W6 · Toolkit— gates27–31 hrs

Week 6 of 28 · Toolkit · due 2026-07-04 · 27–31 hrs

Analysis: Integration + FTC + Algebra: Functional Equations + LA: Inner Products + Comb: Generating Functions + NT: Congruences + Hidden Tools: ODE Recognition + Generating Functions

Analysis: Integration + FTC + Algebra: Functional Equations + LA: Inner Products + Comb: Generating Functions + NT: Congruences + Hidden Tools: ODE Recognition + Generating Functions gate complete

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This week's lesson · 10 min read

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Integrals with leverage: FTC, symmetry tricks, and the generating function idea

Putnam integrals are almost never exercises in finding antiderivatives — they are exercises in refusing to. The three moves that actually appear (reflection, differentiation under the integral, exploiting FTC as a bridge between size and slope) all work AROUND the integrand instead of through it. The same philosophy powers this week's other big opening: generating functions, where you refuse to count directly and let coefficients do the counting.

Choose the representation in which the problem is easy: reflect the integral, encode the sequence, take the inner product. Representation is a decision, not a given.

Key ideas — the week on one card

  1. The reflection $x \mapsto a + b - x$ fixes $\int_a^b$; when the integrand has partner symmetry, ADDING the two copies removes the hard factor.
  2. Functional equations are interrogations: substitute $0$, $x = y$, swaps — every substitution extracts one fact.
  3. $\langle f, g \rangle = \int fg$ is a genuine inner product on $C[a,b]$: Cauchy–Schwarz turns integral constraints into integral bounds.
  4. Generating functions make parts into factors: ordered compositions from allowed parts $S$ are counted by $\frac{1}{1 - \sum_{s \in S} x^s}$.

1The reflection trick and its relatives

The single most profitable line in contest integration: $\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$. Substituting $x \mapsto a+b-x$ costs nothing and often produces an integral that ADDS nicely to the original — the sum simplifies even when neither piece does. If $I = \int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$, reflection swaps sin and cos, so $2I = \int_0^{\pi/2} 1\,dx$ and $I = \pi/4$ with no antiderivative in sight.

The Fundamental Theorem of Calculus earns its name on this exam as a BRIDGE: it converts hypotheses about $f$ into conclusions about $\int f$ and back. If $f$ is continuous and $\int_0^1 f = 0$, then $F(x) = \int_0^x f$ satisfies $F(0) = F(1) = 0$ — and Rolle (last week) gives a point where $F' = f$ vanishes. Integration hypotheses plus MVT-family theorems is a standing combo.

Differentiation under the integral sign — $\frac{d}{dt}\int f(x,t)\,dx = \int \frac{\partial f}{\partial t}\,dx$ under mild conditions — turns a hard integral into a differential equation for it. Introduce a parameter where none exists ($\int_0^1 \frac{x^t - 1}{\ln x} dx$: differentiate in $t$, get $\frac{1}{t+1}$, integrate back). Feynman popularized it; Week 18 drills it.

mirror line x = π/4f(x)f(π/2−x)
The reflection x ↦ π/2 − x sends the solid curve to the dashed one; together they fill the box, so each integral is half of it.

Worked example

Evaluate $\int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x}\,dx$.

  1. Nudge 1

    Apply $x \mapsto \pi - x$ and write the SECOND expression for $I$.

    Reveal step 1

    Reflect with $x \mapsto \pi - x$: $\sin$ and $\cos^2$ are invariant, but $x$ becomes $\pi - x$. So $I = \int_0^\pi \frac{(\pi - x)\sin x}{1 + \cos^2 x} dx$.

  2. Nudge 2

    Add the two copies of $I$ — the lonely $x$ disappears.

    Reveal step 2

    Add the two expressions for $I$: $2I = \pi \int_0^{\pi} \frac{\sin x}{1 + \cos^2 x}\,dx$ — the $x$ is GONE, which was the whole obstruction.

  3. Nudge 3

    $u = \cos x$ turns what remains into an arctangent.

    Reveal step 3

    Substitute $u = \cos x$, $du = -\sin x\,dx$: $2I = \pi \int_{-1}^{1} \frac{du}{1 + u^2} = \pi \left[\arctan u\right]_{-1}^{1} = \pi \cdot \frac{\pi}{2}$. Hence $I = \frac{\pi^2}{4}$.

  4. Answer

    $\frac{\pi^2}{4}$ — reflection kills the linear factor, then $u = \cos x$ finishes.

Pitfall. Reflexively hunting an antiderivative. When the integrand mixes a polynomial factor with a symmetric trig body, reflect FIRST — the polynomial factor is usually the removable part.

2Functional equations: substitution is interrogation

A functional equation is a machine you interrogate with inputs. The standard opening sequence: plug $x = y = 0$; plug $y = 0$ with $x$ free; plug $y = x$ and $y = -x$; hunt for values forcing $f(\text{something}) = 0$ or a fixed constant. Each substitution is a QUESTION, and you should know what you are asking — 'is $f(0)$ forced?', 'is $f$ even?', 'can I isolate $f(x)$?'.

Cauchy's equation $f(x+y) = f(x) + f(y)$ is the ancestral form. Over $\mathbb{Q}$ its solutions are exactly $f(x) = cx$ — provable by building up from $f(1)$ through integers and fractions. Over $\mathbb{R}$, linearity needs ONE regularity crumb: continuity at a single point, monotonicity on an interval, or boundedness near zero all force $f(x) = cx$. Contest problems supply exactly one crumb; your job is to spot which and cite it.

Always finish with the verification loop: candidate solutions found by substitution must be CHECKED in the original equation, and the write-up must argue no others exist. Half credit lives in the difference between 'we found $f(x) = x$' and 'we showed every solution is $f(x) = x$'.

Worked example

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x + y) + f(x - y) = 2f(x)$ for all $x, y$.

