Week 17 of 28 · Mastery · due 2026-09-19 · 26–30 hrs

Exam Craft + Verification Intensive

Exam Craft + Verification Intensive gate complete

This week's lesson · 9 min read

not started · read first, then train the gates

Quadratic residues, and the craft of not losing points

Half this week is number theory's most elegant chapter: quadratic residues, Euler's criterion, and the reciprocity law that Gauss proved eight times. The other half is exam craft — triage and verification — because from here forward, the training assumption flips: you know enough mathematics to score; the remaining work is converting knowledge into points at a table with a clock. Verification is where converted points go to survive.

Points are earned twice: once when you find the argument, once when you verify and write it so a stranger cannot refuse it — this week trains the second earning.

Key ideas — the week on one card

Euler's criterion: for odd prime $p$ and $p \nmid a$, $\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p$; exactly half the nonzero residues are squares.
$-1$ is a QR mod an odd prime $p$ iff $p \equiv 1 \pmod 4$ — so odd prime divisors of $n^2 + 1$ are $\equiv 1 \pmod 4$.
For DISTINCT odd primes $p, q$: reciprocity flips $\left(\frac{p}{q}\right)$ to $\left(\frac{q}{p}\right)$ with a sign only when both are $\equiv 3 \pmod 4$ — compute symbols gcd-style.
Verification is a separate pass: expand every 'clearly', test boundary inputs, audit hypotheses and quantifiers, reverse-read the final line against the question.

1Quadratic residues and Euler's criterion

For an odd prime $p$, the nonzero residues split exactly in half: $\frac{p-1}{2}$ quadratic residues (squares mod $p$) and $\frac{p-1}{2}$ non-residues — because $x \mapsto x^2$ is two-to-one on $(\mathbb{Z}/p)^*$. The Legendre symbol $\left(\frac{a}{p}\right)$ records the answer ($+1$ residue, $-1$ non-residue, $0$ if $p \mid a$), and it is COMPLETELY MULTIPLICATIVE in $a$: the product of two non-residues is a residue — the residues form an index-2 subgroup, and everything follows from that group structure.

Euler's criterion computes it: $\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p$. Proof idea worth owning: $(\mathbb{Z}/p)^*$ is cyclic (primitive root $g$), $a = g^k$ is a square iff $k$ is even, and $g^{(p-1)/2} = -1$ since it squares to 1 without being 1. The instant corollaries do heavy lifting: $-1$ is a QR iff $p \equiv 1 \pmod 4$, and the classic application — any prime dividing $n^2 + 1$ is 2 or $\equiv 1 \pmod 4$ — turns 'residue' facts into 'infinitely many primes of form $4k+1$' arguments.

Quadratic reciprocity finishes the toolkit: for odd primes $p \ne q$, $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$ — the symbols agree unless BOTH primes are $\equiv 3 \pmod 4$. With the supplements ($\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2}$, $\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$: 2 is a QR iff $p \equiv \pm 1 \pmod 8$), any Legendre symbol computes by flip-and-reduce, like a gcd. Putnam problems rarely demand reciprocity itself but often reward the residue COUNT ($\frac{p-1}{2}$ of each) and the $n^2 + 1$ divisor fact.

Mod 17, the squares land on exactly eight classes — the two-to-one squaring map, drawn.

Worked example

Show every prime divisor $p > 2$ of $n^2 + 1$ satisfies $p \equiv 1 \pmod 4$; conclude there are infinitely many such primes.

Nudge 1
If $p \mid n^2+1$, what is the exact multiplicative order of $n$ mod $p$?
Reveal step 1
If $p \mid n^2 + 1$ then $n^2 \equiv -1 \pmod p$: the element $n$ has order exactly 4 (its square is $-1 \ne 1$, its fourth power is 1). By Lagrange, $4 \mid p - 1$.
Nudge 2
Euler's criterion says the same thing — check the parity of $\frac{p-1}{2}$.
Reveal step 2
(Equivalently via Euler: $-1$ is a QR mod $p$, so $(-1)^{(p-1)/2} = 1$, forcing $\frac{p-1}{2}$ even.)
Nudge 3
Euclid the infinitude: build $N$ from the supposed full list so its prime divisors must be new AND $\equiv 1 \pmod 4$.
Reveal step 3
Infinitude, Euclid-style: suppose $p_1, \dots, p_k$ were ALL the primes $\equiv 1 \pmod 4$, and consider $N = (2 p_1 \cdots p_k)^2 + 1$. $N$ is odd, so every prime divisor of $N$ is $\equiv 1 \pmod 4$ by part one — but none of the $p_i$ divides $N$ (each leaves remainder 1). A new prime of the right form exists; contradiction. The order-of-an-element fact powered the whole argument.
Answer
$n$ has order 4 mod $p$, so $4 \mid p-1$; Euclid's construction on $(2\prod p_i)^2 + 1$ manufactures infinitely many primes $\equiv 1 \pmod 4$.

