Week 15 of 28 · Mastery · due 2026-09-05 · 26–30 hrs

A2/B2 Slot Mastery

A2/B2 Slot Mastery gate complete

This week's lesson · 9 min read

not started · read first, then train the gates

A2/B2 slot mastery, and geometry on the integer grid

A2/B2 is where the exam starts charging admission: still solvable slots, but the hidden observation is two moves deep instead of one, and a standard technique must be EXECUTED, not just spotted. This week drills that genre and adds lattice geometry — Pick's theorem and the geometry of integer points — a compact topic the Putnam loves because it welds counting to area in one formula.

One level up from A1, the exam stops hiding the door and starts hiding the key: A2/B2 problems name their topic openly and charge you two correct moves to enter.

Key ideas — the week on one card

A2/B2 anatomy: the first idea TRANSFORMS the problem into a second, standard one — re-classify the transformed problem and finish it as its own task.
Pick's theorem: a simple lattice polygon has $A = I + \frac{B}{2} - 1$; boundary counts are gcd computations, and the smallest lattice-triangle area is $\frac12$.
The two-ways collision: compute one quantity twice (shoelace vs formula) and let incompatible arithmetic natures finish the proof.
Solved-but-slowing is pre-failure drift: log durations on problems you got RIGHT — times are the leading indicator, misses the lagging one.

1The A2/B2 genre: two moves deep

The structural difference from A1/B1: the first observation no longer finishes the problem — it TRANSFORMS it into a second, standard problem. A typical A2 arc: notice the sum telescopes (move one), then evaluate the resulting limit with a comparison or Taylor bound (move two). Solvers who bank A1 but bleed on A2 usually execute move one and then stall, expecting the problem to be over; the genre's convention is that it is not.

Practical consequence: after your first real idea works, immediately re-classify the TRANSFORMED problem as if it were new — ninety-second first-move discipline again, from Week 9. The transformed problem is usually from a different track than the original (an algebra problem becomes an analysis estimate; a counting problem becomes an inequality), which is why the synthesis training of Weeks 12-13 sits directly upstream of this slot.

Time budget: 30-35 minutes including write-up. The write-up burden is heavier than A1 — two moves means two things to justify, and the transition sentence between them ('it therefore suffices to show...') is where self-graded 10s become real-world 6s. Write the reduction explicitly; graders cannot award the second move if the bridge to it is missing.

Worked example

Evaluate $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{k(k+1)\cdots(k+m)}$ for fixed $m \ge 1$.

Nudge 1
The consecutive factors telescope — find the partial-fraction identity whose difference collapses the product.
Reveal step 1
Move one — telescope: the partial-fraction identity $\frac{1}{k(k+1)\cdots(k+m)} = \frac{1}{m}\left(\frac{1}{k(k+1)\cdots(k+m-1)} - \frac{1}{(k+1)\cdots(k+m)}\right)$ (verify by combining the right side over a common denominator: the numerator is $(k+m) - k = m$).
Nudge 2
Sum the telescoping identity and watch only the first and last terms survive.
Reveal step 2
Move two — evaluate the collapsed sum: summing over $k = 1, \dots, n$ leaves $\frac{1}{m}\left(\frac{1}{1 \cdot 2 \cdots m} - \frac{1}{(n+1)\cdots(n+m)}\right)$, and the tail term $\to 0$ as $n \to \infty$ (it is $O(n^{-m})$).
Nudge 3
Bound the surviving tail and let $n \to \infty$.
Reveal step 3
Limit: $\frac{1}{m \cdot m!}$. Note the anatomy: move one is the creative step (finding the telescoping form), move two is standard analysis (tail vanishes), and the transition sentence 'the sum telescopes to...' is what the rubric pays for.
Answer
$\frac{1}{m \cdot m!}$ — telescoping partial fractions, then a vanishing tail. Two moves, one bridge sentence.

Pitfall. Declaring victory after move one. 'It telescopes' is half a solution; the collapsed expression still owes you a limit evaluation with justification. Re-classify the transformed problem and finish it as its own task.

