Week 10 of 28 · Toolkit · due 2026-08-01 · 25–29 hrs

All Tracks: Consolidation + Identify 2 Weakest

All Tracks: Consolidation + Identify 2 Weakest gate complete

This week's lesson · 8 min read

not started · read first, then train the gates

Consolidation: finding the two topics that will cost you points

This week's product is not solved problems — it is a diagnosis. Two of your six tracks are weaker than the others, and until you name them with evidence, they will quietly tax every mock you take. The consolidation format exists to force honest measurement: linear algebra and number theory get maturity passes, every track gets a stress test, and by Sunday you commit — in writing — to the two topics that own your next fortnight.

You cannot fix what you refuse to rank: this week converts vague unease into two named weaknesses with problem-level evidence.

Key ideas — the week on one card

Audit in three layers — recall, execution, recognition — and weight recognition failures triple: unrouted problems never reach your toolkit.
Evidence means problem citations; feelings about a topic are not data.
The NT maturity reflex: for $n$ dividing an exponential expression, take the SMALLEST prime factor of $n$ and let order-divides collapse it.
For prime $p$ with $p \nmid a$: $\operatorname{ord}_p(a)$ divides both $p-1$ and every exponent $m$ with $a^m \equiv 1 \pmod p$ — the gcd of those is where contradictions come from.

1How to audit a topic honestly

A topic audit has three measurable layers. RECALL: can you state the five core theorems of the topic cold, with hypotheses — not 'MVT exists' but 'continuous on $[a,b]$, differentiable on $(a,b)$'? EXECUTION: given a problem you have classified correctly, do you finish it, and how often does the finish leak (algebra slips, unjustified steps)? RECOGNITION: on a mixed set, do you route the topic's problems to it at all? Score each layer per track this week; a topic is 'weak' when two of the three layers fail.

The evidence standard: cite specific problems. 'I feel shaky on number theory' is noise; '$v_p$ arguments stall when the valuations tie, and I misrouted two order-of-element problems as induction' is a training plan. Your misclassification log from Week 9 is exactly this evidence — read it before ranking anything.

Beware the comfort inversion: the topic you ENJOY most is often over-practiced and the one you avoid is under-measured. If a track has produced no wrong answers recently, that is usually insufficient exposure, not mastery — schedule its stress test first.

Worked example

A worked audit: you attempt six linear algebra problems and solve four. Weak or strong?

Nudge 1
Do not count the misses — classify them: execution slip or recognition failure?
Reveal step 1
Layer the misses, don't count them. Miss 1: you set up the right determinant but expanded a $3\times 3$ wrong — execution leak, cheap to fix with a verification habit.
Nudge 2
Which miss would still happen on exam day when nobody names the topic for you?
Reveal step 2
Miss 2: the problem wanted rank-nullity over $\mathbb{F}_2$ and you never considered working over a finite field — recognition failure, expensive, needs targeted reps (this is the Week 9 LA-in-combinatorics seam).
Nudge 3
Weigh the two miss types differently before issuing a verdict.
Reveal step 3
Verdict: 4/6 with one recognition failure in a signature Putnam pattern ranks LA as a deep-dive CANDIDATE — the raw score alone would have hidden that. Recognition failures weigh triple, because on exam day nobody tells you which problems are linear algebra.
Answer
Layered audit: execution leaks are cheap, recognition failures are expensive — rank by failure TYPE, not count.

Pitfall. Ranking topics by how the practice FELT. Feeling fluent and being fluent diverge hardest exactly where problems were familiar; only cold, unseen problems measure anything.

2The LA and NT maturity passes

Linear algebra maturity on this exam means the second-layer reflexes: determinant as signed volume and as a polynomial in entries; rank-nullity as a counting weapon; eigenvalue arguments where the matrix is a device you built, not an object you were given; trace as sum-of-eigenvalues cashed against characteristic-polynomial coefficients. The load-drill-bridge gate structure this week walks exactly this ladder — the bridge problems are where matrices meet combinatorics, and they predict December problems better than any other drill in the program.