Pitfall. Quoting '$-1$ is a QR iff $p \equiv 1 \pmod 4$' without excluding $p = 2$ ($n^2 + 1$ is even for odd $n$). The exam's residue arguments are odd-prime arguments; say so in one clause.

2Triage: the decision that outscores any theorem

The triage drill formalizes what the mixed sets began: for each problem, a fast READ produces three artifacts — a difficulty estimate FOR YOU, a candidate first move, and a confidence tag (bank / fight / park). The December session gives ninety minutes for three problems; a top-200 score needs roughly two banked problems per session, which means the triage read decides more of your score than any single technique in the toolkit.

The tags have operational meanings. BANK: you see the whole arc — solve it FIRST and finish-clean before touching anything else. FIGHT: you see an entry but not the exit — this gets the middle block with a hard checkpoint. PARK: no viable entry — it gets whatever time remains, spent hunting partial credit (prove a special case, establish the key lemma, verify small cases with a stated conjecture; the rubric pays 1-3 for genuine partial progress, 0 for a page of flailing).

The failure mode the drill trains against is PRIDE ORDERING — attacking the interesting problem before the bankable one. Interest is not expected value. You are allowed to find the parked problem fascinating at minute 70, after twenty points are in the boat.

Worked example

Triage read, worked: 'Compute $\left(\frac{3}{97}\right)$' appears as a sub-step in a fight-tagged problem. Do it in under a minute.

Nudge 1
Check both primes mod 4 before flipping — does the sign appear?
Reveal step 1
Reciprocity: $97 \equiv 1 \pmod 4$, so $\left(\frac{3}{97}\right) = \left(\frac{97}{3}\right)$ with no sign flip.
Nudge 2
Reduce the top mod the bottom and flip again if needed.
Reveal step 2
Reduce: $97 \equiv 1 \pmod 3$, so $\left(\frac{97}{3}\right) = \left(\frac{1}{3}\right) = 1$.
Nudge 3
Two moves total: that speed is the point — note the gcd rhythm.
Reveal step 3
So 3 is a quadratic residue mod 97. The flip-and-reduce loop is gcd-fast — two steps here — and having it automatic is what lets a residue sub-question stay a sub-question instead of eating a fight-block's clock.
Answer
$\left(\frac{3}{97}\right) = 1$: one reciprocity flip (no sign, since $97 \equiv 1 \bmod 4$), one reduction.

Pitfall. Forgetting the sign rule mid-flip: the $(-1)$ appears ONLY when both primes are $\equiv 3 \pmod 4$. Write the exponent $\frac{p-1}{2}\cdot\frac{q-1}{2}$ once rather than trusting the remembered slogan.

3Verification: the ritual that keeps solved problems solved

Verification is a distinct pass with its own checklist, run AFTER the write-up feels done: (1) every 'clearly' and 'it follows' — expand or justify; (2) boundary inputs — $n = 0, 1$, empty set, equal variables, the degenerate triangle; (3) every division, log, and square root — is the operand nonzero/positive, and did you SAY so; (4) quantifier audit — does the proof actually prove 'for all', or did it quietly verify one case; (5) reverse read — does the final line answer the QUESTION ASKED (compute vs prove vs determine-all), in its original language.

Run the pass as a hostile stranger, ideally after solving something else in between — the context switch breaks the writer's blindness that makes your own gaps invisible. The certified-lemma gate this week is the same discipline applied upward: a lemma you cite must be one you can prove on demand, or it is not yours to cite.

Budget verification INTO the plan: five minutes per banked problem, non-negotiable, taken from the parked problem's time if necessary. The arithmetic is brutal but simple: a verification pass that catches one dropped case per two sessions is worth more than an entire extra technique in the toolkit, at a twentieth of the training cost.

Worked example

Verification pass, worked: your write-up claims 'for all real $x$, $f(x) = f(-x)$ implies $f'(0) = 0$.' Audit it.

Nudge 1
Audit the hypotheses first: which differentiability did the problem actually GIVE?
Reveal step 1
Checklist item (3)-(4): the claim uses $f'(0)$ — does the problem GIVE differentiability at 0, or did you assume it? If $f$ is merely even, $f' (0)$ may not exist ($f(x) = |x|$). If differentiability is given, the proof is: differentiate $f(x) = f(-x)$... but wait — differentiating both sides needs differentiability EVERYWHERE, or argue directly from the difference quotient at 0.
Nudge 2
Rebuild the claim from the difference quotient at 0 alone.
Reveal step 2
The clean repair: $f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}$; evenness gives $\frac{f(-h) - f(0)}{-h} = -\frac{f(h) - f(0)}{h}$, so the limit equals its own negative, hence 0 — using only differentiability AT ZERO, which is what was given.
Nudge 3
Name what the checklist caught: true conclusion, leaky argument.
Reveal step 3
This is the verification pass working as designed: the conclusion was true, the original argument was leaky (assumed more than given), and the repair is three lines. Graders distinguish exactly this — and 'true conclusion, leaky argument' is the most common shape of lost points at your level.
Answer
The difference-quotient argument needs only differentiability at 0; the checklist's hypothesis audit is what caught the over-assumption.