2Pick's theorem: area as a census

Pick's theorem: a simple polygon with vertices on lattice points has area $A = I + \frac{B}{2} - 1$, where $I$ counts interior lattice points and $B$ counts boundary lattice points. Three quantities, one identity — so any two determine the third, and problems exploit every direction: compute an area by counting points, bound a point count by an area, prove a polygon must contain an interior point because its area is too big to avoid one.

The supporting cast: a segment between lattice points $(0,0)$ and $(p, q)$ contains $\gcd(p, q) - 1$ interior lattice points (divisibility meets geometry — the boundary count $B$ is a gcd computation around the polygon). And the smallest possible lattice triangle area is $\frac12$, achieved exactly by triangles with $I = 0, B = 3$ ('empty' triangles); this is Pick's theorem's most quotable corollary and a frequent lemma.

Recognition cues on the exam: 'lattice points', 'integer coordinates', any polygon-plus-counting hybrid, and Minkowski-flavored statements ('a convex region this large must contain a lattice point'). For the last family, know the one-liner: a convex region symmetric about a lattice point with area $> 4$ (in the classical normalization about the origin) contains another lattice point — Minkowski's theorem, whose proof idea (a volume-pigeonhole on the half-scaled region) is Week 3's pigeonhole running in continuous mode.

Pick's census on a real lattice triangle: bright dots are interior (I), oxblood are boundary (B).

Worked example

Show no equilateral triangle has all three vertices at lattice points.

Nudge 1
Compute the area TWICE: once from integer coordinates, once from the side length.
Reveal step 1
Compute the area two ways. Way one — the shoelace formula: with integer coordinates, $A = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ is RATIONAL (half an integer, in fact).
Nudge 2
Shoelace makes it rational; what does the equilateral formula make it?
Reveal step 2
Way two — the equilateral formula: $A = \frac{\sqrt{3}}{4} s^2$, where $s^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2$ is a positive INTEGER. So $A$ is $\sqrt{3}$ times a positive rational — irrational.
Nudge 3
Two incompatible natures for one number — conclude.
Reveal step 3
A number cannot be both rational and irrational; no such triangle exists. The template — compute one quantity in two ways and let the arithmetic natures collide — is a signature lattice move, and Pick's theorem is often the second 'way'.
Answer
Shoelace makes the area rational; $\frac{\sqrt3}{4}s^2$ makes it irrational. Contradiction — the two-ways-of-counting template in pure form.

Pitfall. Applying Pick to polygons that self-intersect or have holes — the formula as stated needs a SIMPLE polygon. For holes there is a corrected version ($A = I + \frac{B}{2} - 1 + h$), but on the exam the safe move is to decompose into simple pieces first.

3Maintenance economics: keeping A1/B1 warm while you climb

This week introduces the maintenance pattern that runs to December: last week's mastered slot gets a small, timed, weekly dose — enough to keep the recognition circuits warm, cheap enough not to tax the new material. The spaced-repetition deck handles theorem recall; the maintenance sets handle SPEED, which decays faster than knowledge and is invisible until a timer exposes it.

Treat maintenance sets as pure timed performance: exam conditions, target pace (A1/B1 in 20-25), no notes, and a one-line log per problem — time, verdict, and the observation that cracked it. Rising times with correct answers is the early-warning signal that precedes wrong answers by about two weeks; the log makes the drift visible while it is still cheap to correct.

The same economics explain the week's other standing gates — the keep-warm rotation and spiral reinforcement are the same principle applied to topics instead of slots. None of it is busywork: by Week 25's full mocks, you will be simultaneously 'current' in a dozen systems, and the interest payments on each are these thirty-minute doses.

Worked example

Your A1 maintenance log reads: week 1 — 19 min, solved; week 2 — 24 min, solved; week 3 — 31 min, solved. What does it say?