Number theory maturity means the modular toolkit runs without friction: orders of elements ($\operatorname{ord}_p(a)$ divides $p-1$, and divides any exponent that kills $a$), Fermat/Euler as instant reflexes, CRT for splitting a modulus, and the valuation habits from Week 8. The test of maturity: given 'find all $n$ such that $n \mid 2^n - 1$', you should — within a minute — reach for the SMALLEST prime factor of a hypothetical $n > 1$ and let order-divides arguments collapse it.

Both passes end with the same ritual: write the topic's one-page technique card from memory, then check it against your notes. Gaps on the card are recall failures — the cheapest kind to fix and the most embarrassing to discover in December.

Worked example

Find all positive integers $n$ with $n \mid 2^n - 1$.

Nudge 1
Let $p$ be the SMALLEST prime factor of a hypothetical $n > 1$ and consider the order of 2 mod $p$.
Reveal step 1
$n = 1$ works. Suppose $n > 1$ and let $p$ be the SMALLEST prime dividing $n$. Then $2^n \equiv 1 \pmod p$, so $d = \operatorname{ord}_p(2)$ divides $n$. (Note $p$ is odd, since $2^n - 1$ is odd.)
Nudge 2
The order divides both $n$ and $p - 1$ — what is $\gcd(n, p-1)$ given how $p$ was chosen?
Reveal step 2
Also $d \mid p - 1$ by Fermat. So $d$ divides $\gcd(n, p-1)$. But every prime factor of $p - 1$ is smaller than $p$, hence smaller than every prime factor of $n$ — so $\gcd(n, p-1) = 1$.
Nudge 3
Order 1 forces $p \mid 1$ — read off the contradiction.
Reveal step 3
Therefore $d = 1$, meaning $2 \equiv 1 \pmod p$, i.e. $p \mid 1$ — impossible. Answer: only $n = 1$. The smallest-prime move manufactured the coprimality that kills the order.
Answer
Only $n = 1$. Smallest prime factor + order-divides is the signature NT maturity pattern.

Pitfall. Invoking $\operatorname{ord}_p$ without checking $\gcd(2, p) = 1$ (here: $p$ odd since $2^n - 1$ is odd). One line, and graders look for it.

Before you open the gates

Audit in three layers — recall, execution, recognition — and rank failures by type, with recognition weighted triple.
Evidence means problem citations; feelings about a topic are not data.
NT reflex: unknown $n$ dividing an exponential expression → smallest prime factor of $n$, then order-divides.
End each maturity pass by writing the technique card from memory, then diffing against notes.
Commit the two deep-dive topics in writing by Sunday — the next two weeks are built on that decision.

Check yourself

1. In the topic audit, which failure type weighs heaviest?

2. In the proof that $n \mid 2^n - 1$ forces $n = 1$, taking $p$ as the SMALLEST prime factor of $n$ ensures:

3. A track produced zero wrong answers in three weeks. The audit-honest reading is:

Practice ladder — three rungs, rising

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

Rung 1 (Fermat reflex). Find every prime $p$ such that $p \mid 2^{p} + 1$.

One hint

Fermat's little theorem pins $2^{p} \bmod p$ instantly. Reduce $2^{p} + 1$ modulo $p$ and read off the constraint.

Worked resolution

By Fermat's little theorem, $2^{p} \equiv 2 \pmod p$ for every prime $p$. Hence $2^{p} + 1 \equiv 3 \pmod p$, so $p \mid 2^{p}+1$ forces $p \mid 3$, i.e. $p = 3$. Check: $2^{3} + 1 = 9$ and $3 \mid 9$; and $p = 2$ fails since $2^{2}+1 = 5$. So $p = 3$ is the only prime. $\blacksquare$ [Source: this week's number-theory maturity pass — Fermat's little theorem as an instant reflex.]

