Week 4 of 28 · Toolkit · due 2026-06-20 · 26–30 hrs

Analysis: Continuity + Algebra: Complex Numbers + LA: Linear Maps + Comb: IE + Hidden Tools: Finite Differences + Compactness

Analysis: Continuity + Algebra: Complex Numbers + LA: Linear Maps + Comb: IE + Hidden Tools: Finite Differences + Compactness gate complete

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Continuity buys theorems: IVT, compactness, and the complex plane

'Continuous' is the cheapest hypothesis a problem can hand you, and it buys the two most-used theorems on the entire exam: the Intermediate Value Theorem and the Extreme Value Theorem. This week you learn what those theorems cost internally — compactness — while the algebra track opens the complex plane, where every polynomial finally has all its roots and half of trigonometry collapses into one exponential.

Existence without construction: IVT, EVT, and the pigeonhole of dimension all conjure objects you could never write down explicitly. Learning to WANT a nonconstructive proof is this week's real skill.

Key ideas — the week on one card

IVT: continuous on $[a,b]$ with a sign change forces a zero — and 'fixed point' compiles to 'zero of $f(x) - x$'.
EVT: a continuous function on a closed bounded interval attains its extremes; growth at $\pm\infty$ lets you BUILD that interval.
$e^{i\theta} = \cos\theta + i\sin\theta$ turns trig sums into geometric series, and the $n$-th roots of unity sum to zero.
$\Delta$ drops polynomial degree by exactly one, so degree-$n$ data has constant $n$-th differences — extrapolate from the table.

1IVT: the fixed-point factory

The Intermediate Value Theorem says a continuous function on $[a,b]$ hits every value between $f(a)$ and $f(b)$. Its contest costume is almost always the same: to prove SOME point exists with property $P$, build an auxiliary function $g$ whose sign change encodes $P$, then let IVT produce the point. You never find the point; you certify its existence and move on.

The canonical build: to show a continuous $f: [0,1] \to [0,1]$ has a fixed point, set $g(x) = f(x) - x$. Then $g(0) = f(0) \ge 0$ and $g(1) = f(1) - 1 \le 0$, so IVT hands you a zero of $g$ — a fixed point of $f$. The entire art is choosing $g$; the theorem does the rest. Week 7's 'auxiliary functions' gate industrializes this move.

IVT is also why every odd-degree real polynomial has a real root: as $x \to \pm\infty$ the leading term dominates with opposite signs, so the polynomial changes sign on some huge interval. Note the division of labor — limits give the sign change, continuity converts it into a root.

IVT: a continuous curve cannot change sign without paying the toll at zero.

Worked example

A hiker walks up a mountain trail starting at 6am Saturday, arriving at 6pm. Sunday she descends the same trail, 6am to 6pm. Show there is a time of day at which she is at the SAME point of the trail on both days.

Nudge 1
Two walks on the same clock — define $u(t)$ and $d(t)$ and put both days on one time axis.
Reveal step 1
Model: let $u(t)$ be her position (distance along the trail) at time $t$ on Saturday, $d(t)$ on Sunday, both continuous on $[6, 18]$ with $u(6) = 0, u(18) = L$ and $d(6) = L, d(18) = 0$.
Nudge 2
Subtract them: what are the signs of $g$ at 6am and at 6pm?
Reveal step 2
Auxiliary function: $g(t) = u(t) - d(t)$. Then $g(6) = 0 - L < 0$ and $g(18) = L - 0 > 0$.
Nudge 3
A sign change plus continuity is the whole theorem.
Reveal step 3
IVT gives $t^*$ with $g(t^*) = 0$, i.e. $u(t^*) = d(t^*)$ — same time of day, same spot. No information about WHERE; existence is the entire claim.
Answer
Apply IVT to $g(t) = u(t) - d(t)$, which changes sign between 6am and 6pm.

Pitfall. Applying IVT without stating continuity, or on a domain where the function is not continuous everywhere. The hypothesis is one word; graders dock real points when it goes unsaid.

2Compactness and EVT: why maxima exist

The Extreme Value Theorem — a continuous function on a closed bounded interval attains a maximum and minimum — is the license behind every 'consider the point where $f$ is largest' argument. That opening move is one of the strongest in the game: the maximizer is a point with a PROPERTY (nothing beats it), and properties are leverage. But the license has terms: closed AND bounded. On $(0,1]$, the function $1/x$ is continuous and unbounded; on $[0,\infty)$, $x$ attains no max.