Nudge 1
Read the trend, not the verdicts — what is the time series doing?
Reveal step 1
The verdicts say 'fine'; the trend says 'decaying'. A 60% time inflation across three weeks on same-difficulty problems is recognition slowdown — the observation is arriving later each week.
Nudge 2
Split the inflation: is the IDEA arriving later, or the write-up dragging?
Reveal step 2
Diagnose which phase inflated: if the first idea now arrives at minute 15 instead of 5, it is recognition decay (schedule a pattern-catalog re-read plus a five-problem burst); if ideas arrive on time but write-ups crawl, it is fluency decay (drill the write-up templates).
Nudge 3
Intervene at 'slower', not at 'wrong' — say why that ordering is cheaper.
Reveal step 3
Act this week, not after a miss: the sequence 'slower, slower, wrong' is the standard decay arc, and intervening at 'slower' costs one session while intervening at 'wrong' costs three. This is the entire case for logging times on problems you got RIGHT.
Answer
Solved-but-slowing is pre-failure drift; split recognition delay from write-up drag and re-drill the inflated phase now.

Pitfall. Logging only misses. Times on SOLVED problems are the leading indicator; misses are the lagging one. A maintenance log without durations cannot warn you of anything.

Before you open the gates

A2/B2 anatomy: move one transforms, move two finishes — after the first idea lands, re-classify the transformed problem and restart the 90-second discipline.
Write the bridge sentence ('it therefore suffices to...') explicitly; the second move earns nothing if the reduction to it is unstated.
Lattice cues → Pick ($A = I + \frac{B}{2} - 1$), gcd boundary counts, empty triangles have area $\frac12$, and the two-ways area collision template.
Maintenance sets are timed performances with logged durations — solved-but-slowing is the early warning that precedes misses.
Budget: A2/B2 at 30-35 minutes; two moves means two justifications in the write-up.

Check yourself

1. Pick's theorem for a simple lattice polygon states $A =$

2. The number of interior lattice points on the segment from $(0,0)$ to $(p,q)$ is:

3. The lattice-equilateral impossibility proof works because:

Practice ladder — three rungs, rising

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

Rung 1 (Pick, direct). A simple lattice polygon has $12$ interior lattice points and $8$ boundary lattice points. Find its area. Then, using the segment lattice-point count, determine how many lattice points lie strictly between $(0,0)$ and $(24,18)$.

One hint

Pick's theorem: $A=I+\tfrac{B}{2}-1$. The number of lattice points strictly interior to the segment from $(0,0)$ to $(p,q)$ is $\gcd(p,q)-1$.

Worked resolution

By Pick's theorem, $A=I+\tfrac{B}{2}-1=12+\tfrac{8}{2}-1=12+4-1=15$. For the segment from $(0,0)$ to $(24,18)$, the number of strictly-interior lattice points is $\gcd(24,18)-1=6-1=5$. [Source: this week's Pick's-theorem gate — $A=I+\tfrac{B}{2}-1$ and the gcd boundary-count corollary (Putnam and Beyond, section 4: lattice points and Pick's theorem).]

Rung 2 (A2 two-move, telescoping). Evaluate $\displaystyle\sum_{n=1}^{\infty}\arctan\frac{1}{n^2+n+1}$.

One hint

Move one: find angles whose difference has this tangent. Since $\tan(\alpha-\beta)=\tfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$, try $\alpha=\arctan(n+1)$, $\beta=\arctan n$. Move two: telescope and take the limit.

Worked resolution

Move one (transform). For each $n\ge1$, $\tan\big(\arctan(n+1)-\arctan n\big)=\dfrac{(n+1)-n}{1+n(n+1)}=\dfrac{1}{n^2+n+1}$, and $\arctan(n+1)-\arctan n\in(0,\tfrac{\pi}{2})$, so $\arctan\tfrac{1}{n^2+n+1}=\arctan(n+1)-\arctan n$. Move two (finish). The partial sum telescopes: $\sum_{n=1}^{N}\big(\arctan(n+1)-\arctan n\big)=\arctan(N+1)-\arctan 1$. Letting $N\to\infty$, $\arctan(N+1)\to\tfrac{\pi}{2}$, so the series equals $\tfrac{\pi}{2}-\tfrac{\pi}{4}=\tfrac{\pi}{4}$. [Source: this week's A2/B2 slot-mastery genre — move one transforms (the arctangent-difference identity), move two finishes (telescope + limit), the two-move anatomy of the section's worked sum.]

