W9 · Toolkit— gates24–28 hrs

Week 9 of 28 · Toolkit · due 2026-07-25 · 24–28 hrs

All Tracks: Archive Sprint + Hidden Tools: Abel Summation + LA in Combinatorics

All Tracks: Archive Sprint + Hidden Tools: Abel Summation + LA in Combinatorics gate complete

Train this week in the trainer →

This week's lesson · 8 min read

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Abel's summation and the archive sprint: technique meets volume

Eight weeks of tools; this week you swing them at volume. The archive sprint is deliberate: recognition — knowing WHICH tool a problem wants within ninety seconds — is a separate skill from execution, and it only trains on bulk exposure. The one new weapon is Abel summation, discrete integration by parts, which completes your sum-manipulation kit and shows up in analysis clothing more often than any other discrete tool.

Recognition is trained, not owned: the sprint converts eight weeks of technique from things you can do into things you notice.

Key ideas — the week on one card

  1. Abel summation trades $\sum a_k b_k$ for partial sums $A_k$ against differences $b_k - b_{k+1}$ — integration by parts for sums.
  2. Dirichlet's test: bounded partial sums times a weight decreasing to zero converges — the tool that SEES cancellation where comparison tests are blind.
  3. Recognition is a separate skill from execution, and it only trains on unseen problems: ninety seconds to a committed first move.
  4. Pigeonhole design order: name the threshold first, then build boxes so any collision wins the problem outright.

1Abel summation: integration by parts for sums

Abel's summation formula: $\sum_{k=1}^{n} a_k b_k = A_n b_n + \sum_{k=1}^{n-1} A_k (b_k - b_{k+1})$, where $A_k = a_1 + \cdots + a_k$. Structurally it trades the sequence $a_k$ for its partial sums $A_k$, at the price of differencing $b_k$ — exactly as integration by parts trades $f$ for $\int f$ at the price of differentiating $g$. Use it when partial sums of ONE factor are controlled (bounded, telescoping, known in closed form) while the OTHER factor is monotone or smooth.

Its two headline corollaries do real contest work. Dirichlet's test: if $A_n$ is bounded and $b_n$ decreases to 0, then $\sum a_n b_n$ converges — this is why $\sum \frac{\sin n}{n}$ converges (partial sums of $\sin n$ are bounded via the geometric-sum formula for $e^{in}$; there is your Week 4 complex numbers again). Abel's test is the variant with $\sum a_n$ convergent and $b_n$ monotone bounded.

The estimation voice: Abel converts $\sum_{k \le n} \frac{a_k}{k}$-type sums into statements about the growth of $A_k$. Whenever a problem couples an arithmetic sequence with a decaying weight, write the Abel identity FIRST and see which side is friendlier — it is a two-line transformation with no risk.

±1/sin(½) ≈ ±2.09bounded forever — so the 1/n weight can drag the series to convergence
Dirichlet's hypothesis, computed: forty partial sums of sin n, none escaping the ±1/sin(½) band.

Worked example

Show that $\sum_{n=1}^{\infty} \frac{\sin n}{n}$ converges.

  1. Nudge 1

    Bound $\sum \sin k$ by taking the imaginary part of a geometric sum of $e^{ik}$.

    Reveal step 1

    Bound the partial sums of $\sin$: $\sum_{k=1}^{n} \sin k = \operatorname{Im} \sum_{k=1}^n e^{ik} = \operatorname{Im}\left(e^i \frac{e^{in} - 1}{e^i - 1}\right)$, and the modulus is at most $\frac{2}{|e^i - 1|} = \frac{1}{\sin(1/2)}$ — a constant $C$, independent of $n$.

  2. Nudge 2

    Oscillating factor with bounded partial sums, weight decreasing to zero — name the test.

    Reveal step 2

    Apply Dirichlet's test: $a_k = \sin k$ has bounded partial sums ($|A_n| \le C$), and $b_k = \frac{1}{k}$ decreases monotonically to 0.

  3. Nudge 3

    Convergent, but check: is it absolute? Compare $|\sin n|$ against $\sin^2 n$.