Under the hood is Bolzano–Weierstrass: every bounded sequence has a convergent subsequence. The EVT proof is worth internalizing as a pattern — take a sequence $x_n$ with $f(x_n) \to \sup f$, extract a convergent subsequence $x_{n_k} \to c$ by B–W, and continuity forces $f(c) = \sup f$, so the sup is attained. Sequence, subsequence, pass to the limit: this three-beat rhythm reappears constantly.

The contest reflex: whenever a problem says 'minimize', 'maximize', or 'closest', first secure that the extremum EXISTS (compact domain + continuity, or a growth argument pushing the search into a compact set), then interrogate the extremal point. Week 20's extremal combinatorics is this same reflex in a discrete world.

Worked example

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous with $f(x) \to +\infty$ as $x \to \pm\infty$. Show $f$ attains a global minimum.

Nudge 1
Growth at $\pm\infty$ buys a window outside of which $f$ is large — anchor it to the known value $f(0)$.
Reveal step 1
Anchor a value: pick any point, say $f(0) = c$. The growth hypothesis gives $M > 0$ such that $f(x) > c$ whenever $|x| > M$ — outside a compact window, $f$ is worse than at 0.
Nudge 2
Inside the closed window, which theorem hands you a minimum?
Reveal step 2
Inside the window: $[-M, M]$ is closed and bounded, so EVT applies — $f$ attains a minimum on it at some $x_0$, and $f(x_0) \le f(0) = c$ since $0 \in [-M, M]$.
Nudge 3
Check that the window minimum also beats every point outside the window.
Reveal step 3
Stitch: for $|x| > M$ we have $f(x) > c \ge f(x_0)$, and for $|x| \le M$, $f(x) \ge f(x_0)$ by choice of $x_0$. So $x_0$ is a GLOBAL minimizer. Growth confines the hunt; compactness finishes it.
Answer
Growth at infinity confines candidate minimizers to a compact interval $[-M, M]$; EVT attains the minimum there, and it beats everything outside.

Pitfall. Invoking EVT on a non-compact domain. The theorem needs closed and bounded; when the domain is all of $\mathbb{R}$, you must first BUY compactness with a growth argument, as above.

3Complex numbers: trigonometry's better notation

Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ is not a fact to admire — it is a compression algorithm. Every trig identity you have ever half-remembered is the real or imaginary part of an exponential identity you can re-derive in one line: $e^{i(a+b)} = e^{ia}e^{ib}$, expand, and both addition formulas fall out simultaneously. Stop memorizing; start expanding.

The $n$-th roots of unity $\omega_k = e^{2\pi i k/n}$ are the exam's favorite finite structure. Two facts do nearly all the work: they sum to zero ($\sum_{k=0}^{n-1} \omega_k = 0$ for $n \ge 2$, geometric series), and they are exactly the roots of $z^n - 1 = \prod_k (z - \omega_k)$. Evaluating that factorization at clever points produces product identities that look unhittable by other means — a preview of Week 19's root-of-unity filter.

Geometrically, multiplication by $e^{i\theta}$ IS rotation by $\theta$. Problems about rotating configurations, regular polygons, or sums of unit vectors are begging to be written in complex notation, where 'rotate the whole picture' costs one multiplication.

Worked example

Compute $\cos\frac{\pi}{7} - \cos\frac{2\pi}{7} + \cos\frac{3\pi}{7}$ (equivalently $\cos\frac{\pi}{7} + \cos\frac{3\pi}{7} + \cos\frac{5\pi}{7}$, since $\cos\frac{5\pi}{7} = -\cos\frac{2\pi}{7}$).

Nudge 1
Name $\zeta = e^{i\pi/7}$ and write the target as the real part of a short sum of its powers.
Reveal step 1
Set $\zeta = e^{i\pi/7}$, a primitive 14th root of unity. The target is $\operatorname{Re}(\zeta + \zeta^3 + \zeta^5)$ — odd powers of $\zeta$ below 7.
Nudge 2
Which powers of $\zeta$ are exactly the roots of $z^7 = -1$, and what do those roots sum to?
Reveal step 2
The seven odd powers $\zeta, \zeta^3, \zeta^5, \zeta^7, \zeta^9, \zeta^{11}, \zeta^{13}$ are exactly the roots of $z^7 = -1$ (since $(\zeta^{odd})^7 = e^{i\pi \cdot odd} = -1$), so they sum to $0$ (the $z^6$ coefficient of $z^7 + 1$).
Nudge 3
Pair each power with its conjugate — real parts double, imaginary parts cancel.
Reveal step 3
Pair conjugates: $\zeta^{13} = \bar{\zeta}, \zeta^{11} = \bar{\zeta}^3, \zeta^9 = \bar{\zeta}^5$, and $\zeta^7 = -1$. The sum becomes $2\operatorname{Re}(\zeta + \zeta^3 + \zeta^5) - 1 = 0$, so the target equals $\frac{1}{2}$.
Answer
$\frac{1}{2}$ — the seven solutions of $z^7 = -1$ sum to zero; pairing conjugates isolates twice the desired cosine sum minus 1.