Pitfall. Verifying by re-reading in the same mental state that wrote the proof. Writer's blindness survives re-reads; it does not survive a context switch plus a checklist run as a hostile stranger.

Before you open the gates

Residue toolkit: two-to-one squaring (half are QRs), Euler's half-power criterion, $-1$ QR iff $p \equiv 1 \pmod 4$, odd divisors of $n^2+1$ are $1 \bmod 4$.
Legendre symbols compute gcd-style: flip (sign only when both $\equiv 3 \bmod 4$), reduce, repeat.
Triage tags are operational: bank first and finish-clean, fight with a pre-set checkpoint, park for partial credit.
Verification is a separate five-minute pass per banked problem: expand 'clearly', test boundaries, audit hypotheses and quantifiers, reverse-read the final line against the question.
Cite only lemmas you could prove on demand — certified means reproducible, not remembered.

Check yourself

1. Euler's criterion states $\left(\frac{a}{p}\right) \equiv$

2. The reciprocity sign: $\left(\frac{p}{q}\right)$ and $\left(\frac{q}{p}\right)$ DIFFER exactly when:

3. A problem tagged PARK in triage gets its remaining time spent on:

Practice ladder — three rungs, rising

Each rung: attempt cold, one hint if stuck, worked resolution only after a real try.

Rung 1 (Euler's criterion, direct). Using Euler's criterion, decide whether $3$ is a quadratic residue modulo $13$ by computing $3^{6} \bmod 13$; if it is, exhibit a square root.

One hint

Euler's criterion: $3$ is a QR mod $13$ iff $3^{(13-1)/2} = 3^6 \equiv 1 \pmod{13}$. Compute $3^6$ via $3^3 = 27 \equiv 1$.

Worked resolution

$3^3 = 27 \equiv 1 \pmod{13}$, so $3^6 = (3^3)^2 \equiv 1 \pmod{13}$. By Euler's criterion $\left(\frac{3}{13}\right) \equiv 3^6 \equiv 1 \pmod{13}$, so $3$ is a quadratic residue mod $13$. Indeed $4^2 = 16 \equiv 3 \pmod{13}$, so $x \equiv \pm 4$ solves $x^2 \equiv 3$. [Source: Euler's criterion from this week's 'Quadratic residues and Euler's criterion' section (104 NT Ch.5; PnB section 5.2), at its most direct.]

Rung 2 (reciprocity, flip-and-reduce). Compute the Legendre symbol $\left(\frac{7}{13}\right)$ using quadratic reciprocity and the supplements, and state whether $7$ is a quadratic residue mod $13$.

One hint

Check both primes mod 4: $7 \equiv 3$, $13 \equiv 1$ — not both $\equiv 3$, so no sign on the flip. Reduce $13 \bmod 7$, split off $\left(\frac{2}{7}\right)$ (use $7 \equiv -1 \bmod 8$), and flip once more.

Worked resolution

Since $13 \equiv 1 \pmod 4$, reciprocity gives $\left(\frac{7}{13}\right) = \left(\frac{13}{7}\right)$ with no sign. Reduce the top: $13 \equiv 6 \pmod 7$, so $\left(\frac{13}{7}\right) = \left(\frac{6}{7}\right) = \left(\frac{2}{7}\right)\left(\frac{3}{7}\right)$. The 2-supplement: $7 \equiv -1 \pmod 8$ gives $\left(\frac{2}{7}\right) = 1$. For $\left(\frac{3}{7}\right)$, both $3$ and $7$ are $\equiv 3 \pmod 4$, so the flip carries a sign: $\left(\frac{3}{7}\right) = -\left(\frac{7}{3}\right) = -\left(\frac{1}{3}\right) = -1$. Hence $\left(\frac{7}{13}\right) = (1)(-1) = -1$: $7$ is a non-residue mod $13$. (Check: the squares mod $13$ are $\{1,3,4,9,10,12\}$, and $7$ is absent.) [Source: The quadratic-reciprocity flip-and-reduce loop from this week's 'Quadratic residues and Euler's criterion' section and the Triage worked example (104 NT Ch.5, Legendre-symbol problems).]

Rung 3 (the residue count, proved solo). Let $p$ be an odd prime. Prove that $\sum_{a=1}^{p-1} \left(\frac{a}{p}\right) = 0$ — equivalently, that exactly half of the nonzero residues mod $p$ are quadratic residues.