    Reveal step 3

    Conclusion: the series converges. (Not absolutely — $\sum \frac{|\sin n|}{n}$ diverges by comparison with $\sum \frac{\sin^2 n}{n} = \sum \frac{1 - \cos 2n}{2n}$, whose first half diverges while the second converges by the same Dirichlet argument. Conditional convergence, certified from both sides.)

  4. Answer

    Dirichlet's test: geometric-sum bound $|\sum_{k \le n} \sin k| \le \frac{1}{\sin(1/2)}$, weight $\frac1k \downarrow 0$.

Pitfall. Trying the comparison or ratio test on oscillating series. Those tests see only absolute values and are blind to cancellation; Abel/Dirichlet is the tool that SEES the oscillation.

2The recognition drill: ninety seconds to a first move

This week's sprint format: read a problem, commit to a first move within ninety seconds, write ONE line justifying it, then either execute or move to the next. The point is not solving — it is training the classifier. After eight weeks you own roughly a dozen openers: fixed points, auxiliary function, telescope, extremal object, generating function, filter, valuation, pigeonhole-by-dimension, reflect the integral, interrogate the functional equation, Abel, invariant. A trained eye maps problem to opener before conscious reasoning finishes.

Log every misclassification — the problem where you said 'generating function' and the real key was a valuation. Those logs ARE next week's consolidation syllabus: misrecognition, not misexecution, is what separates a 20 from a 40 on this exam, and it is invisible until you measure it.

One honest warning: recognition training on archive problems you have already seen inflates confidence. The sprint gates draw from unseen ranges deliberately. Resist re-solving old friends; the classifier only learns on novel inputs.

Worked example

Classify (do not solve): 'Prove that among any 51 integers chosen from $\{1, \dots, 100\}$, two are coprime.'

  1. Nudge 1

    51 out of 100 is a majority threshold — thresholds scream pigeonhole; ask what the boxes are.

    Reveal step 1

    Scan for structure: 51 out of 100 — a MAJORITY threshold. Thresholds scream pigeonhole; the question is what the boxes are.

  2. Nudge 2

    Design boxes where ANY two members are coprime — adjacent integers are the cheapest such pairs.

    Reveal step 2

    Design boxes so that any two items in one box satisfy the goal: consecutive pairs $\{1,2\}, \{3,4\}, \dots, \{99,100\}$ — 50 boxes, and consecutive integers are always coprime ($\gcd(n, n+1) = 1$, one Euclidean step).

  3. Nudge 3

    Count boxes against picks and let the collision finish it.

    Reveal step 3

    51 integers into 50 boxes force two in one box — coprime. Classification: pigeonhole with designed boxes; total time, under a minute. (The sibling problem — two where one DIVIDES the other — uses boxes indexed by odd parts; same 51/100 threshold, different box design.)

  4. Answer

    Pigeonhole; boxes = consecutive pairs; $\gcd(n, n+1) = 1$ closes it.

Pitfall. Confusing 'I recognize the technique' with 'I remember this problem'. The former transfers to December; the latter does not. Classify unseen problems only.

Before you open the gates

  • Oscillating factor times decaying weight → Abel/Dirichlet before anything else; bound the oscillation's partial sums via complex exponentials.
  • In the sprint: ninety seconds, one committed first move, one justification line — then execute or advance.
  • Log misclassifications verbatim; they are the only honest map of your blind spots.
  • Pigeonhole design order: name the threshold, then design boxes so any collision wins the problem.
  • Volume weeks are for the classifier, not the ego — unseen problems only.

Check yourself

1. $\sum_{n=2}^{\infty} \frac{\cos n}{\ln n}$ — the right convergence tool is:

2. Abel summation trades $\sum a_k b_k$ into partial sums $A_k$ against:

3. 51 integers from $\{1,\dots,100\}$ must contain two where one divides the other. The pigeonhole boxes are indexed by:

Practice ladder — three rungs, rising

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

Rung 1 (Dirichlet, direct). Prove that $\sum_{n=1}^{\infty} \frac{\cos n}{\sqrt{n}}$ converges.

One hint

Bound the partial sums of $\cos n$ using the geometric sum of $e^{in}$ (note $1$ is not a multiple of $2\pi$), then match Dirichlet's test with the weight $1/\sqrt{n}$.