Pitfall. Grinding trig sums with sum-to-product identities when the exponential form is three lines. If a trig expression has angles in arithmetic progression, reach for $e^{i\theta}$ and a geometric series FIRST.

4Hidden tool: finite differences

The finite difference operator $\Delta f(n) = f(n+1) - f(n)$ is discrete differentiation, and it detects polynomials perfectly: $f$ is a polynomial of degree $d$ iff $\Delta^{d+1} f \equiv 0$, and the leading coefficient is $\Delta^d f / d!$. When a problem hands you a mystery function on integers satisfying suspicious identities, difference it until it dies — the number of steps tells you its degree.

The companion basis fact: every polynomial of degree $d$ is a unique linear combination of the binomial coefficients $\binom{x}{0}, \binom{x}{1}, \dots, \binom{x}{d}$ — with integer combination coefficients exactly when the polynomial takes integer values on all integers — and $\Delta \binom{x}{k} = \binom{x}{k-1}$. This is why 'polynomial takes integer values at integers' does NOT mean integer coefficients — $\binom{x}{2} = \frac{x(x-1)}{2}$ is the standard counterexample, and the binomial basis is the standard repair.

Worked example

A polynomial $P$ of degree 3 satisfies $P(0) = 1, P(1) = 2, P(2) = 7, P(3) = 22$. Find $P(4)$ without finding $P$.

Nudge 1
Build the difference table — for degree 3 the third row must be constant.
Reveal step 1
Difference table: first differences $1, 5, 15$; second differences $4, 10$; third differences $6$. Degree 3 means the third differences are CONSTANT — every further entry of that row is 6.
Nudge 2
Extend each row one entry to the right, working bottom-up.
Reveal step 2
Extend backward: next second difference $= 10 + 6 = 16$; next first difference $= 15 + 16 = 31$.
Nudge 3
$P(4)$ is $P(3)$ plus the extended first difference.
Reveal step 3
So $P(4) = P(3) + 31 = 53$. The table replaces interpolation entirely — four values plus constancy of $\Delta^3$ determine the fifth.
Answer
$P(4) = 53$, by extending the difference table using constant third differences.

Pitfall. Assuming integer-valued-at-integers implies integer coefficients. False: $\binom{x}{2}$. The correct normal form is the binomial basis with integer weights.

5Inclusion-exclusion: counting by overcorrection

The principle: $|A_1 \cup \cdots \cup A_n| = \sum |A_i| - \sum |A_i \cap A_j| + \sum |A_i \cap A_j \cap A_k| - \cdots$ — add the singles, subtract the pairs, add the triples, alternating to the end. Why it works is worth one sentence: an element lying in exactly $m$ of the sets is counted $\binom{m}{1} - \binom{m}{2} + \cdots = 1 - (1-1)^m = 1$ time, by the binomial theorem. That one-line proof is also the recognition cue — inclusion-exclusion IS the binomial theorem doing set arithmetic.

The working form on the exam is usually complementary: to count objects avoiding ALL bad properties, take the total and inclusion-exclude the bad sets. The craft is choosing bad sets whose INTERSECTIONS are easy — 'permutations fixing the letters in $S$' has size $(n - |S|)!$ no matter which letters $S$ names, and that symmetry collapses the big sum into a clean series.

Recognition cues: 'no element in its original position', 'at least one of each', 'onto functions' ($\text{surjections} = \sum_k (-1)^k \binom{n}{k}(n-k)^m$), Euler's $\varphi$ via prime divisors, and any count phrased with 'avoid'. When the intersections are NOT symmetric, inclusion-exclusion still applies but stops being pleasant — that asymmetry is the signal to hunt a bijection instead.