Rung 3 (pigeonhole on the lattice, done solo). Prove that among any five lattice points in the plane, some two of them have a midpoint that is again a lattice point.

One hint

The midpoint of $(x,y)$ and $(x',y')$ is a lattice point iff $x\equiv x'$ and $y\equiv y'\pmod 2$. How many parity classes can a lattice point fall into?

Worked resolution

Assign to each lattice point $(x,y)$ its parity class $(x\bmod 2,\ y\bmod 2)\in\{0,1\}^2$. There are only $4$ classes, so among $5$ points two of them — say $(x,y)$ and $(x',y')$ — share a class, meaning $x\equiv x'$ and $y\equiv y'\pmod 2$. Then $x+x'$ and $y+y'$ are both even, so the midpoint $\big(\tfrac{x+x'}{2},\tfrac{y+y'}{2}\big)$ has integer coordinates — a lattice point. [Source: this week's A2/B2 method review (Putnam and Beyond, section 1: the pigeonhole principle) fused with the new lattice-geometry material — pigeonhole on the four parity classes of $\mathbb{Z}^2$.]

Prove it — constructed response

Prove that no equilateral triangle has all three of its vertices at lattice points of $\mathbb{Z}^2$. Then extend the result: prove that no regular hexagon can have all six vertices at lattice points.

Self-grade against the rubric — completion requires the judgment, not the text

Shows every triangle with lattice-point vertices has rational area — e.g. via the shoelace/determinant formula $A=\tfrac12|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$, which is a half-integer when all coordinates are integers.
Shows an equilateral triangle with a lattice-point side has area $\tfrac{\sqrt3}{4}s^2$ where $s^2=(x_1-x_2)^2+(y_1-y_2)^2$ is a positive integer, hence irrational; the two values of the one area collide, so no lattice equilateral triangle exists.
Deduces the hexagon claim: alternate vertices of a regular hexagon are mutually equidistant and so form an equilateral triangle, so a lattice regular hexagon would contain a lattice equilateral triangle — impossible.

Model proof (compare AFTER grading yourself)

(Rational area.) Let a triangle have vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)\in\mathbb{Z}^2$. The shoelace formula gives area $A=\tfrac12\big|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\big|$. The bracketed quantity is an integer, so $A\in\tfrac12\mathbb{Z}$; in particular $A$ is rational. (Irrational area.) Suppose the triangle is equilateral with side length $s$. One side joins two lattice points, so $s^2=(x_1-x_2)^2+(y_1-y_2)^2$ is a positive integer. An equilateral triangle of side $s$ has area $A=\tfrac{\sqrt3}{4}s^2$, which is $\tfrac{\sqrt3}{4}$ times a positive integer — irrational, since $\sqrt3$ is irrational. But $A$ was just shown rational, and a number cannot be both rational and irrational. Hence no equilateral triangle has all vertices in $\mathbb{Z}^2$. (Hexagon.) In a regular hexagon $V_1V_2V_3V_4V_5V_6$, the alternate vertices $V_1,V_3,V_5$ are mutually equidistant (each pair subtends the same central angle), so they form an equilateral triangle. If all six vertices were lattice points, then $V_1,V_3,V_5$ would be a lattice equilateral triangle, which we just proved impossible. Therefore no regular hexagon has all vertices in $\mathbb{Z}^2$. $\blacksquare$

Confidence in your proof (0–5): 2

The gates

A2/B2 Method Review (before drilling)

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- Review gate (depth): A2/B2 Method Review (before drilling)

sources & assignments (5)

