Week 7 of 28 · Toolkit · due 2026-07-11 · 27–31 hrs

Analysis: Uniform Convergence + Algebra: Advanced Inequalities + LA: Spectral Theorem + Comb: Extremal + NT: Diophantine + Geometry begins + Hidden Tools: Auxiliary Functions + Trace/Det Identities + Graph Modeling

Analysis: Uniform Convergence + Algebra: Advanced Inequalities + LA: Spectral Theorem + Comb: Extremal + NT: Diophantine + Geometry begins + Hidden Tools: Auxiliary Functions + Trace/Det Identities + Graph Modeling gate complete

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This week's lesson · 11 min read

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Uniformity, auxiliary functions, and the art of the right object

This is the densest week of the toolkit, and its tracks share a secret: each one is about CONSTRUCTING the right object rather than analyzing the given one. Uniform convergence asks you to control infinitely many functions with one epsilon; auxiliary-function problems ask you to build the function whose zero is your theorem; extremal combinatorics asks you to interrogate the largest example. The given problem is raw material — the object you build is the solution.

When the problem's own objects resist, construct a better one: an auxiliary function, an extremal example, a spectral basis. Putnam rewards builders.

Key ideas — the week on one card

On a fixed bounded interval, uniform convergence — one $N$ working for every $x$ at once — licenses $\lim \int f_n = \int \lim f_n$; pointwise convergence does not.
To prove something about $f'$, build the auxiliary $g$ whose derivative IS that something, then apply Rolle or MVT to $g$.
The extremal object — largest, unextendable, minimal — carries constraints its siblings don't; interrogate it.
Real symmetric matrices behave: real eigenvalues, orthogonal eigenvectors, and the max of $x^T A x$ on the unit sphere is $\lambda_{\max}$.

1Uniform convergence: one epsilon to rule them all

Pointwise convergence lets each point $x$ pick its own $N$; uniform convergence demands one $N$ that works for EVERY $x$ simultaneously: $\sup_x |f_n(x) - f(x)| \to 0$. The distinction is not pedantry — it is exactly what decides whether limits commute with integrals, with continuity, and with each other. $f_n(x) = x^n$ on $[0,1]$ converges pointwise to a discontinuous limit; the convergence cannot be uniform, and the sup-distance (always $1$ near $x = 1$) says so.

The working test is the Weierstrass M-test: if $|f_n(x)| \le M_n$ for all $x$ and $\sum M_n < \infty$, then $\sum f_n$ converges uniformly. It converts a uniformity question into a numerical series question — which is why it settles most contest instances in one line: bound the $n$-th term by its sup, sum the sups.

What uniformity BUYS: the uniform limit of continuous functions is continuous; uniform convergence on $[a,b]$ lets you integrate term by term; and (with one extra hypothesis on derivatives) differentiate term by term. When a problem asks you to evaluate $\lim \int$ or $\int \sum$, the honest solution names the uniformity that licenses the swap — and the sneaky problems are built around swaps that FAIL.

Worked example

Justify and evaluate: $\lim_{n \to \infty} \int_0^1 \frac{n x^{n-1}}{1 + x}\,dx$.

Nudge 1
Test the naive swap first — the mass hiding near $x = 1$ is exactly what it misses.
Reveal step 1
The blind swap fails spectacularly: $nx^{n-1} \to 0$ pointwise on $[0,1)$, so swapping gives 0 — but the mass of $nx^{n-1}$ (which integrates to exactly 1) concentrates at $x = 1$. This is a delta-spike in disguise, the standard counterexample shape.
Nudge 2
Integrate by parts to move the $n$ onto the boundary term.
Reveal step 2
Correct route — integrate by parts to expose the boundary: $\int_0^1 \frac{n x^{n-1}}{1+x} dx = \left[\frac{x^n}{1+x}\right]_0^1 + \int_0^1 \frac{x^n}{(1+x)^2} dx = \frac{1}{2} + \int_0^1 \frac{x^n}{(1+x)^2}\,dx$.
Nudge 3
Squeeze the leftover integral with the crudest bound that still vanishes.
Reveal step 3
The remaining integral is squeezed: $0 \le \int_0^1 \frac{x^n}{(1+x)^2} dx \le \int_0^1 x^n dx = \frac{1}{n+1} \to 0$. So the limit is $\frac{1}{2}$ — the concentrating mass sampled the integrand's value at $x=1$, namely $\frac{1}{1+1}$.
Answer
$\frac{1}{2}$. The kernel $nx^{n-1}$ concentrates at $x=1$ and samples $\frac{1}{1+x}$ there; parts + squeeze makes it rigorous.

Pitfall. Swapping $\lim$ and $\int$ without a license. If the functions concentrate, spike, or escape to infinity, pointwise convergence lies to you. Always ask: where does the mass go?

2Auxiliary functions: build the witness-maker

The auxiliary-function method is IVT and Rolle upgraded from theorems to a design pattern. The recipe: (1) restate the goal as '$g$ vanishes somewhere' for a $g$ you get to design; (2) design $g$ so its endpoint values or its extremum force the vanishing; (3) apply the right existence theorem. The design step is the mathematics — the rest is bookkeeping.

Design heuristics that cover most problems: to compare $f$ against a chord or another function, subtract them. To prove a RATIO equals something, multiply through and subtract instead — cleaner zeros. To force a derivative condition like $f'(c) = \lambda f(c)$, kill the $\lambda$ with an integrating factor: $g(x) = e^{-\lambda x} f(x)$ has $g' = e^{-\lambda x}(f' - \lambda f)$, so Rolle on $g$ produces exactly the point you want. That exponential-multiplier move alone resolves a family of Putnam problems.

