Hard examples — put Alpha to the test

Prove rigorously that n^5 − n is divisible by 30 for every integer n.
Use modular arithmetic for 2, 3 and 5 separately (cite Fermat's little theorem where you use it) and explain why that is enough.

Then answer: how many integers n with 1 ≤ n ≤ 1000 make n² + 1 divisible by 5? Show the residue table mod 5 and the count.

Finally, write a 5-line Python check for both results.

A fair six-sided die is rolled until two sixes appear in a row.
1) Set up the states and the system of equations for the expected number of rolls, and solve it exactly.
2) Generalise to k consecutive sixes and give a closed form.
3) Sanity-check with a short Monte-Carlo simulation in Python (state the expected spread for 100,000 trials).
Explain each step as if teaching a JEE aspirant.

Evaluate I = ∫₀^{π/2} ln(sin x) dx exactly.
Use the substitution x → π/2 − x and the double-angle identity; justify why the improper integral converges.
Then compute I numerically to 6 decimal places with a method of your choice, and confirm it matches your exact answer.
Finally, use the result to evaluate ∫₀^{π} x·ln(sin x) dx.

You are a senior engineer debugging a production Node.js + TypeScript checkout service for an Indian D2C store. Below is a 4-file excerpt.

Customers report:
1) Some UPI totals are off by ₹0.01.
2) Occasionally a coupon is applied twice.
3) Rarely, a shopper sees another shopper's cart.

// money.ts
export const addGst = (rupees: number) => rupees * 1.18;
export const round2 = (n: number) => Math.round(n * 100) / 100;

// cart-cache.ts
const cache = new Map<string, Cart>();
export function getCart(req: Req) {
  const key = req.headers["x-forwarded-for"] ?? "anon";
  if (!cache.has(key)) cache.set(key, { items: [], coupon: null });
  return cache.get(key)!;
}

// coupon.ts
export async function applyCoupon(cart: Cart, code: string) {
  if (cart.coupon) return cart;
  const c = await db.coupons.find(code);   // ~40 ms
  cart.coupon = c;
  cart.discount = (cart.discount ?? 0) + c.amount;
  return cart;
}

// checkout.ts
export async function total(cart: Cart) {
  let sum = 0;
  for (const it of cart.items) sum += round2(addGst(it.price * it.qty));
  return round2(sum - (cart.discount ?? 0));
}

Tasks:
- Find the root cause of EACH symptom; point to exact lines and explain the mechanism (not just "race condition").
- Rank by severity (money / privacy / UX).
- Give minimal diffs that fix them (integer paise, a proper session key, an idempotent coupon apply).
- Write 3 focused Vitest tests that fail before your fix and pass after.
- List anything you would still check in the rest of the codebase that this excerpt cannot show.

I'll paste (or attach) several files from my project. Act like a careful staff engineer joining the codebase.

Step 1 — Map: list each file's responsibility, the main data flow, and where state lives. Do not guess about files I haven't shared; ask for them.
Step 2 — Bug hunt: the bug is: [describe what you see, expected vs actual, steps to reproduce, any error/stack trace].
Trace the request through the files, form 2–3 hypotheses, and say what evidence in the code supports or rules out each.
Step 3 — Fix: give the smallest diff for the most likely cause, plus a test that would have caught it.
Step 4 — Risks: other places in the shared code with the same pattern.

Files:
[paste files here, each starting with its path]

Build a "Goa trip expense splitter" web app.
- Add friends, add expenses (who paid, amount in ₹, split equally or by custom shares).
- Compute balances and then MINIMISE the number of settle-up payments (greedy max-creditor / max-debtor), showing each step.
- Store everything in localStorage; Indian number formatting (₹1,25,000).
- Include a visible "Run tests" panel with at least 8 tests: rounding to paise, custom shares that don't add up, one person paying everything, zero-balance friends, and that total paid always equals total owed.
- Clean, mobile-first UI.