Source Putnam and Beyond — PnB §1 (induction/pigeonhole) + §3 (bounding/estimation) — do 2 A2/B2-level problems (one per section)
Source Engel — Problem-Solving Strategies — Engel — in your strong topic (Ch.4 combinatorics or Ch.12 inequalities), redo 2 medium worked solutions cold
Source Michael Penn — Proof Writing (playlist) — Michael Penn — medium-difficulty A2/B2-style worked solutions (induction+construction, bounding) Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Shahriar Shahriari — Combinatorics, An Invitation (playlist) — Combinatorics lecture series — direct playlist companion to this gate's reading
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Writeup companion for “A2/B2 Method Review (before drilling)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

New Topic

New Topic: Lattice Geometry & Pick's Theorem

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- New-topic extension (continuous learning alongside weak-topic review): New Topic: Lattice Geometry & Pick's Theorem

why this gate: the lesson's §2 Pick's theorem: area as a census is what it trains

sources & assignments (6)

Source Putnam and Beyond — PnB section 4 — lattice points, Pick's theorem, Minkowski's theorem (intro)
Source OlympicMathTutor — Olympiad Geometry Theorems (playlist) — Mathologer — Pick's theorem visual proof (companion)
Source Engel — Problem-Solving Strategies — Engel Ch.4 — lattice-point & geometry-combinatorics problems 1–4 (Pick/lattice)
Source Michael Penn — Putnam Exam Solutions (playlist) — Putnam worked-solutions playlist — companion for this gate's problems.
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Writeup companion for “New Topic: Lattice Geometry & Pick's Theorem” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems PnB — PnB section 4 — 2 lattice/Pick 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 Axler — Linear Algebra Done Right — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — 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 algebra speed reps (open each below) · ~8 min eachAMC/AIME · archive: AMC 12 A AMC 12 B 2021 AMC 10 B

Archive Block — A2/B2

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- Do: 8 Putnam A2 · all topics · 15 min each - Do: 8 Putnam B2 · all topics · 15 min each

sources & assignments (6)

Source Iowa State Putnam Prep Archive — Iowa State Putnam sheets — Induction sheet problems 1–2 + Geometry of Polynomials sheet problem 1
Source Michael Penn — Putnam Exam Solutions (playlist) — Putnam walkthrough playlist — worked solutions companion to this handout
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Putnam problem-walkthrough playlist — second solution-voice companion for slot-mastery practice
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Writeup companion for “Archive Block — A2/B2” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems 8 Putnam A2 · all topics · 15 min each — Putnam 2019 A2, 2017 A2 · mixed · 15 min eachPutnam
Problems 8 Putnam B2 · all topics · 15 min each — Putnam 2024 B2, 2023 B2 · mixed · 15 min eachPutnam

Seminar Bridge

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- Do: MIT sum_integrals.pdf probs 5–8 (Feynman/Abel reinforcement) - Watch: Silver Integration Bee Advanced — 3 problems

sources & assignments (4)

Source Silver — Integration Bee Training (playlist) — Silver Integration Bee Advanced — 3 problems
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Pacing companion for “Seminar Bridge” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Supplemental — Lupu TTU MATH 4000, Recitation 7 (direct) — Writeup companion for “Seminar Bridge” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems MIT sum_integrals.pdf — MIT sum_integrals.pdf probs 5–8 (Feynman/Abel reinforcement)Putnam

A1/B1 Maintenance

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- Archive: 4 Putnam A1/B1 · your 2 weakest topics · 10 min each

why this gate: the lesson's §3 Maintenance economics: keeping A1/B1 warm while you climb is what it trains

sources & assignments (5)

Source Axler — Linear Algebra Done Right — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Writeup companion for “A1/B1 Maintenance” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — 4 Putnam A1/B1 · your 2 weakest topics · 10 min eachPutnam

archive pull: 4 problems · Mixed Putnam/Proof/Analysis/Algebra · A1/B1 · Standard · 10 min →

Spiral Reinforcement

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- Archive: 2 · Algebra A2 + 2 · Combinatorics A2 · any slot · 15 min each

sources & assignments (5)