  1. Nudge 1

    Feed the equation cheap inputs — $x = y = 0$, then $x = y$ — and normalize with $g = f - f(0)$.

    Reveal step 1

    Interrogate: $x = y = 0$ gives $2f(0) = 2f(0)$ — nothing. Try $x = y$: $f(2x) + f(0) = 2f(x)$. Now set $g(x) = f(x) - f(0)$, so $g(0) = 0$ and $g(2x) = 2g(x)$.

  2. Nudge 2

    Change variables so the equation says $g$ is additive (Jensen's shape).

    Reveal step 2

    The original equation in $g$: $g(x+y) + g(x-y) = 2g(x)$ (constants cancel). With $x = \frac{u+v}{2}, y = \frac{u-v}{2}$: $g(u) + g(v) = 2g(\frac{u+v}{2}) = g(u + v)$ using the doubling identity — Cauchy's equation for $g$.

  3. Nudge 3

    State the general solution honestly — regularity is what pins $g$ down to linear.

    Reveal step 3

    Without regularity, solutions over $\mathbb{R}$ are all additive functions: $f(x) = A(x) + c$ with $A$ additive. If the problem grants continuity (or any crumb), $A(x) = ax$ and $f(x) = ax + c$. Verify: $(a(x+y)+c) + (a(x-y)+c) = 2ax + 2c$ ✓. State which regularity you used — it is the difference between the two answers.

  4. Answer

    Exactly the additive-plus-constant functions; with any regularity crumb (continuity at a point suffices), $f(x) = ax + c$.

Pitfall. Announcing $f(x) = ax + c$ from Cauchy WITHOUT a regularity hypothesis. Pathological additive solutions exist (via a Hamel basis); the crumb is not optional, and grading rubrics know it.

3Inner products: geometry you can compute with

An inner product turns a vector space into geometry: lengths ($\|v\|^2 = \langle v, v\rangle$), angles, and orthogonality. The Cauchy–Schwarz inequality $|\langle u, v\rangle| \le \|u\| \|v\|$ you met as a sum inequality is really THIS statement — and choosing a clever inner product space is how it cracks integral inequalities: on $C[0,1]$ with $\langle f, g\rangle = \int_0^1 fg$, Cauchy–Schwarz reads $(\int fg)^2 \le \int f^2 \int g^2$.

Orthogonality is a computation device: if $\{e_i\}$ are orthonormal, coefficients are inner products ($v = \sum \langle v, e_i\rangle e_i$) and Pythagoras generalizes to $\|v\|^2 = \sum \langle v, e_i \rangle^2$ — Bessel/Parseval. The contest use: bound a sum of squares of 'components' by the square of the 'length', often with the trig system on $[0, 2\pi]$ or with rows of a matrix.

The projection formula — the closest point to $v$ in a subspace $W$ is the orthogonal projection, and the error $v - \operatorname{proj}_W v$ is orthogonal to $W$ — converts minimization problems into linear algebra. 'Minimize $\int_0^1 (x^2 - a - bx)^2 dx$ over $a, b$' is not calculus; it is projecting $x^2$ onto the span of $\{1, x\}$.

Worked example

For continuous $f$ on $[0,1]$ with $\int_0^1 f(x)\,dx = 1$, prove $\int_0^1 f(x)^2\,dx \ge 1$.

  1. Nudge 1

    The constraint is an inner product of $f$ with the constant function 1.

    Reveal step 1

    Recognize the inner product: on $C[0,1]$, take $\langle f, g \rangle = \int_0^1 fg$. The hypothesis says $\langle f, 1 \rangle = 1$, where $1$ is the constant function.

  2. Nudge 2

    Apply Cauchy–Schwarz to $\langle f, 1 \rangle$ and compute $\|1\|$.

    Reveal step 2

    Cauchy–Schwarz: $1 = |\langle f, 1\rangle|^2 \le \|f\|^2 \|1\|^2 = \left(\int_0^1 f^2\right)\left(\int_0^1 1\right) = \int_0^1 f^2$.

  3. Nudge 3

    Equality analysis: proportional to a constant means constant.

    Reveal step 3

    Equality iff $f$ is proportional to $1$, i.e. $f \equiv 1$ — consistent with the constraint. One line once the space is named; the entire solution is the CHOICE of inner product.

  4. Answer

    Cauchy–Schwarz against the constant function: $1 = \langle f, 1\rangle^2 \le \int f^2 \cdot \int 1^2$, equality iff $f \equiv 1$.

Pitfall. Verifying inner-product axioms in the write-up but forgetting the actual argument — or the reverse. On the exam: name the space, cite Cauchy–Schwarz, state equality. Three sentences; do not pad, do not skip.

4Generating functions: sequences as coefficients

A generating function packs a sequence into a power series $A(x) = \sum a_n x^n$, converting sequence operations into algebra: shifting multiplies by $x$, convolution of sequences is multiplication of series, and linear recurrences become rational functions. The Fibonacci archetype: $F(x) = \frac{x}{1 - x - x^2}$, read directly from the recurrence, and partial fractions then yields Binet's closed form with zero cleverness.

The identities you need daily: $\frac{1}{1-x} = \sum x^n$, its derivative-powered upgrade $\frac{1}{(1-x)^{k}} = \sum \binom{n+k-1}{k-1} x^n$ (stars and bars, encoded!), and the binomial theorem for arbitrary exponents. Counting problems phrased as 'number of ways to pay $n$ cents with coins of sizes $a, b, c$' are literally asking for a coefficient of $\frac{1}{(1-x^a)(1-x^b)(1-x^c)}$.

The evaluation trick runs the machine backward: identities among coefficients follow from evaluating the SERIES at points. $\sum_k \binom{n}{k} = 2^n$ is $(1+x)^n$ at $x = 1$; alternating sums are $x = -1$; and next week's roots-of-unity filter is 'evaluate at all $n$-th roots of unity and average' to extract every $k$-th coefficient.