Rung 2 (order divides two things). Let $p$ be an odd prime. Prove that every prime divisor $q$ of $2^{p} - 1$ satisfies $q \equiv 1 \pmod p$.

One hint

Let $d$ be the order of $2$ modulo $q$. It divides $p$ (prime!) and also divides $q-1$. Rule out $d = 1$.

Worked resolution

Since $2^{p} - 1$ is odd, $q$ is an odd prime, so $2$ is invertible modulo $q$ and $d = \operatorname{ord}_q(2)$ is defined. From $q \mid 2^{p} - 1$ we get $2^{p} \equiv 1 \pmod q$, so $d \mid p$. As $p$ is prime, $d = 1$ or $d = p$. If $d = 1$ then $2 \equiv 1 \pmod q$, i.e. $q \mid 1$ — impossible. Hence $d = p$. By Fermat's little theorem $d \mid q - 1$, so $p \mid q - 1$, that is $q \equiv 1 \pmod p$. $\blacksquare$ [Source: this week's number-theory maturity pass — the order of an element divides both the exponent and $q-1$.]

Rung 3 (trace as an invariant). Prove that there are no real $n \times n$ matrices $A, B$ with $AB - BA = I_n$.

One hint

Compute the trace of both sides. What is $\operatorname{trace}(AB) - \operatorname{trace}(BA)$, and what is $\operatorname{trace}(I_n)$?

Worked resolution

For any $n \times n$ matrices, $\operatorname{trace}(AB) = \sum_{i}\sum_{k} A_{ik}B_{ki} = \sum_{k}\sum_{i} B_{ki}A_{ik} = \operatorname{trace}(BA)$. Hence $\operatorname{trace}(AB - BA) = 0$. But $\operatorname{trace}(I_n) = n \ne 0$. So $AB - BA = I_n$ is impossible over $\mathbb{R}$. $\blacksquare$ [Source: this week's linear-algebra maturity pass — trace as a similarity invariant and sum of eigenvalues.]

Prove it — constructed response

Let $A$ be an $n \times n$ real matrix satisfying $A^2 = A$ (an idempotent, i.e. a projection). Prove that $\operatorname{rank}(A) = \operatorname{trace}(A)$. Make the eigenvalue / subspace structure explicit; do not appeal to a formula you have not justified.

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

Establishes the idempotent decomposition $\mathbb{R}^n = \operatorname{im}(A) \oplus \ker(A)$ — from $v = Av + (v-Av)$ with $A(v-Av)=0$, and $\operatorname{im}(A) \cap \ker(A) = \{0\}$ — equivalently, that $A$ is diagonalizable with every eigenvalue in $\{0,1\}$.
Shows $A$ acts as the identity on $\operatorname{im}(A)$ (since $A(Au) = A^2u = Au$) and as $0$ on $\ker(A)$, so in a basis adapted to the decomposition the matrix of $A$ is diagonal with exactly $r = \operatorname{rank}(A)$ ones and $n - r$ zeros.
Concludes $\operatorname{trace}(A) = r = \operatorname{rank}(A)$ by justifying trace-invariance under change of basis — from $\operatorname{trace}(XY) = \operatorname{trace}(YX)$ one gets $\operatorname{trace}(P^{-1}AP) = \operatorname{trace}(A)$ — and then summing the $r$ ones.

Model proof (compare AFTER grading yourself)