One hint

Pick a non-residue $g$, so $\left(\frac{g}{p}\right) = -1$. Multiplication by $g$ permutes $(\mathbb{Z}/p)^*$; feed that into the sum and use complete multiplicativity of the Legendre symbol.

Worked resolution

Let $S = \sum_{a=1}^{p-1}\left(\frac{a}{p}\right)$. A non-residue exists (at most $\frac{p-1}{2} < p-1$ nonzero residues are squares), so fix $g$ with $\left(\frac{g}{p}\right) = -1$. As $a$ runs over $(\mathbb{Z}/p)^*$ so does $ga$, hence $S = \sum_{a}\left(\frac{ga}{p}\right) = \left(\frac{g}{p}\right)\sum_{a}\left(\frac{a}{p}\right) = -S$, using complete multiplicativity $\left(\frac{ga}{p}\right) = \left(\frac{g}{p}\right)\left(\frac{a}{p}\right)$. Thus $2S = 0$, so $S = 0$. Since each term is $\pm 1$ and they sum to $0$, there are equally many $+1$'s and $-1$'s: exactly $\frac{p-1}{2}$ quadratic residues and $\frac{p-1}{2}$ non-residues. [Source: The two-to-one squaring / residue-count fact and the multiplicativity of the Legendre symbol from this week's 'Quadratic residues and Euler's criterion' section, re-derived solo (PnB section 5.2).]

Prove it — constructed response

Let $p$ be an odd prime and suppose $p \mid n^2 + 1$ for some integer $n$. Prove that $p \equiv 1 \pmod 4$. Write the full argument as you would on the exam: name the criterion you use, verify its hypotheses, and say explicitly at which step you use that $p$ is odd.

The gates

Exam Craft + Verification (review)

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- Review gate (depth): Exam Craft + Verification (review)

why this gate: the lesson's §3 Verification: the ritual that keeps solved problems solved is what it trains

sources & assignments (8)

Source Stanford 07wk7 (Misc + Strategy) — Stanford Polya 07wk7 — verification & write-up discipline; do the 5 strategy problems
Source Putnam and Beyond — PnB §1 — proof-writing/rigor checklist; re-audit 2 past write-ups for fatal slips
Source Verification checklist — Build your fatal-slip checklist: false claim · missing case · wrong bound · unproven 'clearly' — apply to 2 graded write-ups
Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Po-Shen Loh — checking work & avoiding fatal slips (companion to Stanford 07wk7)
Source Michael Penn — Number Theory v2 (playlist) — Number-theory lecture series — direct playlist companion to this gate's reading
Source Krusemeyer — A Few Thoughts on the Putnam — Krusemeyer *A Few Thoughts on the Putnam* — read fully; list 5 committee patterns, match 1 archive problem to each
Source Stanford Vakil — Putnam Seminar (tiered sheets) — Stanford Vakil tiered sheets — do the 'Essential' set, then 2 'Advanced' problems, self-graded
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “Exam Craft + Verification (review)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

New Topic

New Topic: Quadratic Residues & Reciprocity

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- New-topic extension (continuous learning alongside weak-topic review): New Topic: Quadratic Residues & Reciprocity

why this gate: the lesson's §1 Quadratic residues and Euler's criterion is what it trains

sources & assignments (6)

Source 104 Number Theory Problems (Andreescu) — 104 Number Theory Problems Ch.5 problems 1–4 (Legendre symbol, quadratic residues, reciprocity)
Source Putnam and Beyond — PnB section 5.2 — quadratic residues & orders (distinct from LTE in Wk 21)
Source Michael Penn — Modular Arithmetic and Linear Congruences (playlist) — Michael Penn — quadratic reciprocity (companion)
Source Michael Penn — Number Theory v2 (playlist) — Watch the lecture matching this gate's NT topic; write the modulus-choice or order/valuation trigger it teaches.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “New Topic: Quadratic Residues & Reciprocity” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems 104 NT — 104 Number Theory Problems — 2 quadratic-residue problemsPutnam

AMC/AIME Speed Warmup

AMC/AIME Speed Warmup (build pattern recognition + speed)

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- AMC/AIME speed warmup — fast pattern-recognition reps; not full Putnam difficulty.

sources & assignments (5)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “AMC/AIME Speed Warmup (build pattern recognition + speed)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive (hand-picked) — 3 AMC/AIME combinatorics speed reps (open each below) · ~8 min eachAMC/AIME · archive: AMC 10 B AMC 10 B AMC 10 A

Triage Drill

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- Read: Stanford 07wk7 (Misc + Strategy) — 5 problems (use as triage target) - Protocol: 5-min scan of all 6 problems → written rank order → solve in that order - Do: 1 full Putnam exam year using triage protocol throughout

why this gate: the lesson's §2 Triage: the decision that outscores any theorem is what it trains

sources & assignments (6)