Worked resolution

The partial sums of $a_k = \cos k$ are bounded: $\left|\sum_{k=1}^{n}\cos k\right| \le \left|\sum_{k=1}^{n} e^{ik}\right| \le \frac{2}{|e^{i}-1|} = \frac{1}{|\sin(1/2)|}$, independent of $n$ (here $1 \notin 2\pi\mathbb{Z}$, so $e^{i} \ne 1$). The weight $b_k = \frac{1}{\sqrt{k}}$ decreases monotonically to $0$. By Dirichlet's test, $\sum_{n=1}^{\infty} \frac{\cos n}{\sqrt{n}}$ converges. (It is not absolute: $|\cos n| \ge \cos^2 n = \frac{1+\cos 2n}{2}$, and $\sum \frac{1}{2\sqrt n}$ diverges while $\sum \frac{\cos 2n}{2\sqrt n}$ converges by the same test, so $\sum \frac{|\cos n|}{\sqrt n}$ diverges.) $\blacksquare$ [Source: this week's Abel-summation section — Dirichlet's test on an oscillating factor against a weight decreasing to zero; PnB §3.2.10.]

Rung 2 (pigeonhole, designed boxes). Prove that among any $51$ integers chosen from $\{1, 2, \dots, 100\}$, there are two such that one divides the other.

One hint

Write each integer as $2^{a} m$ with $m$ odd. How many possible odd parts $m$ are there in this range, and what happens when two chosen integers share one?

Worked resolution

Write each chosen integer uniquely as $n = 2^{a} m$ with $m$ odd. The odd part $m$ lies in $\{1, 3, 5, \dots, 99\}$, a set of exactly $50$ elements — these are the boxes. With $51$ integers and $50$ boxes, pigeonhole forces two chosen integers $2^{a} m$ and $2^{b} m$ with $a < b$ to share an odd part $m$. Then $2^{a} m \mid 2^{b} m$, so one divides the other. $\blacksquare$ [Source: this week's recognition-drill section — the same $51/100$ threshold as the worked coprime example, re-boxed by odd part.]

Rung 3 (linear algebra in combinatorics — Oddtown). In a town of $n$ citizens, clubs form so that every club has an odd number of members and every two distinct clubs share an even number of members. Prove that there are at most $n$ clubs.

One hint

Encode each club as its indicator vector in $\mathbb{F}_2^{n}$. Translate 'odd size' and 'even intersection' into dot products modulo $2$, then prove the vectors are linearly independent.

Worked resolution

Let the clubs be $C_1, \dots, C_m$ with indicator vectors $v_1, \dots, v_m \in \mathbb{F}_2^{n}$. Over $\mathbb{F}_2$ the dot product satisfies $v_i \cdot v_j \equiv |C_i \cap C_j| \pmod 2$, so the hypotheses give $v_i \cdot v_i = |C_i| \equiv 1$ and $v_i \cdot v_j \equiv 0$ for $i \ne j$. Suppose $\sum_{i} a_i v_i = 0$ with $a_i \in \mathbb{F}_2$. Dotting with $v_j$: $0 = \sum_i a_i (v_i \cdot v_j) = a_j (v_j \cdot v_j) = a_j$, since the cross terms vanish. Hence every $a_j = 0$, so $v_1, \dots, v_m$ are linearly independent in $\mathbb{F}_2^{n}$, giving $m \le \dim \mathbb{F}_2^{n} = n$. $\blacksquare$ [Source: this week's Linear-Algebra-in-Combinatorics hidden tool — the Oddtown dimension bound from Yufei Zhao's algcomb handout §1–2.]

Prove it — constructed response

Prove that for every real number $x$, the infinite series $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$ converges. Name the convergence test you use, verify each of its hypotheses against this series, and handle the degenerate values of $x$ explicitly.