Worked example

How many ways can $n \ge 0$ be written as an ordered sum of 1s and 2s (compositions with parts 1, 2)?

  1. Nudge 1

    Encode one part as $x + x^2$; a composition with $k$ parts is that factor to the $k$-th power.

    Reveal step 1

    Encode one part: a part is 1 or 2, so its generating function is $x + x^2$. A composition with $k$ parts contributes $(x + x^2)^k$; summing over all $k \ge 0$ gives $\frac{1}{1 - x - x^2}$ — the geometric series over $x + x^2$.

  2. Nudge 2

    Sum the geometric series over $k$ and read the recurrence off the denominator.

    Reveal step 2

    Recognize the series: $\frac{1}{1 - x - x^2} = \sum c_n x^n$ where the denominator relation forces $c_n = c_{n-1} + c_{n-2}$, $c_0 = c_1 = 1$ — Fibonacci shifted.

  3. Nudge 3

    Which famous sequence obeys $c_n = c_{n-1} + c_{n-2}$? Verify $n = 4$ by hand.

    Reveal step 3

    So the count is $F_{n+1}$ (with $F_1 = F_2 = 1$): compositions of 4 number $F_5 = 5$, and listing confirms: $1111, 112, 121, 211, 22$. The generating function DERIVED the recurrence; nothing was guessed.

  4. Answer

    $F_{n+1}$ — the generating function $\frac{1}{1-x-x^2}$ carries Fibonacci coefficients.

Pitfall. Worrying about convergence. Contest generating functions are FORMAL power series — coefficient bookkeeping, no analysis needed — unless you actually evaluate at a real point, at which point convergence becomes your problem again. Know which mode you are in.

Before you open the gates

  • Before integrating anything, test the reflection $x \mapsto a + b - x$; if the integrand simplifies when ADDED to itself, you are done in three lines.
  • Interrogate functional equations in a fixed order ($x=y=0$; one variable zero; $y = \pm x$), and know what question each substitution asks.
  • Every found solution of a functional equation gets verified in the original, and the write-up must exclude others.
  • See an integral inequality between products and squares → name the inner product space and let Cauchy–Schwarz speak.
  • Translate counting to coefficients when parts/choices multiply: encode one object, then take products and geometric series.

Check yourself

1. $\int_0^{\pi/2} \frac{\cos x}{\sin x + \cos x}\,dx$ equals:

2. $f(x+y) = f(x)f(y)$ for all reals, $f$ continuous, $f$ not identically zero. Then $f$ is:

3. The number of solutions of $a + b + c = 20$ in nonnegative integers is the coefficient of $x^{20}$ in:

Practice ladder — three rungs, rising

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

Rung 1 (functional equation, Cauchy reduction). Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ with $f(x+y) = f(x)+f(y)+2xy$ for all real $x,y$.

One hint

Peel off the piece responsible for the $2xy$: set $g(x) = f(x)-x^2$ and show $g$ solves Cauchy's equation, then spend the continuity hypothesis.

Worked resolution

Put $x=y=0$: $f(0) = 2f(0)$, so $f(0)=0$. Let $g(x) = f(x)-x^2$. Then $g(x+y) = f(x+y)-(x+y)^2 = [f(x)+f(y)+2xy]-x^2-2xy-y^2 = g(x)+g(y)$, so $g$ is additive. An additive function that is also continuous is linear, $g(x)=cx$, hence $f(x) = x^2 + cx$ for a constant $c$. Check: $(x+y)^2 + c(x+y) = [x^2+cx]+[y^2+cy]+2xy$. ✓ Continuity is exactly the crumb that excludes the pathological (Hamel-basis) additive solutions. [Source: this week's Track B — Algebra: Functional Equations; the Cauchy-reduction substitution toolkit, PnB section 3.4.1 + Engel Ch. 11.]

Rung 2 (inner-product inequality). Let $f:[0,1]\to\mathbb{R}$ be continuous with $\int_0^1 f(x)\,dx = 0$. Prove that $\left(\int_0^1 x\,f(x)\,dx\right)^2 \le \frac{1}{12}\int_0^1 f(x)^2\,dx$.

One hint

The constraint lets you replace $x$ by $x-\tfrac12$ for free. Then apply the Cauchy–Schwarz inequality $\left(\int gh\right)^2 \le \int g^2 \int h^2$ on $[0,1]$.

Worked resolution

Because $\int_0^1 f = 0$, adding a constant multiple of $f$ changes nothing: $\int_0^1 x f\,dx = \int_0^1 (x-\tfrac12) f\,dx$. Apply Cauchy–Schwarz for the inner product $\langle g,h\rangle = \int_0^1 gh$: $\left(\int_0^1 (x-\tfrac12)f\right)^2 \le \left(\int_0^1 (x-\tfrac12)^2\,dx\right)\left(\int_0^1 f^2\,dx\right)$. The first factor is $\int_0^1 (x-\tfrac12)^2\,dx = \tfrac{1}{12}$, so $\left(\int_0^1 x f\right)^2 \le \tfrac{1}{12}\int_0^1 f^2$. Subtracting the mean is the orthogonal projection off the constants — the geometry the constraint hands you. [Source: this week's Track D — Linear Algebra: Axler Ch. 6 inner products, with the integral Cauchy–Schwarz recognition item in PnB section 3.2.7.]

Rung 3 (generating function, closed form). Let $a_0 = 2$, $a_1 = 1$, and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 2$. Form $A(x) = \sum_{n\ge0} a_n x^n$, show $A(x) = \frac{2-x}{1-x-2x^2}$, and read off a closed form for $a_n$.

One hint

Multiply the recurrence by $x^n$ and sum over $n \ge 2$; write the shifted sums back in terms of $A(x)$. Then factor $1-x-2x^2$ and use partial fractions.