MVT itself is the pattern's first output: apply Rolle to $g(x) = f(x) - f(a) - \frac{f(b)-f(a)}{b-a}(x - a)$. Once you see MVT as a DESIGNED auxiliary function rather than a memorized theorem, you can design variants on demand — Cauchy's MVT ($g = (f(b)-f(a))h(x) - (h(b)-h(a))f(x)$) stops being a separate theorem to memorize.

Worked example

Let $f$ be continuous on $[0,1]$, differentiable on $(0,1)$, with $f(0) = 0$ and $f(1) = 1$. Show there is $c \in (0,1)$ with $f'(c) = 2c$.

Nudge 1
You want a zero of $f'(x) - 2x$. Which $g$ has that as ITS derivative?
Reveal step 1
Restate: we want a zero of $f'(x) - 2x$. That expression is the derivative of $g(x) = f(x) - x^2$ — the auxiliary function designs itself once the goal is written as a derivative.
Nudge 2
Evaluate your $g$ at both endpoints.
Reveal step 2
Check $g$ at the endpoints: $g(0) = f(0) - 0 = 0$ and $g(1) = f(1) - 1 = 0$. Equal values.
Nudge 3
Equal endpoint values — quote the right theorem, with its hypotheses.
Reveal step 3
Rolle on $g$ over $[0,1]$: some $c \in (0,1)$ has $g'(c) = 0$, i.e. $f'(c) = 2c$. Three lines total — the entire difficulty was recognizing $2c$ as $(x^2)'$.
Answer
Rolle applied to $g(x) = f(x) - x^2$, which vanishes at both endpoints.

Pitfall. Hunting for $c$ analytically. Existence theorems do not localize; if your solution computes a specific $c$, you have either solved a special case or solved the wrong problem.

3Extremal reasoning: interrogate the biggest example

The extremal method proves statements about ALL configurations by studying the BEST one. Standard opening: 'consider a configuration maximizing [the relevant quantity]' — legal whenever the search space is finite (or compactness buys it, Week 4). The maximizer has a superpower: any local improvement is impossible, and each impossible improvement is an inequality you now own for free.

The companion move in graph-flavored problems is minimal-counterexample: assume the claim fails, take a SMALLEST failure, and derive a smaller one — contradiction. Both are the same idea (well-ordering) pointed in opposite directions. Choose by asking which object is easier to interrogate: the best example, or the least criminal.

Concrete classic: any graph has a vertex of degree at least the average; a graph where every vertex has degree $\ge d$ contains a path of length $\ge d$ (take a MAXIMAL path — its endpoint's neighbors all live on the path, or the path would extend). 'Maximal, therefore its neighbors are trapped' is a sentence you will write a dozen times before December.

The extremal object at work: a path that cannot be extended traps all of v₀'s neighbors, and the farthest one closes the long cycle.

Worked example

Show that every finite simple graph in which every vertex has degree at least $d \ge 2$ contains a cycle of length at least $d + 1$.

Nudge 1
Take a path that cannot be extended — where must all of $v_0$'s neighbors live?
Reveal step 1
Take a MAXIMAL path $v_0 v_1 \cdots v_k$ (one that cannot be extended at either end). Every neighbor of $v_0$ must already lie ON the path — otherwise the path extends, contradicting maximality.
Nudge 2
Among those neighbors, look at the FARTHEST one along the path.
Reveal step 2
$v_0$ has at least $d$ neighbors, all among $v_1, \dots, v_k$. Let $v_j$ be the FARTHEST neighbor along the path; since the $d$ neighbors occupy distinct positions, $j \ge d$.
Nudge 3
Close the cycle through that far neighbor and count its length.
Reveal step 3
Close the loop: the cycle $v_0 v_1 \cdots v_j v_0$ (path plus the edge back) has length $j + 1 \ge d + 1$ — a genuine cycle since $j \ge d \ge 2$. The whole proof interrogated one extremal object — the unextendable path.
Answer
A maximal path's endpoint has all $\ge d$ neighbors on the path; the farthest one closes a cycle of length $\ge d+1$.

Pitfall. Saying 'take the longest path' without justifying existence. In a FINITE graph, fine — say the word finite. In infinite settings maximality needs an argument (or fails), and graders notice.

4Spectral theorem: symmetric matrices behave

The spectral theorem for real symmetric matrices delivers everything at once: all eigenvalues are real, eigenvectors of distinct eigenvalues are orthogonal, and there is an orthonormal basis of eigenvectors — $A = Q\Lambda Q^T$ with $Q$ orthogonal. In that basis a quadratic form $x^T A x$ becomes a weighted sum of squares $\sum \lambda_i y_i^2$, which makes optimization trivial: on the unit sphere, $x^T A x$ is maximized at $\lambda_{\max}$ and minimized at $\lambda_{\min}$.

That sentence — quadratic form bounds come from extreme eigenvalues — is the contest cash value. 'Show $x^T A x \ge 0$ for all $x$' means 'show eigenvalues are nonnegative'; 'bound this messy quadratic expression' means 'diagonalize the symmetric matrix behind it'. Symmetry is the hypothesis to CHECK first whenever a matrix inequality appears; without it, none of this machinery is licensed.