The gates

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

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

why this gate: the lesson's §2 The recognition drill: ninety seconds to a first move is what it trains

sources & assignments (5)
  • Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source 3Blue1Brown — Essence of Linear Algebra — Intuition companion for “AMC/AIME Speed Warmup (build pattern recognition + speed)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Archive (hand-picked) — 3 AMC/AIME combinatorics speed reps (open each below) · ~8 min eachAMC/AIME · archive: AMC 12 B2021 AMC 10 BAMC 10 A

Track A — Analysis

Track A — Analysis: Archive + Advanced Techniques (1.5 hrs/day)

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- Archive: 8 Putnam A2/B2 · Analysis · mixed (series/convergence, MVT/continuity, integration) · 20 min each - Do: Stanford 05wk6 probs 5–7 - Cold re-solve: 4 Rudin exercises from Weeks 1–8 error list — zero notes

sources & assignments (6)
  • Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.01SC — Single Variable Calculus (OCW) — Intuition companion for “Track A — Analysis: Archive + Advanced Techniques (1.5 hrs/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Stanford 05wk6 hard bridge — Stanford 05wk6 probs 5–7Putnam
  • Problems Archive Browser — Putnam 2004 B2 · Analysis · 20 min eachPutnam

archive pull: 1 problems · Analysis · A2/B2 · Challenge · 20 min →

Hidden Tool

Hidden Tool: Abel Summation / Summation by Parts: Load

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*Prerequisites: series (Rudin Ch. 3) + integration (Rudin Ch. 6)* - Read: PnB section 3.2.10 (Abel summation) — or Rudin Ch. 3 notes on Abel's theorem - Do: MIT sum_integrals.pdf — 2 problems requiring Abel summation - Abel Summation: - Discrete: Σaₙbₙ = AₙBₙ|ₐᵇ − ΣAₙΔbₙ where Aₙ = Σa, Δbₙ = bₙ₊₁−bₙ (analogous to integration by parts) - Use when: bₙ monotone + Aₙ bounded → convergence by Dirichlet/Abel test - Abel's theorem: if Σaₙ converges, then lim_{x→1⁻} Σaₙxⁿ = Σaₙ

why this gate: the lesson's §1 Abel summation: integration by parts for sums is what it trains

sources & assignments (6)

Hidden Tool: Abel Summation / Summation by Parts: Drill

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unlocks after: Hidden Tool: Abel Summation / Summation by Parts: Load

*Prerequisites: series (Rudin Ch. 3) + integration (Rudin Ch. 6)* - Read: PnB section 3.2.10 (Abel summation) — or Rudin Ch. 3 notes on Abel's theorem - Do: MIT sum_integrals.pdf — 2 problems requiring Abel summation - Abel Summation: - Discrete: Σaₙbₙ = AₙBₙ|ₐᵇ − ΣAₙΔbₙ where Aₙ = Σa, Δbₙ = bₙ₊₁−bₙ (analogous to integration by parts) - Use when: bₙ monotone + Aₙ bounded → convergence by Dirichlet/Abel test - Abel's theorem: if Σaₙ converges, then lim_{x→1⁻} Σaₙxⁿ = Σaₙ

why this gate: the lesson's §1 Abel summation: integration by parts for sums is what it trains

sources & assignments (4)
  • Problems MIT sum_integrals.pdf — MIT sum_integrals.pdf: scan the problem list in order and solve the FIRST TWO whose solution requires Abel summation / summation by parts (deterministic rule — the handout contains exactly this pair). Write which two you solved in the reflection.Putnam

Track B — Algebra

Track B — Algebra: Archive Sprint (1 hr/day)

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- Archive: 4 Putnam A2/A3 · Algebra · mixed poly/ineq/FE · 20 min each - Do: 117 Polynomial Problems #75 and #90 - Do: 100 Functional Equations #30, 35

sources & assignments (7)
  • Source Yufei Zhao — Algebraic Techniques in Combinatorics (algcomb.pdf) — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source 3Blue1Brown — Essence of Linear Algebra — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Cezar Lupu — TTU MATH 4000, Lecture 12 — Intuition companion for “Track B — Algebra: Archive Sprint (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems 117 Polynomial — 117 Polynomial Problems #75 and #90Putnam
  • Problems 100 Functional Equations — 100 Functional Equations #30, 35Putnam
  • Problems Archive Browser — Putnam 1967 B3 · Algebra · 20 min eachPutnam

archive pull: 1 problems · Algebra · A2/A3 · Challenge · 20 min →

Track D — Linear Algebra

Track D — Linear Algebra: Cold Re-Solves (1 hr/day)