Source Axler — Linear Algebra Done Right — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — 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 1970 B2 · Algebra · 15 min eachPutnam

archive pull: 1 problems · Algebra · A2 · Challenge · 15 min →

Deep Study — Linear algebra beyond the toolkit

Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10, with a Ch. 9 bridge skim)

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Mastery-phase learning gate: Axler Ch. 8/10 is genuinely new material beyond the Toolkit's Ch. 1–7 spine. Sequencing note: skim Ch. 9 (operators on real vector spaces, ~30 min) before Ch. 10 — the trace/determinant chapter's real-case results lean on it; the skim is part of this gate's budget, not extra.

Ritual: Output: trace-identity card + 2 fully written exercises from Axler 8.A.

sources & assignments (10)

Source Axler — Linear Algebra Done Right — Ch. 8 §8.A (generalized eigenvectors) exercises 8.A 1–4 — Toolkit covered Ch. 1–7; this is the next layer that unlocks hard LA Putnam problems.
Source Axler — Linear Algebra Done Right — Ch. 10 §10.A (trace) exercises 10.A 1–3 — trace identities (tr(AB)=tr(BA), trace = eigenvalue sum) are recurring Putnam weapons.
Source MIT 18.A34 — linalg.pdf (Yufei Zhao) — Problems 5–6 (the hardest pair — the sheet has six problems per the verification doc) cold: rank, determinant, and eigenvalue tricks.
Source CMU 09-Linear Algebra (Putnam seminar) — Problems 3–6 (the sheet has ten; problems 1–2 were the Week 8/10 drill work) — determinant and rank arguments at full depth.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Watch fully, then write the similarity-invariance card: what survives a change of basis (rank, det, trace, eigenvalues).
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Watch fully, then write the trace-equals-eigenvalue-sum card with a 3-line proof sketch.
Source Supplemental — 3Blue1Brown — Essence of Linear Algebra — The whole 16-episode series; episodes 13–14 are assigned above — browse the rest for any geometric gap (esp. 7 inverse matrices, 9 dot products).
Source Supplemental — Lupu TTU MATH 4000, Lecture 2 (direct) — Putnam seminar spine, followed in course order (Lecture 2 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 2 (direct) — Problem session paired with Lecture 2. Solutions are presented and defended live — grade each one against your own writeup standard: would it earn 10/10?

Problems Drill — Gram matrices and PSD — Prove from the definition: for any real matrix A, the Gram matrix A^T A is positive semidefinite, and rank(A^T A) = rank(A). Then use PSD-ness to prove the Cauchy–Schwarz inequality in R^n in two lines. This is the quadratic-form reflex behind many A3/B3 linear-algebra disguises.Standard

USAMO/IMO Bridge — USAMO 2014/1

USAMO/IMO Bridge — USAMO 2014/1 (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 (5)

Source USAMO 2014/1 — AoPS wiki (statement + solutions + discussion link) — Quartic with real roots, b-d ≥ 5: minimize (x₁²+1)(x₂²+1)(x₃²+1)(x₄²+1). Factor x²+1 over C — the symmetric-function/complex-factorization reflex this week trains. Time cap 60 min — do not scroll to the Solutions section until the attempt is over.
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Strategy companion for “USAMO/IMO Bridge — USAMO 2014/1 (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 Supplemental — Lupu TTU MATH 4000, Recitation 7 (direct) — Pacing companion for “USAMO/IMO Bridge — USAMO 2014/1 (stretch)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Mary Wootters (Stanford) — The probabilistic method and Ramsey numbers — Writeup companion for “USAMO/IMO Bridge — USAMO 2014/1 (stretch)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems USAMO 2014/1 — Quartic with real roots, b-d ≥ 5: minimize (x₁²+1)(x₂²+1)(x₃²+1)(x₄²+1) — full writeup to the Evan Chen standard, 60 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 — Comb

Continuation — Comb: Wilf generatingfunctionology §1 (≈2.5 hrs)