Write $r = \operatorname{rank}(A) = \dim \operatorname{im}(A)$. (Decomposition.) For any $v \in \mathbb{R}^n$, $v = Av + (v - Av)$; here $Av \in \operatorname{im}(A)$, and $A(v - Av) = Av - A^2 v = Av - Av = 0$, so $v - Av \in \ker(A)$. Thus $\operatorname{im}(A) + \ker(A) = \mathbb{R}^n$. If $w \in \operatorname{im}(A) \cap \ker(A)$, write $w = Au$; then $w = Au = A^2 u = A(Au) = Aw = 0$. So the sum is direct: $\mathbb{R}^n = \operatorname{im}(A) \oplus \ker(A)$, and $\dim\ker(A) = n - r$. (Action.) On $\operatorname{im}(A)$, $A$ is the identity: any $w = Au$ satisfies $Aw = A^2 u = Au = w$. On $\ker(A)$, $A$ is $0$. Take a basis of $\mathbb{R}^n$ formed by $r$ vectors spanning $\operatorname{im}(A)$ followed by $n - r$ vectors spanning $\ker(A)$; in this basis the matrix of $A$ is diagonal with exactly $r$ entries equal to $1$ and $n - r$ entries equal to $0$. (Trace.) Trace is invariant under change of basis: for any invertible $P$, $\operatorname{trace}(P^{-1}AP) = \operatorname{trace}\big((P^{-1}A)P\big) = \operatorname{trace}\big(P(P^{-1}A)\big) = \operatorname{trace}(A)$, using the identity $\operatorname{trace}(XY) = \operatorname{trace}(YX)$ (immediate from $\sum_{i}\sum_{k} X_{ik}Y_{ki} = \sum_{k}\sum_{i} Y_{ki}X_{ik}$). The matrix of $A$ in the adapted basis is $P^{-1}AP$ for the change-of-basis matrix $P$, so $\operatorname{trace}(A)$ equals the trace of that diagonal matrix, namely $r$. Therefore $\operatorname{trace}(A) = r = \operatorname{rank}(A)$. $\blacksquare$

Confidence in your proof (0–5): 2

The gates

Consolidation Resources — LA + NT maturity (study before the consolidation problems)

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Teaching/maturity support so this problem-heavy week has the resource quota: learn/consolidate the methods before (and after) the problems.

why this gate: the lesson's §2 The LA and NT maturity passes is what it trains

sources & assignments (4)

Source Sheldon Axler — Linear Algebra Done Right (lecture videos) — Author-led Linear Algebra Done Right series — consolidate Ch.1–7 before the LA archive problems
Source Michael Penn — Number Theory v2 (playlist) — Number-theory problem sessions — consolidate modular arithmetic/orders before the NT archive problems
Source Yufei Zhao — Linear Algebra Tricks for the Putnam — Yufei Zhao putnam_linear_algebra.pdf — re-derive the trace/det/eigenvalue identities from memory; redo any problem missed in Wk7–8
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — Writeup companion for “Consolidation Resources — LA + NT maturity (study before the consolidation problems)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

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 Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). 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 (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Change of basis (Essence of LA, Ch. 13) — 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 geometry speed reps (open each below) · ~8 min eachAMC/AIME · archive: 2021 AMC 10 B 2021 AMC 10 B 2021 AMC 10 B

Track A — Analysis

Track A — Analysis (1 hr/day)

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- Archive: 6 Putnam A2/B2 · Analysis · any slot · 20 min each - Cold re-solve: 3 Rudin exercises Ch. 3–7 from memory - Do: Abel summation follow-up from Week 9 · Stanford 06wk3 final problem

sources & assignments (6)

Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). 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 (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Calculus — Intuition companion for “Track A — Analysis (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Abel summation follow-up from Week 9 · Stanford 06wk3 final — Abel summation follow-up from Week 9 · Stanford 06wk3 final problemPutnam
Problems Archive Browser — Putnam 1999 B2 · Analysis · 20 min eachPutnam

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

Track B — Algebra

Track B — Algebra (1 hr/day)