Worked example

Find the maximum of $2x^2 + 2xy + 2y^2$ subject to $x^2 + y^2 = 1$.

Nudge 1
Write the form as $x^T A x$ with a SYMMETRIC $A$ — split the cross term evenly.
Reveal step 1
Write the form as $\begin{pmatrix} x & y\end{pmatrix} A \begin{pmatrix} x \\ y \end{pmatrix}$ with symmetric $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ — the off-diagonal SPLITS the $xy$ coefficient.
Nudge 2
Two invariants, two eigenvalues: solve the characteristic quadratic.
Reveal step 2
Eigenvalues: $\det(\lambda I - A) = (\lambda - 2)^2 - 1 = 0$ gives $\lambda = 3, 1$.
Nudge 3
On the unit sphere, the form's maximum is the top eigenvalue.
Reveal step 3
Maximum on the unit circle is $\lambda_{\max} = 3$, attained at the eigenvector direction $\frac{1}{\sqrt2}(1, 1)$ — check: $x = y = \frac{1}{\sqrt2}$ gives $1 + 1 + 1 = 3$ ✓.
Answer
$3$, at $(x,y) = \pm\frac{1}{\sqrt{2}}(1,1)$ — the top eigenvalue of the symmetric matrix of the form.

Pitfall. Writing the $xy$ coefficient into ONE off-diagonal entry. Quadratic forms need the symmetric split — coefficient $c$ of $xy$ becomes $c/2$ in each of the two off-diagonal slots, or every eigenvalue you compute afterward is wrong.

Before you open the gates

Before swapping limit and integral, ask where the mass goes; concentration means the swap lies.
Write the target identity as 'derivative of WHAT vanishes' — the auxiliary function then designs itself; use $e^{\lambda x}$ multipliers to absorb linear terms.
Open extremal problems by naming the extremal object and the quantity it maximizes; every blocked improvement is a free inequality.
Check symmetry before touching matrix machinery; split cross terms evenly into the matrix.
This week's gates are heavy — read the lesson once, then let each gate re-teach its slice under time pressure.

Check yourself

1. $f_n(x) = \frac{x}{1 + n x^2}$ on $\mathbb{R}$. The convergence $f_n \to 0$ is:

2. To produce $c$ with $f'(c) = 3f(c)$ for suitable boundary data, the auxiliary function to Rolle is:

3. In a finite graph where every vertex has degree $\ge 3$, a maximal path guarantees:

Practice ladder — three rungs, rising

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

Rung 1 (spectral theorem, direct). Find the maximum and minimum of $3x^2 + 4xy + 3y^2$ subject to $x^2 + y^2 = 1$.

One hint

Write the form as $\mathbf{v}^T A \mathbf{v}$ with $A$ symmetric (split the $4xy$ evenly off the diagonal). On the unit circle the extremes are the largest and smallest eigenvalues of $A$.

Worked resolution

The form is $\mathbf{v}^T A \mathbf{v}$ with $A = \begin{pmatrix} 3 & 2 \\ 2 & 3 \end{pmatrix}$ (the cross-term coefficient $4$ splits as $2+2$). Its eigenvalues solve $\det(A-\lambda I) = (3-\lambda)^2 - 4 = 0$, so $3-\lambda = \pm 2$ and $\lambda = 5, 1$. On the unit circle the maximum is $\lambda_{\max} = 5$ (attained at the eigenvector $\tfrac{1}{\sqrt2}(1,1)$) and the minimum is $\lambda_{\min} = 1$ (at $\tfrac{1}{\sqrt2}(1,-1)$). Check: at $\tfrac{1}{\sqrt2}(1,1)$, $3\cdot\tfrac12 + 4\cdot\tfrac12 + 3\cdot\tfrac12 = 5$. ✓ [Source: this week's Track D — Linear Algebra: Axler Ch. 7 spectral theorem; on the unit sphere a quadratic form's extremes are the extreme eigenvalues of its symmetric matrix.]

Rung 2 (uniform convergence, Weierstrass M-test). Prove that $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$ converges uniformly on $\mathbb{R}$, and deduce that its sum is continuous.

One hint

Bound the $n$-th term by something independent of $x$ whose series converges, then quote the theorem that a uniform limit of continuous functions is continuous.

Worked resolution

For every real $x$ and every $n$, $\left|\frac{\sin(nx)}{n^2}\right| \le \frac{1}{n^2} =: M_n$, and $\sum_{n\ge1} M_n = \sum_{n\ge1} \frac{1}{n^2} < \infty$. By the Weierstrass M-test the series converges uniformly on all of $\mathbb{R}$. Each partial sum is a finite sum of continuous functions, hence continuous, and a uniform limit of continuous functions is continuous; therefore the sum $\sum_{n\ge1}\frac{\sin(nx)}{n^2}$ is continuous on $\mathbb{R}$. [Source: this week's Track A — Analysis: Rudin Ch. 7 (uniform convergence, the Weierstrass M-test, and continuity of the uniform limit).]

Rung 3 (extremal principle). In the plane are $n$ red points and $n$ blue points, no three collinear. Prove that the red and blue points can be matched in pairs so that the $n$ segments joining matched pairs are pairwise non-crossing.

One hint

Among all perfect red–blue matchings, take one of minimum total segment length. Show that if two of its segments crossed, an uncrossing swap would strictly shorten the total — impossible.