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Ritual: Thread: Combinatorics. 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 Wilf — generatingfunctionology (full free PDF) — Continuation-lane companion for: Comb: Wilf generatingfunctionology §1 (≈2.5 hrs). Reused from the verified pool — watch/read with this week's lens.
Source Michael Penn — Generating Functions and Combinatorial Identities — Continuation-lane companion for: Comb: Wilf generatingfunctionology §1 (≈2.5 hrs). Reused from the verified pool — watch/read with this week's lens.
Source Supplemental — Lupu TTU MATH 4000, Lecture 7 (direct) — Pacing companion for “Continuation — Comb: Wilf generatingfunctionology §1 (≈2.5 hrs)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Supplemental — Lupu TTU MATH 4000, Recitation 7 (direct) — Writeup companion for “Continuation — Comb: Wilf generatingfunctionology §1 (≈2.5 hrs)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Continuation lane — Wilf §1.1–1.3 (the method of generating functions, ~14 pp.). Do: rederive the Fibonacci GF from scratch; solve Wilf's two warm-up recurrences by GF; write the 3-step GF recipe card (define, translate, extract).Standard

Keep-Warm Rotation — abstract algebra · polynomials · functional equations

Keep-Warm Rotation — abstract algebra · polynomials · functional equations (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.

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 — state the subgroup test and exhibit one subgroup of ℤ/nℤ with proof, OR compute the order of one element in a small group; (2) polynomials — one Vieta or interpolation rep, or re-derive the finite-difference degree test; (3) functional equations — one A1/B1-band FE: substitutions first, injectivity/surjectivity second. Dormancy rule: no tool family sleeps longer than 2 weeks — these are the most overdue right now (AbA now rotates in).Standard

archive pull: 1 problems · Abstract Algebra/Algebra/Functional Equations · A1/B1 · Putnam · 10 min →

Track A∗ — Analysis

Track A∗ — Analysis (parallel mastery track): Infinite series — convergence tests, comparison, Abel/Dirichlet

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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: Infinite series — convergence tests, comparison, Abel/Dirichlet. 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 2007 B2 — work it with the reading open; one clean write-up.Putnam
Problems Analysis (reach) — Putnam 1985 B5 — 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): Determinants II — circulant, block, matrix-determinant identities

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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.

sources & assignments (4)

Source Prasolov — Problems and Theorems in Linear Algebra — This week: Determinants II — circulant, block, matrix-determinant identities. 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 2015 B3 — work it with the reading open; one clean write-up.Putnam
Problems Linear Algebra (reach) — Putnam 1986 A4 — 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): Multiplicative functions — σ, τ, φ, Möbius inversion

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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.

sources & assignments (4)

Source 104 Number Theory Problems (Andreescu, Andrica, Feng) — Advanced — This week: Multiplicative functions — σ, τ, φ, Möbius inversion. 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 1966 B2 — work it with the reading open; one clean write-up.Putnam
Problems Number Theory (reach) — Putnam 1986 B3 — 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): Inclusion–exclusion & the sieve

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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 — Combinatorics text — This week: Inclusion–exclusion & the sieve. Inclusion–exclusion, derangements, the sieve you can re-derive. 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 1987 B2 — cold re-solve with the reading open; one clean write-up.Putnam
Problems Combinatorics (reach) — Putnam 1990 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): Combinatorics — Bijections & double counting

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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.

why this gate: the lesson's §3 Maintenance economics: keeping A1/B1 warm while you climb is what it trains

sources & assignments (3)

Source Putnam and Beyond — Combinatorics — This week’s focus: Bijections & double counting. Count one set two ways; build an explicit bijection; the reflection/ballot trick.
Source Combinatorics keep-warm — Lower-cadence topic on a guaranteed weekly rotation (Toolkit Track F model). 1 problem this week; Combinatorics comes round again every 3rd week.