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- Archive: 4 Putnam A2/A3 · Algebra · mixed · 20 min each - Do: MIT 18.A34 roots handout — 2 problems (root-of-unity filter reinforcement) - Do: PnB section 2.2 Newton's identities — 2 power-sum problems

sources & assignments (7)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track D — Linear Algebra (1 hr/day): 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 “Track D — Linear Algebra (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Drill reference — this gate trains the material of “Track D — Linear Algebra (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Newton's Sums — Intuition companion for “Track B — Algebra (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems MIT 18.A34 roots handout — MIT 18.A34 roots handout: solve, in order, the FIRST TWO problems solvable by the roots-of-unity filter (deterministic rule). State the filter identity you used for each.Putnam
Problems PnB — PnB section 2.2 Newton's identities — 2 power-sum problemsPutnam
Problems Archive Browser — Putnam 1981 B3 · Algebra · 20 min eachPutnam

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

Track D — Linear Algebra

Track D — Linear Algebra (1 hr/day): Load

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- Read: PnB section 2.3.7 (inner products / Gram matrices) - Do: PnB section 2.3.7 probs 1–2 · CMU 09-Linear exercise sheet probs 1–2 - Archive: 2 Putnam A1/B1 · Linear Algebra · any slot · 15 min each

sources & assignments (4)

Source Putnam and Beyond — PnB section 2.3.7 (inner products / Gram matrices)
Source 3Blue1Brown — Essence of Linear Algebra — 3Blue1Brown — Essence of Linear Algebra (refresher before consolidation) (opens the official Essence of Linear Algebra series)
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Watch fully before the eigenvalue exercises; write the geometric picture in one sentence.
Source Michael Penn — Newton's Sums — Intuition companion for “Track D — Linear Algebra (1 hr/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track D — Linear Algebra (1 hr/day): Drill

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unlocks after: Track D — Linear Algebra (1 hr/day): Load

sources & assignments (5)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track D — Linear Algebra (1 hr/day): 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 “Track D — Linear Algebra (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Drill reference — this gate trains the material of “Track D — Linear Algebra (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Newton's Sums — Intuition companion for “Track D — Linear Algebra (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems PnB — PnB section 2.3.7 probs 1–2 · CMU 09-Linear exercise sheet probs 1–2Putnam

Track D — Linear Algebra (1 hr/day): Archive bridge

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unlocks after: Track D — Linear Algebra (1 hr/day): Drill

sources & assignments (5)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track D — Linear Algebra (1 hr/day): 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 “Track D — Linear Algebra (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source 3Blue1Brown — Eigenvectors and eigenvalues (Essence of LA, Ch. 14) — Drill reference — this gate trains the material of “Track D — Linear Algebra (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Newton's Sums — Intuition companion for “Track D — Linear Algebra (1 hr/day): Archive bridge” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Archive Browser — Putnam 2000 B1 · Linear Algebra · 15 min eachPutnam

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

Track C — Combinatorics

Track C — Combinatorics (1 hr/day)

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- Archive: 4 Putnam A2/B2 · Combinatorics · any slot · 20 min each - Cold re-solve: 3 combinatorics problems from Weeks 6–8 from memory

sources & assignments (5)

Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). 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 (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source Maths 505 — Complex Analysis Lectures (playlist) — Intuition companion for “Track C — Combinatorics (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Archive Browser — Putnam 1995 B3 · Combinatorics · 20 min eachPutnam

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

Track E — Number Theory

Track E — Number Theory (0.75 hrs/day)

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- Archive: 5 Putnam A1/A2 · NT · all types (mod/Diophantine/descent/valuation) · 15 min each - Cold re-solve: 104 NT Problems #14, 16, 18 — reconstruct proof steps

sources & assignments (4)

Source Michael Penn — Modular Arithmetic and Linear Congruences (playlist) — Michael Penn — number theory problem sessions (modular arithmetic, orders, NT olympiad reps)
Source Math Geeks Method of Infinite Descent — Problem-session companion for “Track E — Number Theory (0.75 hrs/day)” — watch one worked problem, stop, finish it yourself on paper, then compare.
Source CMU 03-Number Theory (Putnam seminar) — Intuition companion for “Track E — Number Theory (0.75 hrs/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Archive Browser — Putnam 2007 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)

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- Archive: 3 Putnam A1/B1 · Probability + 2 · Abstract Algebra · 15 min each - Do: CMU 15-Elementary exercise sheet — 2 problems - Do: Stanford 07wk6 probs 1–3

sources & assignments (9)

Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). 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 (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Intuition companion for “Track F — Geometry + Probability + Abstract (0.5 hrs/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Source Michael Penn — Abstract Algebra (playlist) — Companion lecture series for this week's abstract-algebra rep — watch the relevant 1–2 videos, then do the written rep below. 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.