Worked resolution

Summing $a_n x^n$ over $n\ge2$ gives $A(x) - a_0 - a_1 x = x\big(A(x)-a_0\big) + 2x^2 A(x)$, i.e. $A - 2 - x = x(A-2) + 2x^2 A$. Solving, $A(x)(1 - x - 2x^2) = 2 - x$, so $A(x) = \frac{2-x}{1-x-2x^2}$. Factor $1-x-2x^2 = (1-2x)(1+x)$; partial fractions give $\frac{2-x}{(1-2x)(1+x)} = \frac{1}{1-2x} + \frac{1}{1+x} = \sum_{n\ge0} 2^n x^n + \sum_{n\ge0}(-1)^n x^n$. Hence $a_n = 2^n + (-1)^n$. Check: $a_0 = 2$, $a_1 = 1$, $a_2 = 5 = a_1 + 2a_0$. ✓ [Source: this week's Hidden Tool — Generating Functions (Wilf, generatingfunctionology); this is the linear-recurrence canon assigned in the GF drill gate.]

Prove it — constructed response

Prove that $\int_0^{\pi/2} \frac{dx}{1 + (\tan x)^{\sqrt{2}}} = \frac{\pi}{4}$. Reflection is your only real move: use it, then explain why the exponent $\sqrt{2}$ never enters the answer.

The gates

Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load

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- Read: Rudin *PMA* Ch. 6 (~38 pp.) — Riemann-Stieltjes integral, integrability, linearity, FTC (both forms), integration by parts, change of variables - Watch: Michael Penn Interesting Integrals — 3 videos - Watch: Silver Integration Bee Intermediate — 2 videos - Do: Rudin Ch. 6 ex 1, 3, 5, 7, 9, 11, 13, 16 - Do: PnB section 3.2.7 probs 1–3 - Do: MIT sum_integrals.pdf probs 1–5

sources & assignments (4)

Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Drill

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unlocks after: Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load

- Read: Rudin *PMA* Ch. 6 (~38 pp.) — Riemann-Stieltjes integral, integrability, linearity, FTC (both forms), integration by parts, change of variables - Watch: Michael Penn Interesting Integrals — 3 videos - Watch: Silver Integration Bee Intermediate — 2 videos - Do: Rudin Ch. 6 ex 1, 3, 5, 7, 9, 11, 13, 16 - Do: PnB section 3.2.7 probs 1–3 - Do: MIT sum_integrals.pdf probs 1–5 (write each as an explicit Riemann sum — partition + sample points — before converting)

sources & assignments (7)
  • Source Rudin — Principles of Mathematical Analysis (PMA) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 6 (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 — Interesting Integrals (playlist) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source Silver — Integration Bee Training (playlist) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 6 (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 — Intuition companion for “Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Rudin — Rudin Ch. 6 ex 1, 3, 5, 7, 9, 11, 13, 16Putnam
  • Problems PnB — PnB section 3.2.7 probs 1–3 Also, recognition-level: (a) l'Hôpital for 0/0 and ∞/∞ (and one case where it loops or fails); (b) ∫₁^∞ dx/xᵖ converges iff p > 1; (c) one integral inequality (Cauchy–Schwarz for integrals, ∫fg ≤ √(∫f²)√(∫g²)); (d) the Gamma function Γ(n) = (n−1)! and the Beta–Gamma relation B(x,y) = Γ(x)Γ(y)/Γ(x+y).Putnam
  • Problems MIT sum_integrals.pdf — MIT sum_integrals.pdf probs 1–5 — for each sum-to-integral conversion, write the limit explicitly as a Riemann sum first (name the partition and the sample points), then one line on why integrability justifies passing to the limitPutnam

Hidden Tool

Hidden Tool: ODE Recognition / Integral Equation → ODE: Load

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*Prerequisites: FTC (Track A above) + Leibniz rule (Week 5)* - Read: Paul's Online Math Notes — "Separable Equations" section + "Linear First Order" section only (~20 pp., free online) - Watch: Professor Leonard "Separable Differential Equations" (first 20 min) + "Integrating Factor Method" (first 15 min) - Do — 3 targeted problems: 1. Integral-to-ODE: given F(x) = ∫₀ˣ f(t)eˣ⁻ᵗ dt + x, differentiate both sides (Leibniz) → F'(x)−F(x)=1 → integrating factor μ=e^{−x} → F(x)=Ce^x−1. Work from scratch. 2. Paul's Online — Separable Equations exercises 1, 3, 5. Write dy/f(y) = g(x)dx explicitly before integrating. 3. Paul's Online — Linear First Order exercises 1, 3, 5. Write μ(x) = e^{∫P(x)dx} and verify d/dx[μf] = μQ before integrating. - ODE Recognition: - f' = kf → f = Ce^{kx} (recognize immediately — do not re-derive) - Integral equation F(x) = ∫₀ˣ f(t)h(x,t) dt → differentiate via Leibniz → first-order ODE in F - Separable: dy/f(y) = g(x) dx → integrate both sides independently - Integrating factor: f' + P(x)f = Q(x) → μ = e^{∫P dx} → (μf)' = μQ → integrate - Power series ODE: f = Σaₙxⁿ → substitute → coefficient recurrence (Week 8)

sources & assignments (5)

Hidden Tool: ODE Recognition / Integral Equation → ODE: Drill

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unlocks after: Hidden Tool: ODE Recognition / Integral Equation → ODE: Load