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- Cold re-solve: 5 Axler exercises from error list — zero notes - Archive: 3 Putnam A1/B1 · Linear Algebra · any slot · 15 min each

sources & assignments (5)
  • Source Yufei Zhao — Algebraic Techniques in Combinatorics (algcomb.pdf) — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source 3Blue1Brown — Essence of Linear Algebra — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Cezar Lupu — TTU MATH 4000, Lecture 12 — Intuition companion for “Track D — Linear Algebra: Cold Re-Solves (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Archive Browser — Putnam 2006 B1 · Linear Algebra · 15 min eachPutnam

archive pull: 1 problems · Linear Algebra · A1/B1 · Standard · 15 min →

Hidden Tool

Hidden Tool: Linear Algebra in Combinatorics: Load

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*Prerequisites: Axler Ch. 1–7 complete + combinatorics foundations* - Read: Yufei Zhao algebraic combinatorics handout (yufeizhao.com/olympiad/algcomb.pdf) — intro section - Do: 1 problem from Yufei Zhao handout - LA in Combinatorics: - Incidence matrix: A_{ij} = 1 if element i in set j; AᵀA counts intersections - Dimension argument: k vectors in d-dimensional space → k ≥ d → combinatorial bound - Linear independence over 𝔽₂: parity constraints → rank over 𝔽₂ - Polynomial method: embed set in polynomial ring; dim of polynomial space bounds set size

sources & assignments (5)

Hidden Tool: Linear Algebra in Combinatorics: Drill

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unlocks after: Hidden Tool: Linear Algebra in Combinatorics: Load

*Prerequisites: Axler Ch. 1–7 complete + combinatorics foundations* - Read: Yufei Zhao algebraic combinatorics handout (yufeizhao.com/olympiad/algcomb.pdf) — intro section - Do: 1 problem from Yufei Zhao handout - LA in Combinatorics: - Incidence matrix: A_{ij} = 1 if element i in set j; AᵀA counts intersections - Dimension argument: k vectors in d-dimensional space → k ≥ d → combinatorial bound - Linear independence over 𝔽₂: parity constraints → rank over 𝔽₂ - Polynomial method: embed set in polynomial ring; dim of polynomial space bounds set size

sources & assignments (5)
  • Source Yufei Zhao — Algebraic Techniques in Combinatorics (algcomb.pdf) — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source 3Blue1Brown — Essence of Linear Algebra — Drill reference — this gate trains the material of “Hidden Tool: Linear Algebra in Combinatorics: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Cezar Lupu — TTU MATH 4000, Lecture 12 — Intuition companion for “Hidden Tool: Linear Algebra in Combinatorics: Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems 1 — 1 problem from Yufei Zhao handout + adjacency-matrix micro-rep (~10 min, funded by trimming Track F's Stanford sheet 3 → 2): prove by induction that (A^k)_ij counts walks of length k from i to j; deduce trace(A²) = 2·#edges and trace(A³) = 6·#triangles.Putnam

Track C — Combinatorics

Track C — Combinatorics: Game Theory + Archive (1 hr/day)

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- Do: Engel Ch. 8 game-type probs remaining - Archive: 4 Putnam A1/A2 · Combinatorics · any slot · 15 min each (include game-type if available)

sources & assignments (6)
  • Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source 3Blue1Brown — Essence of Linear Algebra — Intuition companion for “Track C — Combinatorics: Game Theory + Archive (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Engel — Engel Ch. 8 game-type probs remaining — for each game prob, name the tool from this fixed list before reading any solution: invariant/monovariant · pairing strategy · strategy stealing · symmetry/Nim-value. Close with the Nim statement (first player wins iff the XOR of pile sizes is nonzero) — recognition only, no Nim reps.Putnam
  • Problems Archive Browser — Putnam 1979 B1 · Combinatorics · 15 min eachPutnam

archive pull: 1 problems · Combinatorics · A1/A2 · Challenge · 15 min →

Track E — Number Theory

Track E — Number Theory: Archive Sprint (0.75 hrs/day): Load

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- Read: PnB section 5.3.4 (Diophantine advanced pp. 279–290) - Do: 104 NT Problems #21, 23, 25, 27, 51, 55 (if not done) - Archive: 3 Putnam A1/A2 · NT · any slot · 15 min each - Do: MIT ss7.pdf probs 77–79

sources & assignments (3)