Source Stanford 07wk7 (Misc + Strategy) — Use the strategy notes to run the triage drill: 3-minute ranking pass over a 6-problem set, commit your solve order in writing, then solve only your top two.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Dr. Ebrahimian — triage: which problems to attempt first (companion to Stanford 07wk7)
Source Putnam and Beyond — PnB — problem-selection notes; rank a past A-session by expected points before solving any
Source Michael Penn — Putnam Exam Solutions (playlist) — Putnam worked-solutions playlist — companion for this gate's problems.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “Triage Drill” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems 1 full Putnam exam year using triage protocol throughout — 1 full Putnam exam year using triage protocol throughoutPutnam

Archive Block

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- Do: 10 mixed A1–B2 · all topics · triage protocol each time · 20 min each

sources & assignments (5)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “Archive Block” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems 10 mixed A1–B2 · all topics · triage protocol each time · 20 min each — Putnam 2010 A1, 2005 A1 · mixed · 20 min eachPutnam

Geometry Block

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- Archive: 4 Putnam A1/B1 · Geometry · any slot · 12 min each

sources & assignments (5)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source MIT 6.041SC — Probabilistic Systems Analysis (OCW) — Intuition companion for “Geometry Block” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Archive Browser — Putnam 1998 A1, 1997 A1 · Geometry · 12 min each — solve ONE of the two by deliberate coordinates: write down your coordinate choice first (origin, axis alignment, scaling) and one line on why it kills the symmetry; carry the computation to the end without switching methodsPutnam

archive pull: 2 problems · Geometry · A1/B1 · Standard · 12 min →

Certified Lemma

Certified Lemma (backup only)

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Note: Putnam partial credit is near-binary (0–2 or 8–10). Primary goal is full proofs on A1/A2/B1/B2. - Do: on 2 A3/B3 problems — write 1 clean sub-lemma each

sources & assignments (5)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “Certified Lemma (backup only)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems on 2 A3/B3 — on 2 A3/B3 problems — write 1 clean sub-lemma eachNightmare

Spiral Reinforcement

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- Archive: 2 · NT A2 + 2 · LA/Geometry A1 · any slot

sources & assignments (5)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Writeup companion for “Spiral Reinforcement” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — Putnam 2009 B1 · Number TheoryPutnam

archive pull: 1 problems · Number Theory · A2/A1 · Challenge · 20 min →

Deep Study — Proof rigor

Deep Study — Proof rigor: write like the official solutions

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Mastery-phase learning gate: exam craft is a skill with source material, not just practice.

Ritual: Output: upgraded writeup template + the 2019 A1–B2 diff log. Also: write your personal BAIL RULE as one sentence (e.g. "If at minute 15 I cannot name the next concrete step, I rotate strategy; at minute 30 with no certified lemma, I move on."). Pin it to your exam-craft checklist — it is graded in the W20 half-mock time map.

sources & assignments (6)

Source Velleman — How to Prove It — Ch. 3 (Proofs) §3.1–3.4: audit 3 of your own Phase-2 writeups against Velleman's templates; mark every unjustified jump.
Source Putnam and Beyond — Ch. 1 (Methods of Proof): re-do 4 worked examples closed-book — one each of contradiction, induction, pigeonhole, invariant.
Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Putnam 2019 A1–B2: write full solutions cold, then diff yours line-by-line against the official ones; log every gap in the fatal-slip list.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Live solution presentations — copy 5 'justification phrases' presenters use when defending steps into your own writeup template.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Putnam seminar spine, followed in course order (Lecture 4 of 11 assigned). Livestream pace — watch at 1.5x; pause before each solution and commit to a first move yourself before he reveals his.
Source Supplemental — Lupu TTU MATH 4000, Recitation 4 (direct) — Problem session paired with Lecture 4. Solutions are presented and defended live — grade each one against your own writeup standard: would it earn 10/10?

USAMO/IMO Bridge — USAMO 2003/1

USAMO/IMO Bridge — USAMO 2003/1 (stretch) accelerated only

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Ritual: Self-grade 0–7 against the AoPS/official solution: 7 = complete and rigorous · 5–6 = right idea, rigor gaps · 3–4 = key lemma proven · 1–2 = nontrivial progress · 0 = none. Log the score and the single biggest missing idea in the verify box.

sources & assignments (6)