Problems CMU 15-Elementary exercise sheet — CMU 15-Elementary exercise sheet — 2 problemsPutnam
Problems Stanford 07wk6 — Stanford 07wk6 probs 1–3Putnam
Problems Archive Browser — Putnam 2002 B1 · Probability · 15 min eachPutnam
Problems Abstract algebra foundations check (closed-book) — Six questions, closed-book, self-graded: (1) state the subgroup test; (2) is {0,2,4} a subgroup of ℤ/6ℤ? prove or disprove; (3) list all subgroups of ℤ/8ℤ; (4) compute the order of 5 in the unit group (ℤ/12ℤ)*; (5) for φ: ℤ/12ℤ → ℤ/12ℤ, x ↦ 4x, find ker φ and im φ; (6) state Lagrange's theorem and one consequence. Anything shaky seeds the W11 weak-topic audit.Putnam

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

Geometry Power Tool

Geometry Power Tool: Coordinate / Trig / Vector Bashing

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Geometry is ~16% of the Putnam archive but was recognition/coordinate-capped. This names + drills the highest-yield method: most Putnam geometry falls to a clean coordinate/trig/vector setup, no synthetic theorems needed.

sources & assignments (6)

Source Putnam and Beyond — Geometry & Trigonometry — Coordinate geometry, vectors, dot/cross products, conics — set up axes, reduce a synthetic claim to algebra. Read the coordinate + vector sections, work probs 1–6.
Source When to bash (and how) — Pick the origin/axes that kill the most symmetry; distance²/dot-product for perpendicularity; Shoelace for areas; parametrize the locus. Putnam geometry is usually crackable WITHOUT synthetic theorems.
Source Coordinate geometry / vector methods for competition — Setting coordinates, vectors, and the Shoelace/area toolkit on contest problems.

Problems PnB — Putnam and Beyond — Geometry & Trigonometry: coordinate + vector probs 1–6Putnam
Problems MAA archive — Putnam 1974 B1 · 2011 A1 · 2012 A1 (set coordinates, reduce to algebra)Putnam
Problems MAA archive — Putnam 2002 B2 · 2010 B2 (vector / dot-product geometry)Putnam

archive pull: 3 problems · Geometry · Putnam · 45 min →

FE Micro-Spine — A1/B1 archive rep

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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.

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 — Putnam 2008 A1 — 20-minute attempt; output the substitution table before reading any solution.Putnam

Geometry Power Tool — cyclic + power-of-a-point recognition

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Explicit W10 geometry recognition patch so cyclic/power-of-point appears before the Mastery geometry track, not only later in W20.

Ritual: Before solving, write: cyclic? power? radical axis? coordinate bash? Pick one and give the reason.

sources & assignments (5)

Source Power of a Point / Radical Axis recognition card — Write the three power-of-a-point configurations from memory: two chords, two secants, tangent-secant. Add the radical-axis trigger: equal powers with respect to two circles.
Source Coxeter & Greitzer — Geometry Revisited — Ch. 2 cyclic quadrilaterals / circle power recognition: do problems 1–2 only, then return to the Putnam archive rep.
Source Mathematical Circles (Russian Experience) — Aligned circle companion: Ch. 14 Geometry. Use as a capped geometry replay before cyclic/power recognition; output one diagram-to-model rule.