*Prerequisites: FTC (Track A above) + Leibniz rule (Week 5)* - Read: Paul's Online Math Notes — "Separable Equations" section + "Linear First Order" section only (~20 pp., free online) - Watch: Professor Leonard "Separable Differential Equations" (first 20 min) + "Integrating Factor Method" (first 15 min) - Do — 3 targeted problems: 1. Integral-to-ODE: given F(x) = ∫₀ˣ f(t)eˣ⁻ᵗ dt + x, differentiate both sides (Leibniz) → F'(x)−F(x)=1 → integrating factor μ=e^{−x} → F(x)=Ce^x−1. Work from scratch. 2. Paul's Online — Separable Equations exercises 1, 3, 5. Write dy/f(y) = g(x)dx explicitly before integrating. 3. Paul's Online — Linear First Order exercises 1, 3, 5. Write μ(x) = e^{∫P(x)dx} and verify d/dx[μf] = μQ before integrating. - ODE Recognition: - f' = kf → f = Ce^{kx} (recognize immediately — do not re-derive) - Integral equation F(x) = ∫₀ˣ f(t)h(x,t) dt → differentiate via Leibniz → first-order ODE in F - Separable: dy/f(y) = g(x) dx → integrate both sides independently - Integrating factor: f' + P(x)f = Q(x) → μ = e^{∫P dx} → (μf)' = μQ → integrate - Power series ODE: f = Σaₙxⁿ → substitute → coefficient recurrence (Week 8)

sources & assignments (4)
  • Source Paul's Online Math Notes — Differential Equations — Re-watch the worked-solution segment only AFTER attempting this gate's drill problems; note where your route diverged.
  • Source MIT 18.100A — Problem-session companion for “Hidden Tool: ODE Recognition / Integral Equation → ODE: Drill” — watch one worked problem, stop, finish it yourself on paper, then compare.
  • Source MathDoctorBob — Intuition companion for “Hidden Tool: ODE Recognition / Integral Equation → ODE: Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Targeted drill (built on this gate's reading) — (1) Integral-to-ODE: given F(x) = ∫₀ˣ f(t)eˣ⁻ᵗ dt + x, differentiate both sides (Leibniz) → F'(x)−F(x)=1 → integrating factor μ=e^{−x} → F(x)=Ce^x−1. Work from scratch. (2) Paul's Online — Separable Equations exercises 1, 3, 5. Write dy/f(y) = g(x)dx explicitly before integrating. (3) Paul's Online — Linear First Order exercises 1, 3, 5. Write μ(x) = e^{∫P(x)dx} and verify d/dx[μf] = μQ before integrating.Drill

Track B — Algebra

Track B — Algebra: Functional Equations (1.5 hrs/day): Load

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- Read: PnB section 3.4.1 (pp. 185–200) — functional equations fully with examples - Read: Engel Ch. 11 — intro + probs 1–10 - Watch: little fermat FE Tutorial — lessons 1–4 - Do: 100 Functional Equations #1, 5, 10, 15, 20, 25, 30 (record every substitution tried) - Do: CMU 05-FE exercise sheet probs 1–4 - Archive: 3 Putnam A1/A2 · Algebra/FE · any slot · 18 min each - FE Substitution: - x = y = 0: find f(0) — often forces f(0) = 0 or 1 - y = x: get f(2x) in terms of f(x) - y = −x: relate f(x) and f(−x) — parity - x = 1/x: use when domain is ℝ>0 - y = f(x): iterate — f(f(x)) appears - Injectivity/surjectivity: prove before using if domain argument requires it - Domain trap: does the FE force f onto all of ℝ, or only ℝ>0? FE substitution toolkit (work each problem through this list): try x=0, y=0, x=y, y=−x, and swapping x↔y; test injectivity (f(a)=f(b)⇒a=b) and surjectivity; iterate f(f(x)); guess f(x)=x, cx, x+c, then verify; pin down f(0) and f(1) first; only use continuity/monotonicity if the problem grants it.

why this gate: the lesson's §2 Functional equations: substitution is interrogation is what it trains

sources & assignments (7)

Track B — Algebra: Functional Equations (1.5 hrs/day): Drill

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unlocks after: Track B — Algebra: Functional Equations (1.5 hrs/day): Load

- Read: PnB section 3.4.1 (pp. 185–200) — functional equations fully with examples - Read: Engel Ch. 11 — intro + probs 1–10 - Watch: little fermat FE Tutorial — lessons 1–4 - Do: 100 Functional Equations #1, 5, 10, 15, 20, 25 + the iteration/fixed-point rep (tag techniques on every rep: special values · injective · surjective · symmetry · Cauchy reduction · fixed point · iteration · monotonicity/continuity · domain) - Do: CMU 05-FE exercise sheet probs 1–4 - Archive: 3 Putnam A1/A2 · Algebra/FE · any slot · 18 min each - FE Substitution: - x = y = 0: find f(0) — often forces f(0) = 0 or 1 - y = x: get f(2x) in terms of f(x) - y = −x: relate f(x) and f(−x) — parity - x = 1/x: use when domain is ℝ>0 - y = f(x): iterate — f(f(x)) appears - Injectivity/surjectivity: prove before using if domain argument requires it - Domain trap: does the FE force f onto all of ℝ, or only ℝ>0?