Track E — Number Theory: Archive Sprint (0.75 hrs/day): Drill

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unlocks after: Track E — Number Theory: Archive Sprint (0.75 hrs/day): Load

- Read: PnB section 5.3.4 (Diophantine advanced pp. 279–290) - Do: 104 NT Problems #21, 23, 25, 27, 51, 55 (if not done) - Archive: 3 Putnam A1/A2 · NT · any slot · 15 min each - Do: MIT ss7.pdf probs 77–79

sources & assignments (5)
  • Source Putnam and Beyond — Drill reference — this gate trains the material of “Track E — Number Theory: Archive Sprint (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Michael Penn — Number Theory v2 (playlist) — Drill reference — this gate trains the material of “Track E — Number Theory: Archive Sprint (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Michael Penn — Modular Arithmetic and Linear Congruences (playlist) — Intuition companion for “Track E — Number Theory: Archive Sprint (0.75 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems 104 NT — 104 NT Problems #21, 23, 25, 27, 51, 55 (if not done)Putnam
  • Problems MIT ss7.pdf — MIT ss7.pdf probs 77–79Putnam

Track E — Number Theory: Archive Sprint (0.75 hrs/day): Archive bridge

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unlocks after: Track E — Number Theory: Archive Sprint (0.75 hrs/day): Drill

- Read: PnB section 5.3.4 (Diophantine advanced pp. 279–290) - Do: 104 NT Problems #21, 23, 25, 27, 51, 55 (if not done) - Archive: 3 Putnam A1/A2 · NT · any slot · 15 min each - Do: MIT ss7.pdf probs 77–79

sources & assignments (4)
  • Source Putnam and Beyond — Drill reference — this gate trains the material of “Track E — Number Theory: Archive Sprint (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Michael Penn — Number Theory v2 (playlist) — Drill reference — this gate trains the material of “Track E — Number Theory: Archive Sprint (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Michael Penn — Modular Arithmetic and Linear Congruences (playlist) — Intuition companion for “Track E — Number Theory: Archive Sprint (0.75 hrs/day): Archive bridge” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Archive Browser — Putnam 2004 B1 · Number Theory · 15 min eachPutnam

archive pull: 1 problems · Number Theory · A1/A2 · Challenge · 15 min →

Track F — Geometry + Probability + Abstract

Track F — Geometry + Probability + Abstract (0.5 hrs/day): Load

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- Do: MIT probability.pdf probs 1–5 - Do: MIT indep.pdf probs 3–4 - Read: Dummit & Foote Ch. 1–2 — Lagrange, cosets, cyclic groups (now a written kernel/quotient rep, not recognition-only) - Do: MIT algebra.pdf probs 1–6 - Do: Stanford 07wk6 probs 1–3 - Probability: Linearity of expectation · Indicator variables · Law of total probability · Conditioning · Geometric probability · Symmetry/uniformity

sources & assignments (10)
  • Problems Abstract algebra — homomorphisms, kernels, quotients (self-contained) — For φ: ℤ → ℤ/nℤ, compute ker φ and im φ; state that the kernel is a (normal) subgroup and the image is a subgroup. Build the quotient table for ℤ/12ℤ modulo ⟨4⟩ and identify it as ℤ/4ℤ. State the first isomorphism theorem as a recognition trigger: G/ker φ ≅ im φ. (Replaces W9's 'Dummit & Foote — recognition only' line with concrete computation.)Putnam

Track F — Geometry + Probability + Abstract (0.5 hrs/day): Drill

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unlocks after: Track F — Geometry + Probability + Abstract (0.5 hrs/day): Load