Worked resolution

There are finitely many perfect matchings of the reds to the blues; choose one, $M$, minimizing the total length of its $n$ segments. Suppose two of its segments $R_1B_1$ and $R_2B_2$ crossed at a point $P$. Rematch them as $R_1B_2$ and $R_2B_1$ (still a perfect matching). Since $P$ lies on both old segments, $|R_1B_1| + |R_2B_2| = (|R_1P|+|PB_2|) + (|R_2P|+|PB_1|)$, and the triangle inequality gives $|R_1B_2| \le |R_1P|+|PB_2|$ and $|R_2B_1| \le |R_2P|+|PB_1|$, so $|R_1B_2|+|R_2B_1| \le |R_1B_1|+|R_2B_2|$. The inequality is strict, because $P$ (a genuine crossing) does not lie on segment $R_1B_2$ — otherwise $R_1, B_1, B_2$ would be collinear, as $P$ already lies on $R_1B_1$. So the rematch is strictly shorter, contradicting the minimality of $M$. Hence no two segments of $M$ cross. [Source: this week's Track C — Combinatorics: Extremal + Graph Theory (Engel Ch. 8); the minimize-a-quantity extremal argument, where each blocked improvement is a strict inequality.]

Prove it — constructed response

Suppose $f:[0,\infty)\to\mathbb{R}$ is differentiable and $f'(x) \ge c\,f(x)$ for all $x \ge 0$, where $c$ is a real constant. Prove that $f(x) \ge f(0)\,e^{cx}$ for all $x \ge 0$. Build the auxiliary function explicitly, differentiate it, and drive the conclusion from its sign.

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.

sources & assignments (5)

Source Rudin — Principles of Mathematical Analysis (PMA) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100A — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100B Lecture 8 — Convergence Tests; Power Series (OCW) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source Anulus Smaragdinus — Polynomials (playlist) — 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. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.

Problems Archive (hand-picked) — 3 AMC/AIME algebra speed reps (open each below) · ~8 min eachAMC/AIME · archive: 2021 AMC 10 B AMC 12 A AMC 12 B

Track A — Analysis

Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load

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- Read: Rudin *PMA* Ch. 7 (~35 pp.) — uniform convergence, Cauchy criterion, uniform convergence and continuity/integration/differentiation, equicontinuity, Arzelà-Ascoli, term-by-term differentiation of power series - Watch: MIT 18.100A — sequences of functions lecture (OCW) - Do: Rudin Ch. 7 ex 1, 3, 6, 8, 10, 14, 17, 20 - Do: PnB section 3.2.11 probs 1–5 - Do: MIT sum_integrals.pdf probs 6–8 - Do: CMU 07-Convergence — advanced section - Do: Stanford 06wk3 probs 5–8

sources & assignments (5)

Source Rudin — Principles of Mathematical Analysis (PMA) — Rudin *PMA* Ch. 7 (~35 pp.) — uniform convergence, Cauchy criterion, uniform convergence and continuity/integration/differentiation, equicontinuity, Arzelà-Ascoli, term-by-term differentiation of power series
Source MIT 18.100A — MIT 18.100A — sequences of functions lecture (OCW)
Source Radulescu — Problems in Real Analysis — Rădulescu *Problems in Real Analysis* Ch. 2 problems 1–5 + Ch. 4 problems 1–5
Source MIT 18.100B Lecture 8 — Convergence Tests; Power Series (OCW) — Full lecture — the series machinery Rudin Ch. 7 builds on.
Source Anulus Smaragdinus — Polynomials (playlist) — Intuition companion for “Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.

Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Drill

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unlocks after: Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load

sources & assignments (9)

Source Rudin — Principles of Mathematical Analysis (PMA) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.100A — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.100B Lecture 8 — Convergence Tests; Power Series (OCW) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Anulus Smaragdinus — Polynomials (playlist) — Intuition companion for “Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.

Problems Rudin — Rudin Ch. 7 ex 1, 3, 6, 8, 10, 14, 17, 20Putnam
Problems PnB — PnB section 3.2.11 probs 1–5Putnam
Problems MIT sum_integrals.pdf — MIT sum_integrals.pdf probs 6–8Putnam
Problems CMU 07-Convergence — CMU 07-Convergence — advanced section Also state for recognition: (a) limsup/liminf of a sequence (the sup/inf of its subsequential limits) and that lim exists iff limsup = liminf; (b) ℚ is countable but ℝ is uncountable (Cantor diagonal), recognition only; (c) a subset of ℝ is connected iff it is an interval.Putnam
Problems Stanford 06wk3 — Stanford 06wk3 probs 5–8Putnam

Hidden Tool

Hidden Tool: Auxiliary Functions

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*Prerequisites: MVT (Rudin Ch. 5) + convexity/Jensen (Week 5)* - Do — 3 targeted problems: 1. Prove: if f convex (f'' ≥ 0) and f(0) = 0, then g(x) = f(x)/x is non-decreasing for x > 0. Define g, compute g'(x) = [xf'(x)−f(x)]/x², apply MVT + convexity to show g' ≥ 0. 2. Prove: if f'(x) ≥ cf(x) for all x ≥ 0, then f(x) ≥ f(0)e^{cx}. Define g(x) = f(x)e^{−cx}, compute g'(x) ≥ 0, conclude g non-decreasing. 3. Engel Ch. 12 — locate one inequality solved via auxiliary function (search solutions for "let g(x) = ..."). Reproduce the technique on the following problem. - Auxiliary Function: - Define: g = f·e^{−cx} · · or g = f(x)/x · · or g = f − linearization · · or g = log f (if f > 0) - Compute: g'(x) analytically; simplify fully - Sign: show g' ≥ 0 or ≤ 0 using hypotheses on f - Conclude: g monotone → g(b) ≥ g(a) → inequality about f - Connection: "g' ≥ cf" is a differential inequality — same structure as the ODE card

why this gate: the lesson's §2 Auxiliary functions: build the witness-maker is what it trains

sources & assignments (6)