why this gate: the lesson's §2 Functional equations: substitution is interrogation is what it trains

sources & assignments (7)
  • Source Putnam and Beyond — Drill reference — this gate trains the material of “Track B — Algebra: Functional Equations (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 — Drill reference — this gate trains the material of “Track B — Algebra: Functional Equations (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: Functional Equations (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) — Intuition companion for “Track B — Algebra: Functional Equations (1.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems 100 Functional Equations — 100 Functional Equations #1, 5, 10, 15, 20, 25 (#30's slot displaced by the iteration rep below). For EVERY rep, before reading any solution, tag which techniques you used from this fixed list: special values · injectivity · surjectivity · symmetry/swap · Cauchy reduction · fixed point · iteration · monotonicity/continuity · domain restriction (Z/Q/R). Record every substitution tried.Putnam
  • Problems CMU 05-FE exercise sheet — CMU 05-FE exercise sheet probs 1–3 (prob 4's slot displaced to the polynomial-FE micro-rep) — same technique-tagging requirement on eachPutnam
  • Problems Iteration / fixed-point rep (self-contained) — (a) Describe all f: R → R with f(f(x)) = x (involutions): show injectivity is forced, give a non-identity example, and note the role of the fixed-point set. (b) Prove no CONTINUOUS f: R → R satisfies f(f(x)) = −x: continuous + injective ⟹ strictly monotone ⟹ f∘f increasing, but −x is decreasing. Tag both with the technique list. (c) Polynomial FE micro-rep: find all polynomials with P(x²) = P(x)² (answer: P(x) = xⁿ for n ≥ 0, plus P ≡ 0 — argue via leading coefficient + a too-many-roots/degree count, not by inspection). (d) Jensen-equation recognition, one sentence: f((x+y)/2) = (f(x)+f(y))/2 reduces to Cauchy's equation via g(x) = f(x) − f(0); additive + measurable/monotone ⟹ linear.Putnam

Track B — Algebra: Functional Equations (1.5 hrs/day): Archive bridge

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unlocks after: Track B — Algebra: Functional Equations (1.5 hrs/day): Drill

- Read: PnB section 3.4.1 (pp. 185–200) — functional equations fully with examples - Read: Engel Ch. 11 — intro + probs 1–10 - Watch: little fermat FE Tutorial — lessons 1–4 - Do: 100 Functional Equations #1, 5, 10, 15, 20, 25, 30 (record every substitution tried) - Do: CMU 05-FE exercise sheet probs 1–4 - Archive: 3 Putnam A1/A2 · Algebra/FE · any slot · 18 min each - FE Substitution: - x = y = 0: find f(0) — often forces f(0) = 0 or 1 - y = x: get f(2x) in terms of f(x) - y = −x: relate f(x) and f(−x) — parity - x = 1/x: use when domain is ℝ>0 - y = f(x): iterate — f(f(x)) appears - Injectivity/surjectivity: prove before using if domain argument requires it - Domain trap: does the FE force f onto all of ℝ, or only ℝ>0?

why this gate: the lesson's §2 Functional equations: substitution is interrogation is what it trains

sources & assignments (5)
  • Source Putnam and Beyond — Drill reference — this gate trains the material of “Track B — Algebra: Functional Equations (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 — Drill reference — this gate trains the material of “Track B — Algebra: Functional Equations (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: Functional Equations (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) — Intuition companion for “Track B — Algebra: Functional Equations (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 2008 A1, 2001 A1, 2000 A1, 1999 A1 · Algebra/FE/Algebra · 18 min eachPutnam

archive pull: 4 problems · Functional Equations/Algebra · A1/A2 · Challenge · 18 min →

Track D — Linear Algebra

Track D — Linear Algebra: Axler Ch. 5C + Ch. 6 (1 hr/day): Load

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- Read: Axler Ch. 5C (~20 pp.) — real vector spaces, complexification - Read: Axler Ch. 6A–6B (~40 pp.) — inner products, norms, Gram-Schmidt, orthogonal complements - Watch: Sheldon Axler LADR lectures — lectures 11–13 - Watch: MIT 18.06 Strang — orthogonality lecture (OCW) - Do: Axler 5C: 1, 3, 5 · 6A: 1, 5, 9, 12, 15, 18, 21 · 6B: 1, 5, 9, 13 - Do: MIT linalg.pdf probs 3–4

sources & assignments (4)

Track D — Linear Algebra: Axler Ch. 5C + Ch. 6 (1 hr/day): Drill

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unlocks after: Track D — Linear Algebra: Axler Ch. 5C + Ch. 6 (1 hr/day): Load

- Read: Axler Ch. 5C (~20 pp.) — real vector spaces, complexification - Read: Axler Ch. 6A–6B (~40 pp.) — inner products, norms, Gram-Schmidt, orthogonal complements - Watch: Sheldon Axler LADR lectures — lectures 11–13 - Watch: MIT 18.06 Strang — orthogonality lecture (OCW) - Do: Axler 5C: 1, 3, 5 · 6A: 1, 5, 9, 12, 15, 18, 21 · 6B: 1, 5, 9, 13 - Do: MIT linalg.pdf probs 3–4

sources & assignments (6)
  • Source Axler — Linear Algebra Done Right — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 5C + Ch. 6 (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. 5C + Ch. 6 (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. 5C + Ch. 6 (1 hr/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) — Intuition companion for “Track D — Linear Algebra: Axler Ch. 5C + Ch. 6 (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems Axler 5C — Axler 5C: 1, 3, 5 · 6A: 1, 5, 9, 12, 15, 18, 21 · 6B: 1, 5, 9, 13Putnam
  • Problems MIT linalg.pdf — MIT linalg.pdf probs 3–4Putnam

Hidden Tool

Hidden Tool: Generating Functions: Load

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*Prerequisites: sequences (Track A Week 3) + bijection/recursion (Track C Week 5)* - Read: Wilf *generatingfunctionology* intro through section 1.4 (~30 pp.) — OGF notation, coefficient extraction - Watch: Mohamed Omar Generating Series — lessons 1–2 - Do: MIT genfunc.pdf probs 1–6 - Do: CMU 08-Recursions exercise sheet probs 1–3 - Generating Function: - aₙ → F(x) = Σaₙxⁿ; extract: aₙ = [xⁿ]F(x) - Shift: aₙ₊₁ ↔ (F(x)−a₀)/x - Convolution: aₙ = Σbₖcₙ₋ₖ ↔ F(x) = B(x)C(x) - Recurrence → algebraic equation for F(x) → solve → partial fractions → [xⁿ] - EGF: aₙ/n! → use for labeled counting

why this gate: the lesson's §4 Generating functions: sequences as coefficients is what it trains

sources & assignments (4)

