the GitHub repository hamzabellouch/agent-skills (8 stars, last pushed today), licensed MIT. It adds 70 tokens to every session and 532 once invoked, about $0.0003 per session on Opus 5.5. A static security scan graded it A with 0 findings. No closer match exists in the catalogue, so it is treated as the original; first seen 2026-10-17.
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