Problems Combinatorics (F∗) — Putnam 1966 B5 — bijections & double counting; full attempt + one clean write-up.Putnam

Probability Micro-Spine

Probability Micro-Spine: Expectation & linearity

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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: Expectation & linearity — indicators, linearity of expectation, the "count by E[X]" move. Read the matching section, then drill the problem.
Source Probability / combinatorics handbook — Companion: worked probability methods (prob-comb).

Problems Probability (drill) — Putnam 2002 B4 — 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 (12)

Problems Engel — Engel Ch. 7 probs 1–4 (Inequalities)Putnam
Problems MOC — Mathematical Olympiad Challenges — Algebra ch. probs 1–4Putnam
Problems Kedlaya — Putnam 1995 A2,B2 · 1996 A2,B2Putnam
Problems Putnam Archive — Putnam 2004 A2 - 18 min clean-solve; write the final proof and fatal-slip check.Putnam
Problems Putnam Archive — Putnam 2005 B2 - 22 min clean-solve; write the final proof and fatal-slip check.Putnam
Problems Putnam Archive — Putnam 2004 A3 - 35 min lemma hunt; record one rigorous lemma even if the full proof does not close.Nightmare
Problems Putnam Archive — Putnam 2003 B3 - 35 min lemma hunt; compare to official solution after the attempt.Nightmare
Problems Putnam Slot 2 Equalizer — Slot-2 Equalizer Set 2: Putnam 1993 A2, Putnam 1994 A2, Putnam 1996 A2 - clean-solve focus; write final proofs and fatal-slip checks.Putnam
Problems Putnam Slot 4 Equalizer — Slot-4 Equalizer Set 2: Putnam 2016 A4, Putnam 2017 A4, Putnam 2018 A4 - reach/lemma focus; extract the first invariant or structure before reading solutions.Nightmare
Problems Slot 2 Pair Balancer — Putnam 2015 A2, Putnam 2015 B2, Putnam 2016 A2, Putnam 2016 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 07: Putnam 2001 A1, Putnam 2001 A2, Putnam 2001 A3, Putnam 2001 A4, Putnam 2001 A5, Putnam 2001 A6, Putnam 2001 B1, Putnam 2001 B2, Putnam 2001 B3, Putnam 2001 B4, Putnam 2001 B5, Putnam 2001 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 08: Putnam 2002 A1, Putnam 2002 A2, Putnam 2002 A3, Putnam 2002 A4, Putnam 2002 A5, Putnam 2002 A6, Putnam 2002 B1, Putnam 2002 B2, Putnam 2002 B3, Putnam 2002 B4, Putnam 2002 B5, Putnam 2002 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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- Standard ---

Ritual: Pick an A2/B2 you solved 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 Axler — Linear Algebra Done Right — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — 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

- Standard ---

Ritual: Pick one solution. Re-check for: (1) a false claim, (2) a missing case, (3) a wrong bound. Extra check for A2/B2: is the key lemma actually proven, or just asserted?

sources & assignments (4)

Source Axler — Linear Algebra Done Right — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Week reference — this gate applies this week's lead material (“Deep Study — Linear algebra beyond the toolkit (Axler Ch. 8 + 10)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — 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.

Complex numbers keep-warm

Complex numbers keep-warm (20 min)

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Retention keep-warm (2026-07-02 audit): complex numbers went silent for 3-6 week stretches. ~20 minutes: two Lerma problems + one memory recall. Source: Lerma Training 2023 §5 (local PDF, hints/solutions in later parts).

Ritual: Recall (closed book): State |z|^2 = z·z̄ and derive the triangle inequality from it.

sources & assignments (2)

Source Lerma Putnam Training 2023 §5 Complex Numbers — Work problems 5.7–5.8. Roots of unity / De Moivre stay warm between full complex weeks.

Problems Lerma Training 2023 §5 Complex Numbers — Problems 5.7–5.8 (~15 min at conversion speed). Then the recall prompt below — closed book.Putnam

Exit contract — Week 15

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W16 · Mixed A1–B2 Timed Sets + Finish-Clean Drills
the exam follows the final taper week

Train Week 15 in the trainer →