Hidden Tool: Generating Functions: Drill

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unlocks after: Hidden Tool: Generating Functions: Load

*Prerequisites: sequences (Track A Week 3) + bijection/recursion (Track C Week 5)* - Read: Wilf *generatingfunctionology* intro through section 1.4 (~30 pp.) — OGF notation, coefficient extraction - Watch: Mohamed Omar Generating Series — lessons 1–2 - Do: MIT genfunc.pdf probs 1–6 - Do: CMU 08-Recursions exercise sheet probs 1–3 - Generating Function: - aₙ → F(x) = Σaₙxⁿ; extract: aₙ = [xⁿ]F(x) - Shift: aₙ₊₁ ↔ (F(x)−a₀)/x - Convolution: aₙ = Σbₖcₙ₋ₖ ↔ F(x) = B(x)C(x) - Recurrence → algebraic equation for F(x) → solve → partial fractions → [xⁿ] - EGF: aₙ/n! → use for labeled counting

why this gate: the lesson's §4 Generating functions: sequences as coefficients is what it trains

sources & assignments (6)
  • Source Wilf — generatingfunctionology (full free PDF) — Wilf §1–2 — the generating-function method this drill trains. (Replaced a wrong-resource carryover: Paul's ODE notes had been attached to this GF gate.)
  • Source Michael Penn — Interesting Integrals (playlist) — Drill reference — this gate trains the material of “Hidden Tool: ODE Recognition / Integral Equation → ODE: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
  • Source MIT 18.100A — Intuition companion for “Hidden Tool: Generating Functions: Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems MIT genfunc.pdf — MIT genfunc.pdf probs 1–6Putnam
  • Problems CMU 08-Recursions exercise sheet — CMU 08-Recursions exercise sheet probs 1–2 (prob 3's slot displaced to the recurrence canon below)Putnam
  • Problems Linear-recurrence canon (self-contained) — Solve a_n = a_{n-1} + 2a_{n-2}, a_0 = 2, a_1 = 1, THREE ways and reconcile: (1) characteristic equation x² = x + 2 ⟹ roots 2, −1 ⟹ a_n = A·2^n + B·(−1)^n, fit constants; (2) generating function A(x) = Σ a_n x^n: derive A(x) = (2 − x)/(1 − x − 2x²), partial-fraction it over the roots, and read off the same closed form; (3) a nonhomogeneous variant a_n = a_{n-1} + 2a_{n-2} + 1 — find one particular solution (constant) and add the homogeneous part. Then one line: state the Fibonacci GF Σ F_n x^n = x/(1 − x − x²) and where the golden ratio enters as the dominant root (ties to the W8 asymptotics gate). Also note telescoping: Σ_{k=1}^n (b_k − b_{k-1}) = b_n − b_0 as the simplest 'recurrence' you should always spot. Also: (a) state the product/convolution rule on formal power series (convergence irrelevant) — if A(x)=Σaₙxⁿ and B(x)=Σbₙxⁿ then A(x)B(x) has coefficients cₙ = Σₖ aₖ bₙ₋ₖ (convolution); (b) the Catalan generating function C(x) = (1 − √(1−4x))/(2x) satisfies C = 1 + xC² — derive that functional equation from the Catalan recurrence; (c) matrix recurrence: write the Fibonacci recurrence as the power [[1,1],[1,0]]ⁿ and read F_n off the top-right entry.Putnam

Track C — Combinatorics

Track C — Combinatorics: Generating Functions + Recurrences (1 hr/day)

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- Do: 102 Combinatorial Problems #45, 50, 55 - Do: Stanford 06wk2 probs 1–6

why this gate: the lesson's §4 Generating functions: sequences as coefficients is what it trains

sources & assignments (6)
  • Source Rudin — Principles of Mathematical Analysis (PMA) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Interesting Integrals (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
  • Source Silver — Integration Bee Training (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Real Analysis (playlist) — Intuition companion for “Track C — Combinatorics: Generating Functions + Recurrences (1 hr/day)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
  • Problems 102 Combinatorial — 102 Combinatorial Problems #45, 50, 55Putnam
  • Problems Stanford 06wk2 — Stanford 06wk2 probs 1–6Putnam

Track E — Number Theory

Track E — Number Theory: Modular Arithmetic (0.75 hrs/day): Load

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- Read: PnB section 5.2.1–5.2.3 (pp. 257–265) — residue classes, CRT, Fermat, Euler - Read: Engel Ch. 13 probs 9–15 - Watch: Mu Prime Math modular arithmetic — 2 videos - Do: 104 NT Problems #12, 14, 16, 18 - Do: MIT cong.pdf probs 1–3 - Do: Stanford 07wk2 probs 1–2

sources & assignments (7)

Track E — Number Theory: Modular Arithmetic (0.75 hrs/day): Drill

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unlocks after: Track E — Number Theory: Modular Arithmetic (0.75 hrs/day): Load

- Read: PnB section 5.2.1–5.2.3 (pp. 257–265) — residue classes, CRT, Fermat, Euler - Read: Engel Ch. 13 probs 9–15 - Watch: Mu Prime Math modular arithmetic — 2 videos - Do: 104 NT Problems #12, 14, 16, 18 - Do: MIT cong.pdf probs 1–3 - Do: Stanford 07wk2 probs 1–2

sources & assignments (8)
  • Source Putnam and Beyond — Drill reference — this gate trains the material of “Track E — Number Theory: Modular Arithmetic (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) — Drill reference — this gate trains the material of “Track E — Number Theory: Modular Arithmetic (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: Modular Arithmetic (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 — Intuition companion for “Track E — Number Theory: Modular Arithmetic (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 #12, 14, 16, 18Putnam
  • Problems MIT cong.pdf — MIT cong.pdf probs 1–3Putnam
  • Problems Stanford 07wk2 — Stanford 07wk2 probs 1–2Putnam
  • Problems AoPS Floor Function article — Prove Hermite's identity sum_{k=0}^{n-1} floor(x + k/n) = floor(nx) using the article's splitting method, then write the fractional-part version of the statement. Also run the extended Euclidean algorithm on gcd(252, 198) and write explicit Bézout coefficients x, y with 252x + 198y = gcd (Bézout's identity).AMC