Problems Geometry recognition drill — Prove from scratch: intersecting chords theorem, secant-secant theorem, tangent-secant theorem. Then write the one-line trigger for when each appears.Tutorial
Problems Putnam archive geometry replay — Putnam 2012 B2 — redo after the recognition card; explicitly decide whether cyclic/power-of-point helps or coordinate bash is cleaner.Putnam

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 Newman — Newman — problems 9–18Putnam
Problems Kedlaya — Putnam 1987 A1,A2,B1 · 1988 A1,B1Putnam
Problems Putnam Full Past Exam — Full Past Exam 01: Putnam 1995 A1, Putnam 1995 A2, Putnam 1995 A3, Putnam 1995 A4, Putnam 1995 A5, Putnam 1995 A6, Putnam 1995 B1, Putnam 1995 B2, Putnam 1995 B3, Putnam 1995 B4, Putnam 1995 B5, Putnam 1995 B6 - Scout protocol: Day 1 A1-A2-B1-B2, 80 minutes total; Day 2 A3-B3, 60 minutes lemma hunt; Day 3 A4-A6-B4-B6, 60 minutes first-move scan; Day 4 score all 12 slots and write three miss tags; Day 5 do one same-family repair pull before the next paper.Putnam

Mini-Mock — first calibration before deep dives

Mini-Mock — first calibration before deep dives (3 problems / 90 min)

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Moves the first serious mock pressure to Week 10 instead of waiting until the end-of-toolkit audit.

Ritual: After the block, classify each miss as concept gap, first-move gap, proof gap, or time-control gap.

sources & assignments (2)

Source Putnam early calibration protocol — Run one 90-minute block under exam law: no notes, no pause, write submit-ready proofs or clear abandon notes.

Problems Mini-mock set — Putnam 2016 A1, Putnam 2018 A1, Putnam 1982 B3 — 90 minutes continuous; score clean/partial/miss and log first-move failures.Putnam

Reach Bridge — A3 classification drill

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Bridges W8–W10 A3 contact into W14 systematic reach practice.

Ritual: For all three statements write: topic, likely tool, first move, bail condition.

sources & assignments (2)

Source Putnam A3 classification drill — Look at three A3 statements and classify topic, first move, and expected obstruction before attempting one.

Problems A3 classification drill — Classify Putnam 2014 A3, Putnam 2015 A3, Putnam 1995 B4; then attempt Putnam 1966 B4 for 25 minutes, lemma goal only.Nightmare

Reflect

Reflect: reconstruct and log the pattern

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- Reflect: identify 2 weakest topics across all tracks. Log them. Phase 3 will target these in spiral gates. - Verify: one NT proof — every divisibility claim explicitly proved? ---

Ritual: identify 2 weakest topics across all tracks. Log them. Phase 3 will target these in spiral gates.

sources & assignments (4)

Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). 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 (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source Silver — Integration Bee Training (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: identify 2 weakest topics across all tracks. Log them. Phase 3 will target these in spiral gates. - Verify: one NT proof — every divisibility claim explicitly proved? ---

Ritual: one NT proof — every divisibility claim explicitly proved?

why this gate: the lesson's §1 How to audit a topic honestly is what it trains

sources & assignments (4)

Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Week reference — this gate applies this week's lead material (“Track D — Linear Algebra (1 hr/day): Load”). 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 (“Track D — Linear Algebra (1 hr/day): Load”). If an attempt stalls, the repair source is here.
Source Silver — Integration Bee Training (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.

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): Write the sum of all n-th roots of unity and WHY it vanishes (n≥2).

sources & assignments (2)

Source Lerma Putnam Training 2023 §5 Complex Numbers — Work problems 5.3–5.4. Roots of unity / De Moivre stay warm between full complex weeks.

Problems Lerma Training 2023 §5 Complex Numbers — Problems 5.3–5.4 (~15 min at conversion speed). Then the recall prompt below — closed book.Putnam

Exit contract — Week 10

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W11 · Deep-Dive Week 1: Your 2 Weakest Topics
the exam follows the final taper week

Train Week 10 in the trainer →