Source USAMO 2003/1 — AoPS wiki (statement + solutions + discussion link) — Every n has an n-digit multiple of 5^n with all digits odd. A clean induction whose WRITEUP is the point — proof-rigor week's exam piece. Time cap 45 min — do not scroll to the Solutions section until the attempt is over.
Source Evan Chen — How to Write a Solution (the writeup standard) — Every bridge writeup is graded against this standard. Re-skim before writing; steal its phrasing discipline.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Strategy companion for “USAMO/IMO Bridge — USAMO 2003/1 (stretch)” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
Source Macauley — Visual Group Theory 5.3: Examples of group actions — Pacing companion for “USAMO/IMO Bridge — USAMO 2003/1 (stretch)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Michael Penn — One of my favorites from the Putnam — Writeup companion for “USAMO/IMO Bridge — USAMO 2003/1 (stretch)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems USAMO 2003/1 — Every n has an n-digit multiple of 5^n with all digits odd — full writeup to the Evan Chen standard, 45 min cap. Self-grade 0–7 against the AoPS/official solution: 7 = complete and rigorous · 5–6 = right idea, rigor gaps · 3–4 = key lemma proven · 1–2 = nontrivial progress · 0 = none. Log the score and the single biggest missing idea in the verify box.USAMO bridge

Continuation — NT

Continuation — NT: orders + primitive roots (≈2 hrs)

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Ritual: Thread: Number Theory. This lane keeps a second course moving during mastery weeks — new material, not review. Required artifact is written; videos are reinforcement only.

sources & assignments (5)

Source Michael Penn — Number Theory v2 (playlist) — Continuation-lane companion for: NT: orders + primitive roots (≈2 hrs). Reused from the verified pool — watch/read with this week's lens.
Source MIT 18.A34 — cong.pdf (Yufei Zhao) — Continuation-lane companion for: NT: orders + primitive roots (≈2 hrs). Reused from the verified pool — watch/read with this week's lens.
Source Macauley — Visual Group Theory 5.1: Groups acting on sets — Pacing companion for “Continuation — NT: orders + primitive roots (≈2 hrs)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Macauley — Visual Group Theory 5.3: Examples of group actions — Writeup companion for “Continuation — NT: orders + primitive roots (≈2 hrs)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Continuation lane — Work the order/primitive-root arc: prove ord_p(a) | p−1; find all primitive roots mod 11 and mod 13 by hand; then 2 archive reps where the key is 'consider the order of a mod p'. Notes card: the three signs a problem wants orders.Standard

Keep-Warm Rotation — groups / Burnside · valuations / LTE · quadratic residues

Keep-Warm Rotation — groups / Burnside · valuations / LTE · quadratic residues (30 min)

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Ritual: If any micro-rep took over 10 minutes or failed, that family goes on this week's repair list — it was rusting.

why this gate: the lesson's §1 Quadratic residues and Euler's criterion is what it trains

sources & assignments (2)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Pull keep-warm reps here when the archive button's pick doesn't match the rotation topic.

Problems Keep-warm rotation — Three 10-minute micro-reps, no notes: (1) abstract algebra (guaranteed) — one orbit-count rep (necklace-style) OR an orbit-stabilizer reconstruction with a concrete example (e.g. cube faces); (2) valuations / LTE — one v_p computation (factorial or LTE form), with the p=2 caveat stated; (3) quadratic residues — decide one solvability question x² ≡ a (mod p) via Euler's criterion. The dormancy rule: no tool family sleeps longer than 2 weeks — these are the three most overdue right now.Standard

archive pull: 1 problems · Abstract Algebra/Number Theory · A1/B1 · Putnam · 10 min →

Geometry Archive Intensive

Geometry Archive Intensive (mixed difficulty, timed)

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A graded, mixed-difficulty geometry sitting to convert the coordinate/complex/synthetic toolkit into exam points across the full Standard→Boss range.

sources & assignments (5)

Source Geometry decision tree — First 2 min: can I coordinate-bash? complex-number it? is it really disguised counting/inequality? Only then reach for synthetic (PoP, Ceva/Menelaus, inversion). Commit to one lane, set a 20-min bail.
Source Engel — Problem-Solving Strategies, Geometry chapter — Mixed-method geometry problems for spiral review.

Problems MAA archive — Putnam 1983 B1 · 2021 A1 · 2024 B2 (Standard — timed 25 min each)Putnam
Problems MAA archive — Putnam 2008 B3 · 2000 A3 (Challenge — full attempt + autopsy)Nightmare
Problems MAA archive — Putnam 2004 B4 (Boss — read solution if stuck, then write it cold)Nightmare

archive pull: 3 problems · Geometry · Putnam,Nightmare · 75 min →

Track A∗ — Analysis

Track A∗ — Analysis (parallel mastery track): Integrals — comparison to sums, mean-value estimates

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Runs EVERY mastery week so Analysis never goes cold — the Toolkit parallel-track model carried into Mastery. One depth problem + one A3–A6 reach problem each week.

sources & assignments (4)

Source Pólya & Szegő — Problems and Theorems in Analysis I — This week: Integrals — comparison to sums, mean-value estimates. Read the matching section, then do the two problems below.
Source Radulescu — Problems in Real Analysis — Companion reference for the same material.