AMC/AIME Warmup

AMC/AIME Warmup (keep speed-solving active during heavy reading weeks)

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- Archive: 2 AMC-band · any topic · any slot · 8 min each

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. 6 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Interesting Integrals (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
  • Source Silver — Integration Bee Training (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 6 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
  • Source MIT 18.100A — Intuition companion for “AMC/AIME Warmup (keep speed-solving active during heavy reading weeks)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Hidden Tool

Hidden Tool: Chinese Remainder Theorem

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CRT was unnamed in the curriculum (≈1% of Putnam NT, but a basic, fast tool). This names + drills it.

sources & assignments (5)
  • Source CRT — statement & reconstruction — State CRT for coprime moduli; reconstruct x mod mn from x mod m and x mod n via the standard formula.
  • Source 104 Number Theory Problems — CRT / simultaneous-congruence problems — read the congruences section.
  • Source Michael Penn — Chinese Remainder Theorem — CRT statement, proof, and a worked reconstruction.
  • Problems 104 NT — 104 Number Theory Problems — CRT problems 1–2 (simultaneous congruences)Putnam
  • Problems CRT drill — Reconstruct x: x≡2 (mod 3), x≡3 (mod 5), x≡2 (mod 7); then state the general reconstruction recipe.AMC

archive pull: 1 problems · Number Theory · Putnam · 20 min →

Reach Contact — A3 official-solution autopsy

Reach Contact — A3 official-solution autopsy (lemma only)

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Introduces A3 contact immediately after the main FE launch so reach problems do not suddenly appear in Mastery.

Ritual: Goal is one certified first lemma, not a full solve. Stop at 35 minutes total.

sources & assignments (2)
  • Source Putnam archive — A3 autopsy model — Use Putnam 2016 A3 as a functional-equation flavored reach example: read after a 15-minute classification attempt, then extract the first forced substitution/lemma.
  • Problems Putnam A3 autopsy — Putnam 1991 B2 — 15-minute classification attempt, then solution autopsy. Output: first lemma/substitution and why it is forced.Nightmare

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 Engel — Engel Ch. 11 probs 1–8 (functional equations)Putnam
  • Problems Wilf — Generatingfunctionology Ch. 1–2 exercises 1–4Putnam
  • Problems 104 NT — 104 Number Theory Problems — Introductory probs 1–4 (congruences)Putnam

Reflect

Reflect: reconstruct and log the pattern

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- Reconstruct: 1 Rudin Ch. 6 exercise → close → rewrite → pattern note - Reflect: FTC both forms from memory · FE substitution card verified · GF coefficient extraction procedure - Verify: one FE — is injectivity/surjectivity claim proved, not assumed? ---

Ritual: 1 Rudin Ch. 6 exercise → close → rewrite → pattern note | FTC both forms from memory · FE substitution card verified · GF coefficient extraction procedure

sources & assignments (4)

Verify: audit one proof before closing

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unlocks after: Reflect: reconstruct and log the pattern

- Reconstruct: 1 Rudin Ch. 6 exercise → close → rewrite → pattern note - Reflect: FTC both forms from memory · FE substitution card verified · GF coefficient extraction procedure - Verify: one FE — is injectivity/surjectivity claim proved, not assumed? ---

Ritual: one FE — is injectivity/surjectivity claim proved, not assumed?

sources & assignments (4)

Lerma Training

Lerma Training: Combinatorics set (Northwestern)

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Source: Miguel A. Lerma, Northwestern Putnam team training problems (2023). Local PDF: /library/Lerma/putnam-training-2023.pdf. Hints + full solutions are in the PDF's later parts.

sources & assignments (5)
  • Problems Lerma Training 2023 §6 Generating Functions — Work problems 6.1–6.7 (4–6 per session). Encode a count as a coefficient; manipulate the closed form. Self-check against the PDF's hints/solutions parts.Putnam
  • Problems Lerma Training 2023 §9 Pigeonhole Principle — Work problems 9.1–9.11 (4–6 per session). Name the boxes and the objects before claiming a collision. Self-check against the PDF's hints/solutions parts.Putnam
  • Problems Lerma Training 2023 §12 Inclusion–Exclusion — Work problems 12.1–12.3 (4–6 per session). Alternate over- and under-counts; watch the sign pattern. Self-check against the PDF's hints/solutions parts.Putnam
  • Problems Lerma Training 2023 §13 Combinatorics and Probability — Work problems 13.1–13.8 (4–6 per session). Linearity of expectation, symmetry, complementary counting. Self-check against the PDF's hints/solutions parts.Putnam

Complex numbers keep-warm

Complex numbers keep-warm (20 min)

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

Ritual: Recall (closed book): State De Moivre's formula and compute (1+i)^8 mentally.

sources & assignments (2)
  • Problems Lerma Training 2023 §5 Complex Numbers — Problems 5.1–5.2 (~15 min at conversion speed). Then the recall prompt below — closed book.Putnam

Exit contract — Week 6

Verification remaining

  • reading progress…

Carry-forward repairs

  • reading queue…

Next week opens with

W7 · Analysis: Uniform Convergence + Algebra: Advanced Inequalities + LA: Spectral Theorem + Comb: Extremal + NT: Diophantine + Geometry begins + Hidden Tools: Auxiliary Functions + Trace/Det Identities + Graph Modeling
the exam follows the final taper week

Train Week 6 in the trainer →