Explain why the daytime sky is blue but sunsets over the Arabian Sea look orange-red.
1) Derive the λ⁻⁴ dependence of Rayleigh scattering using dimensional analysis (state what the scattered intensity can depend on).
2) Compute how many times more strongly 450 nm light scatters than 700 nm light.
3) Explain why the sky isn't violet even though violet scatters even more.
4) Explain why big particles (dust, pollution haze in Delhi winter) make the sky look whiter instead.
State every assumption.

Estimate the energy used to heat water/milk for all the chai India drinks in one day.
- List every assumption (people, cups per day, ml per cup, temperature rise, stove efficiency) with a low / likely / high value.
- Show the calculation with units at each step, in kWh.
- Compare the likely value to the daily output of a 1 GW power plant.
- Say which single assumption moves the answer most, and by how much.

Design a school/college-lab experiment to test whether turmeric (curcumin) inhibits growth of E. coli on agar.
Include: hypothesis and null hypothesis, independent/dependent/controlled variables, positive and negative controls, how to handle curcumin's poor water solubility (and a solvent-only control), replicates and a rough sample-size argument, the statistical test you'd use and why, safety notes, and what result would FALSIFY the hypothesis.
End with 3 ways the experiment could give a misleading result.

I've attached a long document. Produce a structured brief:
1) 10-line executive summary.
2) Table of the 10 most important numbers: value | what it measures | page/section.
3) Key risks or limitations the document itself mentions.
4) Any statements in different sections that contradict or don't quite match each other.
5) 5 important questions this document does NOT answer.
Rules: quote the exact sentence for every factual claim, cite the page/section, and write "not in document" instead of guessing.

I've attached an agreement (rental agreement / job offer / vendor contract) from India. Read all of it, then give me:
- Each party's obligations, as two lists.
- Every deadline, notice period, lock-in, deposit and penalty, with clause numbers.
- Clauses that look unusual or one-sided compared with common Indian practice — explain why, quoting the clause.
- Things that are missing that I would normally expect.
- 8 questions I should ask before signing.
This is for understanding only, not legal advice — flag anything where I should consult a lawyer.

Read https://en.wikipedia.org/wiki/Chandrayaan-3 in full.
Then give me:
1) A dated timeline of the mission from launch to the end of surface operations.
2) A table of the mission's payloads: name | on lander or rover | what it measured | key finding (if reported).
3) What the page says went differently from Chandrayaan-2, and why that mattered.
Only use what the page says, and mark anything you add from general knowledge.

I run a small D2C store in India doing about ₹2 lakh GMV per month (mostly UPI, some cards).
Research the current published pricing and terms of Razorpay, Cashfree Payments and PhonePe Payment Gateway:
- standard fee for UPI, domestic cards and international cards, settlement time, setup/annual fees, and anything notable for small merchants.
Search the web, read each provider's official pricing page, and cite the source URL for every number.
Then build a comparison table, estimate my monthly fees on each assuming 80% UPI / 20% domestic cards, and recommend one with reasons.
End with a list of things I must confirm with the provider, because pricing changes.

Plan a 4-day trip from Bengaluru to Hampi for 3 college students, total budget ₹12,000 for all three (travel, stay, food, entry tickets).
Steps: search current train/bus options and approximate fares, find 2–3 budget stays with indicative prices, list the must-see sites with entry fees and opening times, and check the weather for this season.
Then produce a day-by-day plan with a cost table that stays under budget, and a plan B if it rains. Cite sources for prices and mark anything that's an estimate.

Build a GST invoice generator for a small kirana / retail shop in India.
- Shop profile (name, GSTIN, state) saved in localStorage.
- Line items: name, HSN code, qty, rate, GST rate (0/5/12/18/28%).
- If buyer state = shop state → split tax into CGST + SGST; otherwise IGST. Show tax per rate slab.
- Grand total in figures (₹1,23,456.00) and in words using Indian numbering (lakh/crore), e.g. "Rupees One Lakh Twenty-Three Thousand ...".
- Validate GSTIN format (15 chars) and show friendly errors.
- Invoice history, and a clean print-ready A4 layout (window.print).
- Mobile-first, works offline.