Problems Analysis (depth) — Putnam 1990 B2 — work it with the reading open; one clean write-up.Putnam
Problems Analysis (reach) — Putnam 1986 B4 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track D∗ — Linear Algebra

Track D∗ — Linear Algebra (parallel mastery track): Diagonalization & the spectral theorem

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Runs EVERY mastery week so Linear Algebra never goes cold — the Toolkit parallel-track model carried into Mastery. One depth problem + one A3–A6 reach problem each week.

why this gate: the lesson's §2 Triage: the decision that outscores any theorem is what it trains

sources & assignments (4)

Source Prasolov — Problems and Theorems in Linear Algebra — This week: Diagonalization & the spectral theorem. Read the matching section, then do the two problems below.
Source Axler — Linear Algebra Done Right (Ch. 8 & 10) — Companion reference for the same material.

Problems Linear Algebra (depth) — Putnam 2014 A2 — work it with the reading open; one clean write-up.Putnam
Problems Linear Algebra (reach) — Putnam 1986 B6 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track E∗ — Number Theory

Track E∗ — Number Theory (parallel mastery track): Quadratic residues & reciprocity (Jacobi symbol)

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Runs EVERY mastery week so Number Theory never goes cold — the Toolkit parallel-track model carried into Mastery. One depth problem + one A3–A6 reach problem each week.

why this gate: the lesson's §1 Quadratic residues and Euler's criterion is what it trains

sources & assignments (4)

Source 104 Number Theory Problems (Andreescu, Andrica, Feng) — Advanced — This week: Quadratic residues & reciprocity (Jacobi symbol). Read the matching section, then do the two problems below.
Source Niven, Zuckerman & Montgomery — Theory of Numbers — Companion reference for the same material.

Problems Number Theory (depth) — Putnam 2000 B2 — work it with the reading open; one clean write-up.Putnam
Problems Number Theory (reach) — Putnam 1987 B6 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track C∗ — Combinatorics

Track C∗ — Combinatorics (parallel mastery track): Generating functions II — exponential GFs & convolutions

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Continuous Combinatorics mastery spine (Track C∗). Combinatorics is the #1 Putnam genre (~20%); A1–B2 combinatorics is already drilled hard in the slots, so this spine adds the continuous theory ladder plus a weekly A3–A6 reach problem (the real gap). One depth re-solve + one fresh reach each week.

sources & assignments (4)

Source Combinatorics spine — GFUNC text — This week: Generating functions II — exponential GFs & convolutions. EGFs, the product/convolution rule, labelled structures. Read the matching section, then do the two problems below.
Source Combinatorics continuous spine — Runs every Mastery week (parallel-track model) so Combinatorics — the single most frequent Putnam genre (~20%) — never goes cold. 1 depth re-solve + 1 fresh reach problem.

Problems Combinatorics (depth) — Putnam 1995 B1 — cold re-solve with the reading open; one clean write-up.Putnam
Problems Combinatorics (reach) — Putnam 1993 A6 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track F∗ — Geometry · Combinatorics · Probability

Track F∗ — Geometry · Combinatorics · Probability (parallel maintenance): Geometry — Complex numbers in geometry

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Bundled parallel maintenance track so Geometry, Combinatorics, and Probability each get a fixed weekly cadence in Mastery (probability previously had multi-week gaps). Rotates focus; ~1.5 hrs.

sources & assignments (3)

Source Evan Chen — EGMO, Complex Numbers chapter — This week’s focus: Complex numbers in geometry. Rotation z↦p+(z−p)e^{iθ}; collinear ⇔ ratio∈ℝ; concyclic ⇔ cross-ratio∈ℝ.
Source Geometry keep-warm — Lower-cadence topic on a guaranteed weekly rotation (Toolkit Track F model). 1 problem this week; Geometry comes round again every 3rd week.

Problems Geometry (F∗) — Putnam 1986 B1 — complex numbers in geometry; full attempt + one clean write-up.Putnam

Probability Micro-Spine

Probability Micro-Spine: Geometric & continuous probability

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Probability micro-spine: probability is ~5% of Putnam but was only bundled into rotating tracks (study-minutes ran ~1%). Three dedicated drills lift it to its real frequency without over-weighting a low-frequency genre.

sources & assignments (3)

Source Probability & Expectation (olympiad) — This week: Geometric & continuous probability — set up the sample space, integrate, symmetry. Read the matching section, then drill the problem.
Source Probability / combinatorics handbook — Companion: worked probability methods (prob-comb).