Source Larson — Problem-Solving Through Problems — Larson *Problem-Solving Through Problems* §6.4 (auxiliary functions / MVT) — do problems 6.4.1–6.4.5
Source Rudin — Principles of Mathematical Analysis (PMA) — Rudin Ch. 5 — MVT and derivative applications (revisit)
Source Michael Penn — Putnam Exam Solutions (playlist) — Michael Penn — MVT & auxiliary-function inequality technique (companion to Larson/Rudin Ch. 5)
Source MIT 18.100B Lecture 16 — Rolle, MVT, Taylor (OCW) — Full lecture; auxiliary-function constructions all run through MVT.
Source Anulus Smaragdinus — Polynomials (playlist) — Intuition companion for “Hidden Tool: Auxiliary Functions” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.

Problems Targeted drill (built on this gate's reading) — (1) Prove: if f convex (f'' ≥ 0) and f(0) = 0, then g(x) = f(x)/x is non-decreasing for x > 0. Define g, compute g'(x) = [xf'(x)−f(x)]/x², apply MVT + convexity to show g' ≥ 0. (2) Prove: if f'(x) ≥ cf(x) for all x ≥ 0, then f(x) ≥ f(0)e^{cx}. Define g(x) = f(x)e^{−cx}, compute g'(x) ≥ 0, conclude g non-decreasing. (3) Engel Ch. 12 — locate one inequality solved via auxiliary function (search solutions for "let g(x) = ..."). Reproduce the technique on the following problem.Drill

Track B — Algebra

Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load

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- Read: Engel Ch. 12 probs 16–25 (SOS, Schur, weighted AM-GM) - Read: Larson *Problem-Solving Through Problems* — Inequalities section: 3 harder problems - Do: Stanford 06wk6 probs 4–6 - Do: 100 Functional Equations #35 (FE reinforcement) - Archive: 3 Putnam A2/B2 · Algebra · mixed ineq+FE · 20 min each

sources & assignments (6)

Source Engel — Problem-Solving Strategies — Engel Ch. 12 probs 16–25 (SOS, Schur, weighted AM-GM)
Source Larson *Problem-Solving Through — Larson *Problem-Solving Through Problems* §6.4 (auxiliary functions / MVT) — do problems 6.4.1–6.4.5
Source little fermat — Olympiad Inequalities Tutorial (playlist) — little fermat — olympiad inequalities: SOS, Schur, weighted AM-GM (companion to Engel Ch. 12)
Source Evan Chen — Olympiad Inequalities — Evan Chen *Olympiad Inequalities* §1–3 (SOS, smoothing, normalization) — read + do worked problems 1–4
Source Michael Penn — Putnam Exam Solutions (playlist) — Lecture series — direct playlist companion to this gate's reading
Source little fermat FE Tutorial — Intuition companion for “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Drill

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unlocks after: Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load

sources & assignments (6)

Source Engel — Problem-Solving Strategies — Drill reference — this gate trains the material of “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Putnam Exam Solutions (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat FE Tutorial — Intuition companion for “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Stanford 06wk6 — Stanford 06wk6 probs 4–6Putnam
Problems 100 Functional Equations — 100 Functional Equations #35 (FE reinforcement)Putnam

Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Archive bridge

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unlocks after: Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Drill

sources & assignments (5)

Source Engel — Problem-Solving Strategies — Drill reference — this gate trains the material of “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Putnam Exam Solutions (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Advanced Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat FE Tutorial — Intuition companion for “Track B — Algebra: Advanced Inequalities (1.5 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 2024 A2, 2023 A2, 2022 A2, 2021 A2 · Algebra · 20 min eachPutnam

archive pull: 4 problems · Algebra · A2/B2 · Challenge · 20 min →

Track D — Linear Algebra

Track D — Linear Algebra: Axler Ch. 7 (1 hr/day): Load

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- Read: Axler Ch. 7A–7C (~55 pp.) — self-adjoint operators, normal operators, spectral theorem (ℂ: normal ↔ unitarily diagonalizable; ℝ: self-adjoint ↔ orthonormally diagonalizable), positive operators, Gram matrices - Watch: Sheldon Axler LADR lectures — lectures 14–16 - Watch: MIT 18.06 Strang — symmetric matrices and positive definiteness lecture - Do: Axler 7A: 1, 3, 5, 8 · 7B: 1, 5, 9 · 7C: 1, 3, 7 - Do: MIT linalg.pdf probs 5–6 - Do: Berkeley Ch. 7 section 7.8 (bilinear forms, Gram matrices, p. 134): 2 problems - Do: MIT ss5.pdf prob 53

sources & assignments (4)

Source Axler — Linear Algebra Done Right — Axler Ch. 7A–7C (~55 pp.) — self-adjoint operators, normal operators, spectral theorem (ℂ: normal ↔ unitarily diagonalizable; ℝ: self-adjoint ↔ orthonormally diagonalizable), positive operators, Gram matrices
Source Sheldon Axler — Linear Algebra Done Right (lecture videos) — Sheldon Axler LADR lectures — lectures 14–16
Source MIT 18.06 Strang — MIT 18.06 Strang — symmetric matrices and positive definiteness lecture
Source Michael Penn — Putnam Exam Solutions (playlist) — Intuition companion for “Track D — Linear Algebra: Axler Ch. 7 (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: Axler Ch. 7 (1 hr/day): Drill