- Do: MIT probability.pdf probs 1–4 + one self-contained Bayes/total-probability rep (false-positive computation) - Do: MIT indep.pdf probs 3–4 - Read: Dummit & Foote Ch. 1–2 — Lagrange, cosets, cyclic groups, Burnside (recognition only) - Do: MIT algebra.pdf probs 1–6 - Do: Stanford 07wk6 probs 1–3 - Probability: Linearity of expectation · Indicator variables · Law of total probability · Bayes' theorem · Conditioning · Geometric probability · Symmetry/uniformity

sources & assignments (8)
  • Source Dummit & Foote — Drill reference — this gate trains the material of “Track F — Geometry + Probability + Abstract (0.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Michael Penn — Abstract Algebra (playlist) — Drill reference — this gate trains the material of “Track F — Geometry + Probability + Abstract (0.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing. Watch-for: this playlist opens with set-theory / proof-writing videos — skip to the 'Abstract Algebra |' entries for the group/ring/field content this gate needs.
  • Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Intuition companion for “Track F — Geometry + Probability + Abstract (0.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems MIT — MIT probability.pdf probs 1–4Putnam
  • Problems Bayes drill (self-contained) — One false-positive computation built from scratch (displaces probability.pdf #5): a condition has 1% prevalence; a test is 95% sensitive and 95% specific; compute P(condition | positive test). Name the partition first, state the law of total probability, then apply Bayes. Close with one sentence on why the base rate dominates the answer.Standard
  • Problems MIT indep.pdf — MIT indep.pdf probs 3–4Putnam
  • Problems MIT algebra.pdf — MIT algebra.pdf probs 1–6Putnam
  • Problems Stanford 07wk6 — Stanford 07wk6 probs 1–2 (prob 3's slot displaced to the adjacency-matrix micro-rep in Track D)Putnam

FE Micro-Spine — Cauchy/Jensen recognition

FE Micro-Spine — Cauchy/Jensen recognition

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Added because FE cannot be a one-week topic; pattern exposure must recur after the W6 load.

Ritual: Write the substitution table before any solution check. Every FE rep must name the first move and the trap it avoided.

why this gate: the lesson's §2 The recognition drill: ninety seconds to a first move is what it trains

sources & assignments (2)
  • Source 100 Functional Equations / Putnam FE repair spine — Micro-rep source for repeated FE pattern exposure: special values, symmetry, injective/surjective forcing, Cauchy/Jensen, iteration, and polynomial-degree comparison.
  • Problems Functional equations micro-rep — 100 Functional Equations #40–41 — decide whether the equation is additive, Jensen-like, multiplicative, or polynomial before solving.Putnam

Problem-Solving Reps — weekly homework

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.

sources & assignments (3)
  • Problems PnB — PnB section 6.1–6.2 probs 1–6Putnam
  • Problems 102 Comb — 102 Combinatorial Problems — Introductory probs 5–8Putnam
  • Problems Kedlaya — Putnam 1985 A1,B1 · 1986 A1,B1Putnam

Reflect

Reflect: reconstruct and log the pattern

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- Reconstruct: 1 archived analysis problem → close → rewrite → pattern note - Reflect: full notes review — one line per tool ("when I see ___, first move is ___") - Verify: one combinatorics problem — prove the bijection inverse explicitly ---

Ritual: 1 archived analysis problem → close → rewrite → pattern note | full notes review — one line per tool ("when I see ___, first move is ___")

sources & assignments (4)
  • Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Interesting Integrals (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

- Reconstruct: 1 archived analysis problem → close → rewrite → pattern note - Reflect: full notes review — one line per tool ("when I see ___, first move is ___") - Verify: one combinatorics problem — prove the bijection inverse explicitly ---

Ritual: one combinatorics problem — prove the bijection inverse explicitly

sources & assignments (4)
  • Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Week reference — this gate applies this week's lead material (“Hidden Tool: Abel Summation / Summation by Parts: Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Interesting Integrals (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 9

Verification remaining

  • reading progress…

Carry-forward repairs

  • reading queue…

Next week opens with

W10 · All Tracks: Consolidation + Identify 2 Weakest
the exam follows the final taper week

Train Week 9 in the trainer →