Problems Probability (drill) — Putnam 2004 A5 — set up the model carefully; one clean write-up.Putnam

archive pull: 1 problems · Probability · A2/B2/A4 · Putnam · 25 min →

Problem-Solving Reps — weekly homework

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Standing weekly homework — the problem-solving book stack, every week. Solve ALL listed; write ONE full clean solution (the rest may stay scratch). Up to 2 due re-solves from your review queue surface first. Slot reinforcement supplement: A2/B2 are deliberately overtrained so they become bankable, not occasional exposure. Slot reinforcement supplement: A3/B3 now get repeated lemma-hunt reps so slot 3 is trained as a strong reach tier.

sources & assignments (11)

Problems Newman — Newman — re-solve problems 19–22 (elegant write-ups)Putnam
Problems Kedlaya — Putnam 1998 A1,A2,B1,B2 (verification focus)Putnam
Problems Putnam Archive — Putnam 2008 A2 - 18 min clean-solve; write the final proof and fatal-slip check.Putnam
Problems Putnam Archive — Putnam 2016 B2 - 22 min clean-solve; write the final proof and fatal-slip check.Putnam
Problems Putnam Archive — Putnam 2007 A3 - 35 min lemma hunt; record one rigorous lemma even if the full proof does not close.Nightmare
Problems Putnam Archive — Putnam 2006 B3 - 35 min lemma hunt; compare to official solution after the attempt.Nightmare
Problems Putnam Slot 2 Equalizer — Slot-2 Equalizer Set 4: Putnam 2015 A2, Putnam 2017 A2, Putnam 2018 A2 - clean-solve focus; write final proofs and fatal-slip checks.Putnam
Problems Putnam Slot 4 Equalizer — Slot-4 Equalizer Set 4: Putnam 2022 A4, Putnam 1987 A4, Putnam 2010 B4 - reach/lemma focus; extract the first invariant or structure before reading solutions.Nightmare
Problems Slot 2 Pair Balancer — Putnam 2019 A2, Putnam 2019 B2, Putnam 2020 A2, Putnam 2020 B2 - timed A2/B2 score-growth reps; write one clean proof and one official-solution autopsy note.Putnam
Problems Putnam Full Past Exam — Full Past Exam 11: Putnam 2005 A1, Putnam 2005 A2, Putnam 2005 A3, Putnam 2005 A4, Putnam 2005 A5, Putnam 2005 A6, Putnam 2005 B1, Putnam 2005 B2, Putnam 2005 B3, Putnam 2005 B4, Putnam 2005 B5, Putnam 2005 B6 - Primary paper protocol: A-session A1-A6 in one 180-minute block; B-session B1-B6 in one 180-minute block within 48 hours; next day score all 12 slots, tag every miss, and choose three repair pulls.Putnam
Problems Putnam Full Past Exam — Full Past Exam 12: Putnam 2006 A1, Putnam 2006 A2, Putnam 2006 A3, Putnam 2006 A4, Putnam 2006 A5, Putnam 2006 A6, Putnam 2006 B1, Putnam 2006 B2, Putnam 2006 B3, Putnam 2006 B4, Putnam 2006 B5, Putnam 2006 B6 - Secondary paper protocol: Day 1 bankable sweep A1-A2-B1-B2 in 100 minutes; Day 2 medium/hard sweep A3-A4-B3-B4 in 100 minutes with certified-lemma goal; Day 3 reach scan A5-A6-B5-B6 in 50 minutes; Day 4 score all 12 slots and re-solve one same-family miss.Putnam

Reflect

Reflect: reconstruct and log the pattern

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- Reflect: exam-craft checklist — what did you violate? - Verify: (1) false claim? (2) missing case? (3) wrong bound? ---

Ritual: Pick the cleanest writeup you produced this week. Close all notes and the solution. Rewrite the full solution from memory, then compare with the original — what did you miss or reorder? Write one pattern note: "When I see [pattern], do [first move]."

sources & assignments (4)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Michael Penn — Number Theory v2 (playlist) — Intuition companion for “Reflect: reconstruct and log the pattern” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Verify: audit one proof before closing

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unlocks after: Reflect: reconstruct and log the pattern

- Reflect: exam-craft checklist — what did you violate? - Verify: (1) false claim? (2) missing case? (3) wrong bound? ---

Ritual: Pick one solution. Re-check for: (1) a false claim, (2) a missing case, (3) a wrong bound. Extra check: run your own exam-craft checklist on it — every violation goes in the fatal-slip list.

sources & assignments (4)

Source Velleman — How to Prove It — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Cezar Lupu — TTU MATH 4000, Recitation 13 — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Week reference — this gate applies this week's lead material (“Deep Study — Proof rigor: write like the official solutions”). If an attempt stalls, the repair source is here.
Source Michael Penn — Number Theory v2 (playlist) — Intuition companion for “Verify: audit one proof before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Exit contract — Week 17

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W18 · Advanced Analysis: Feynman + Abel
the exam follows the final taper week

Train Week 17 in the trainer →