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unlocks after: Track D — Linear Algebra: Axler Ch. 7 (1 hr/day): Load

sources & assignments (8)

Source Axler — Linear Algebra Done Right — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 7 (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Sheldon Axler — Linear Algebra Done Right (lecture videos) — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 7 (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.06 Strang — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 7 (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat FE Tutorial — Intuition companion for “Track D — Linear Algebra: Axler Ch. 7 (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Axler 7A — Axler 7A: 1, 3, 5, 8 · 7B: 1, 5, 9 · 7C: 1, 3, 7Putnam
Problems MIT linalg.pdf — MIT linalg.pdf probs 5–6Putnam
Problems Berkeley — Berkeley Ch. 7 section 7.8 (bilinear forms, Gram matrices, p. 134): 2 problemsPutnam
Problems MIT ss5.pdf — MIT ss5.pdf prob 53Putnam

Hidden Tool

Hidden Tool: Trace + Determinant Identities: Load

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*Prerequisites: Axler Ch. 5 (eigenvalues) + Ch. 7 (spectral)* - Read: PnB section 2.3.2 (determinants pp. 60–66) + section 2.3.6 (eigenvalue/trace/det relations) - Do: PnB section 2.3.2 probs 1–2 · section 2.3.6 probs 1–3 - Trace/Det: - tr(A) = Σλᵢ · det(A) = Πλᵢ - det(AB) = det(A)det(B) — use to factor; det(A⁻¹) = 1/det(A) - Cayley-Hamilton: A satisfies its own characteristic polynomial - tr(AᵀA) = Σaᵢⱼ² — if tr(AᵀA) = 0 and A real then A = 0 - Rank 1: A = uvᵀ → tr(A) = u·v, det(A) = 0

sources & assignments (9)

Source Putnam and Beyond — PnB section 2.3.2 (determinants pp. 60–66) + section 2.3.6 (eigenvalue/trace/det relations)
Source Axler — Linear Algebra Done Right — Axler Ch. 10 — trace and determinant of operators
Source Yufei Zhao — Linear Algebra Tricks for the Putnam — Yufei Zhao *Linear Algebra Tricks for the Putnam* (8 pp.) — read fully; do problems 1–4 (trace/det/eigenvalue)
Source 3Blue1Brown — Essence of Linear Algebra — 3Blue1Brown — determinant & change of basis (Essence of Linear Algebra; companion to Axler Ch. 10) (opens the official Essence of Linear Algebra series)
Source MIT 18.06SC — Linear Algebra (Strang) — Self-contained OCW: four subspaces, projections, spectral decomposition
Source MathTheBeautiful (Grinfeld) — Linear Algebra — Optional — Rigorous, intuition-first linear algebra — deeper grounding when Axler gets abstract
Source MIT 18.A34 — linalg.pdf (Yufei Zhao) — Yufei Zhao 18.A34 linalg.pdf — work the linear-algebra Putnam set, problems 1–5
Source MathTheBeautiful — Linear Algebra (Part 1) — Optional — In-depth proof-based LA series (companion to Axler)
Source MIT 18.06SC — Linear Algebra (video playlist) — Optional — Strang lectures + problem-solving videos

Hidden Tool: Trace + Determinant Identities: Drill

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unlocks after: Hidden Tool: Trace + Determinant Identities: Load

sources & assignments (4)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Hidden Tool: Trace + Determinant Identities: 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: Trace + Determinant Identities: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MathDoctorBob — Intuition companion for “Hidden Tool: Trace + Determinant Identities: Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems PnB — PnB section 2.3.2 probs 1–2 · section 2.3.6 probs 1–3 Also: (a) nilpotent recognition — if Aᵏ = 0 then every eigenvalue is 0, so tr(A) = 0 and det(A) = 0; idempotent — if A² = A then eigenvalues ∈ {0,1} and rank(A) = tr(A); (b) prove the n×n Vandermonde determinant with nodes x₁,…,xₙ equals Π_{i<j}(xⱼ − xᵢ).Putnam

Track C — Combinatorics

Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Load

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- Read: PnB section 6.2 — graph theory subsection (~8–10 pp.: trees, Euler paths, bipartite) - Read: Engel Ch. 8 — extremal + graph sections - Watch: Po-Shen Loh Extremal Combinatorics — CMU 21-738 lecture 1 - Do: 102 Combinatorial Problems #60, 64, 68, 72, 74 - Do: CMU 10-Combinatorics exercise sheet probs 1–2 - Do: MIT comb.pdf probs 1–8

why this gate: the lesson's §3 Extremal reasoning: interrogate the biggest example is what it trains

sources & assignments (6)

Source Putnam and Beyond — PnB section 6.2 — graph theory subsection (~8–10 pp.: trees, Euler paths, bipartite)
Source Engel — Problem-Solving Strategies — Engel Ch. 8 — extremal + graph sections
Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Po-Shen Loh Extremal Combinatorics — CMU 21-738 lecture 1
Source Shahriar Shahriari — Combinatorics, An Invitation (playlist) — Combinatorics lecture series — direct playlist companion to this gate's reading.
Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Intuition companion for “Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Source Mathematical Circles (Russian Experience) — Aligned circle companion: Ch. 5 Graphs-1. Use before extremal/graph reps; output one graph-model translation.

Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Drill

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unlocks after: Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Load

why this gate: the lesson's §3 Extremal reasoning: interrogate the biggest example is what it trains

sources & assignments (8)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): 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 “Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Shahriar Shahriari — Combinatorics, An Invitation (playlist) — Drill reference — this gate trains the material of “Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.100B Lecture 17 — Taylor Remainder; Riemann Integrals (OCW) — Intuition companion for “Track C — Combinatorics: Extremal + Graph Theory (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems 102 Combinatorial — 102 Combinatorial Problems #60, 64, 68, 72 (#74's slot displaced to the planarity reps)Putnam
Problems CMU 10-Combinatorics exercise sheet — CMU 10-Combinatorics exercise sheet prob 1 (prob 2's slot displaced to the Erdős–Szekeres rep)Putnam
Problems MIT comb.pdf — MIT comb.pdf probs 1–7 (prob 8's slot displaced to the planarity reps)Putnam
Problems Erdős–Szekeres rep (self-contained) — Prove Erdős–Szekeres: any sequence of n² + 1 distinct reals contains a monotone subsequence of length n + 1. Label each term with the pair (length of longest increasing run ending there, longest decreasing run ending there), show the labels are distinct, and pigeonhole. One sentence on the recognition trigger: 'long sequence, forced order structure'.Putnam

Hidden Tool

Hidden Tool: Graph Modeling

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*Prerequisites: combinatorics foundations (Weeks 3–5) + graph theory reading (Track C above)* - Do: MIT ss10.pdf probs 119–120 (graph coloring/traversal) - Do: MIT ss11.pdf probs 1–2 (Euler path) - Graph Modeling: - First: "What are the vertices? What are the edges?" before any calculation - Euler circuit: connected + all vertices even degree - Euler path (not circuit): exactly 2 vertices odd degree - Bipartite ↔ no odd cycle; 2-colorable - Tree: connected + |E| = |V|−1 - χ(bipartite) ≤ 2; χ(planar) ≤ 4 (cite; don't prove)

sources & assignments (9)

Source Engel — Problem-Solving Strategies — Engel Ch. 8 — graph theory and extremal sections
Source Putnam and Beyond — PnB section 6.2 — graph theory subsection (trees, Euler paths, bipartite)
Source MIT 6.042J Combinatorial Games lecture (OCW, ~45 min) — MIT 6.042J — graph theory lecture (OCW)
Source Shahriar Shahriari — Combinatorics, An Invitation (playlist) — Combinatorics lecture series — direct playlist companion to this gate's reading.
Source MathDoctorBob — Intuition companion for “Hidden Tool: Graph Modeling” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Source Mathematical Circles (Russian Experience) — Aligned circle companion: Ch. 13 Graphs-2. Use as graph-model reinforcement; output one vertex/edge invariant or extremal trigger.

Problems MIT ss10.pdf — MIT ss10.pdf probs 119–120 (graph coloring/traversal)Putnam
Problems MIT ss11.pdf — MIT ss11.pdf probs 1–2 (Euler path)Putnam
Problems Planarity / Euler-formula reps (self-contained) — (1) Prove Euler's formula V − E + F = 2 for connected planar graphs by induction on edges (tree base case). (2) Derive E ≤ 3V − 6 from it (triangle-bound double count of face-edge incidences) and conclude K₅ is not planar; then derive E ≤ 2V − 4 for bipartite planar graphs (girth ≥ 4) and conclude K₃,₃ is not planar. Time funded by trimming the Track C drill (102 Comb #74 and comb.pdf prob 8 displaced).Putnam

Track E — Number Theory

Track E — Number Theory: Diophantine + Descent (0.75 hrs/day): Load

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- Read: PnB section 5.1.2 (Infinite Descent pp. 248–252) + section 5.3 (Diophantine pp. 270–285) - Read: Engel Ch. 13 probs 16–25 - Watch: Math Geeks Method of Infinite Descent — 1 video - Do: 104 NT Problems #21, 23, 25, 27 - Do: MIT cong.pdf probs 4–5 - Do: Stanford 07wk2 probs 3–4 - Do: MIT ss6.pdf probs 67–70

sources & assignments (5)

Source Putnam and Beyond — PnB section 5.1.2 (Infinite Descent pp. 248–252) + section 5.3 (Diophantine pp. 270–285)
Source Engel — Problem-Solving Strategies — Engel Ch. 13 probs 16–25
Source Math Geeks Method of Infinite Descent — Math Geeks Method of Infinite Descent — 1 video
Source Michael Penn — Number Theory v2 (playlist) — Number-theory lecture series — direct playlist companion to this gate's reading.
Source CMU 03-Number Theory (Putnam seminar) — Intuition companion for “Track E — Number Theory: Diophantine + Descent (0.75 hrs/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track E — Number Theory: Diophantine + Descent (0.75 hrs/day): Drill

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unlocks after: Track E — Number Theory: Diophantine + Descent (0.75 hrs/day): Load

sources & assignments (8)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track E — Number Theory: Diophantine + Descent (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Math Geeks Method of Infinite Descent — Drill reference — this gate trains the material of “Track E — Number Theory: Diophantine + Descent (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: Diophantine + Descent (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source CMU 03-Number Theory (Putnam seminar) — Intuition companion for “Track E — Number Theory: Diophantine + Descent (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, 27Putnam
Problems MIT cong.pdf — MIT cong.pdf probs 4–5Putnam
Problems Stanford 07wk2 — Stanford 07wk2 probs 3–4Putnam
Problems MIT ss6.pdf — MIT ss6.pdf probs 67–70Putnam

Track F — Geometry + Probability + Abstract

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

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- Read: PnB section 4.1–4.2 (pp. 201–240) — coordinate geometry (distance, midpoint, shoelace), dot product, complex plane (|z|, arg, rotation by e^{iθ}), Pick's theorem - Read: CMU 14-Geometry — lecture notes (scan, ~15 min) - Do: PnB section 4.1 probs 1–3

sources & assignments (9)

Source Putnam and Beyond — PnB section 4.1–4.2 (pp. 201–240) — coordinate geometry (distance, midpoint, shoelace), dot product, complex plane (|z|, arg, rotation by e^{iθ}), Pick's theorem
Source CMU 14-Geometry — CMU 14-Geometry — lecture notes (scan, ~15 min)
Source Michael Penn — Abstract Algebra (playlist) — Abstract-algebra lecture series — direct playlist companion to this gate's reading 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 West — Introduction to Graph Theory — West *Introduction to Graph Theory* Ch. 1 (trees & paths) + §3.1 (matchings) — read + do 5 exercises
Source Coxeter & Greitzer — Geometry Revisited — Coxeter–Greitzer *Geometry Revisited* Ch. 1 §1.1–1.4 problems 1–3 (low priority for Putnam)
Source AoPS Wiki — Power of a Point Theorem (statement, proofs, exercises) — The one synthetic tool Putnam geometry actually rewards. Read the three configurations (two secants / tangent-secant / two chords), then do: prove all three from similar triangles in your own writing, no peeking.
Source 3Blue1Brown — Essence of Linear Algebra — Intuition companion for “Track F — Geometry + Probability + Abstract: begins (0.5 hrs/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Source Mathematical Circles (Russian Experience) — Aligned circle companion: Ch. 14 Geometry. Use for coordinate/vector/synthetic model choice; output when to abandon synthetic geometry.

Problems Abstract algebra — groups & subgroups (self-contained) — State the subgroup test (nonempty, closed under the operation and inverses). Prove every subgroup of a cyclic group is cyclic. Classify all subgroups of ℤ/12ℤ by the divisors of 12 and draw the subgroup lattice. Compute the cyclic subgroup ⟨3⟩ in ℤ/12ℤ and its order. (Uses the Track F Abstract slot — the recognition-only reading becomes a written rep.)Putnam

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

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

sources & assignments (5)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track F — Geometry + Probability + Abstract: begins (0.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source CMU 14-Geometry — Drill reference — this gate trains the material of “Track F — Geometry + Probability + Abstract: begins (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: begins (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 — Essence of Linear Algebra — Intuition companion for “Track F — Geometry + Probability + Abstract: begins (0.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems PnB — PnB section 4.1 probs 1–3Putnam

Reach Contact — first-lemma attempt on A3

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Keeps A3 reach exposure alive while W7 teaches extremal/graph tools.

Ritual: A defended lemma or a precise obstruction counts. A vague “I was stuck” does not.

sources & assignments (2)

Source Putnam archive — A3 first-lemma contact — Use Putnam 2002 A3 as a combinatorics/extremal reach contact. Do not chase a full solution; produce a viable invariant/extremal first lemma.

Problems Putnam A3 first lemma — Putnam 1990 B4 — 20-minute attempt; submit only the first lemma, obstruction, or extremal setup you can defend.Nightmare

FE Micro-Spine — injective/surjective forcing

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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 — 100 Functional Equations #36–37 — before solving, write the special values tried: 0, 1, x=y, y=0; then identify whether injectivity or surjectivity is forced.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 Zeitz — Zeitz Ch. 4 graph-crossover probs 1–6Putnam
Problems 102 Comb — 102 Combinatorial Problems — Advanced probs 1–4 (extremal/graph)Putnam
Problems 104 NT — 104 Number Theory Problems — Introductory probs 5–8 (Diophantine)Putnam

Reflect

Reflect: reconstruct and log the pattern

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- Reconstruct: 1 Rudin Ch. 7 exercise → close → rewrite → pattern note - Reflect: uniform convergence trigger — when does the limit commute with ∫ and d/dx? - Reflect: extremal object name the object before claiming what it implies - Reflect: descent template from memory - Verify: one extremal proof — did you name the extremal object explicitly? ---

Ritual: 1 Rudin Ch. 7 exercise → close → rewrite → pattern note | uniform convergence trigger — when does the limit commute with ∫ and d/dx? | extremal object name the object before claiming what it implies | descent template from memory

sources & assignments (4)

Source Rudin — Principles of Mathematical Analysis (PMA) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100A — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100B Lecture 8 — Convergence Tests; Power Series (OCW) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.01SC — Single Variable Calculus (OCW) — 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

Ritual: one extremal proof — did you name the extremal object explicitly?

sources & assignments (4)

Source Rudin — Principles of Mathematical Analysis (PMA) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100A — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100B Lecture 8 — Convergence Tests; Power Series (OCW) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 7 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.01SC — Single Variable Calculus (OCW) — 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 7

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W8 · Analysis: Power Series + Algebra: Polynomial Advanced + LA: Applications + Comb: Recurrences + NT: Valuations + Hidden Tools: Feynman's Trick + Root-of-Unity Filter + Valuations
the exam follows the final taper week

Train Week 7 in the trainer →