Worked example

Count the derangements of $n$ letters — permutations with no fixed point — and show $D_n$ is the nearest integer to $\frac{n!}{e}$ for $n \ge 1$.

Nudge 1
Name the bad sets: $A_i$ = permutations fixing position $i$ — what is the size of a $k$-fold intersection?
Reveal step 1
Let $A_i$ be the permutations fixing position $i$. By symmetry $|A_{i_1} \cap \cdots \cap A_{i_k}| = (n-k)!$ — fix those $k$ positions, permute the rest freely — and there are $\binom{n}{k}$ such intersections.
Nudge 2
Inclusion-exclude the complement and watch $\binom{n}{k}(n-k)!$ simplify against $n!$.
Reveal step 2
Inclusion-exclusion on the complement: $D_n = n! - |A_1 \cup \cdots \cup A_n| = \sum_{k=0}^{n} (-1)^k \binom{n}{k}(n-k)! = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$ — the symmetric intersections collapsed the sum into a partial exponential series.
Nudge 3
Compare against the full series for $e^{-1}$ and bound the tail with the alternating-series estimate.
Reveal step 3
The tail bound finishes it: $\left| D_n - \frac{n!}{e} \right| = n! \left| \sum_{k=n+1}^{\infty} \frac{(-1)^k}{k!} \right| < \frac{n!}{(n+1)!} = \frac{1}{n+1} \le \frac12$ for $n \ge 1$ (alternating series with strictly decreasing terms: the tail is STRICTLY below its first term, so the nearest integer is unique). So $D_n$ is the integer nearest $\frac{n!}{e}$ — a counting problem answered by an analysis estimate, which is this week's whole theme.
Answer
$D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$, and the alternating tail is strictly below $\frac{1}{n+1} \le \frac12$, so $D_n = \left[\frac{n!}{e}\right]$ (nearest integer) for $n \ge 1$.

Pitfall. Trying to inclusion-exclude with asymmetric bad sets whose intersections you cannot name in closed form. The method's power is symmetry; without it the sum is formally correct and practically useless — reach for a bijection or recursion instead.

Before you open the gates

To prove a point exists, build the auxiliary function whose sign change or extremum encodes it — then let IVT/EVT work.
Before interrogating a maximizer, verify it exists: compact domain, or growth that buys one.
Angles in arithmetic progression → complex exponentials + geometric series, always.
Mystery integer function → difference table; constancy depth reveals polynomial degree.
Say the hypothesis. 'By continuity of $f$ on the closed interval...' costs six words and protects full credit.

Check yourself

1. $f$ is continuous on $(0, 1)$ with $f(x) \to +\infty$ at both endpoints. Must $f$ attain a minimum on $(0,1)$?

2. The sum of all 5th roots of unity multiplied by their squares — that is, $\sum_{k=0}^{4} \omega_k^3$ where $\omega_k = e^{2\pi i k/5}$ — equals:

3. A function on integers has constant nonzero 4th differences. It is:

4. How many derangements of 4 letters are there?

Practice ladder — three rungs, rising

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

Rung 1 (IVT, direct). Let $f : [0,1] \to [0,1]$ be continuous. Prove $f$ has a fixed point: some $c$ with $f(c) = c$.

One hint

Apply the IVT to $g(x) = f(x) - x$. Use $0 \le f \le 1$ to sign $g(0)$ and $g(1)$.

Worked resolution

Let $g(x) = f(x) - x$, continuous on $[0,1]$. Since $f(0) \in [0,1]$, $g(0) = f(0) \ge 0$; since $f(1) \in [0,1]$, $g(1) = f(1) - 1 \le 0$. If either endpoint gives $0$ it is already a fixed point; otherwise $g(0) > 0 > g(1)$, so by the IVT some $c \in (0,1)$ has $g(c) = 0$, i.e. $f(c) = c$. [Source: this week's IVT section — the fixed-point factory.]

Rung 2 (complex as notation). Let $\omega = e^{2\pi i / n}$ with $n \ge 2$. Prove $1 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0$, and deduce $\sum_{k=0}^{n-1} \cos\tfrac{2\pi k}{n} = 0$.

One hint

Geometric series: $\sum_{k=0}^{n-1}\omega^k = \tfrac{\omega^n - 1}{\omega - 1}$ with $\omega^n = 1$, $\omega \ne 1$. Then take real parts.

Worked resolution

As $\omega \ne 1$, $\sum_{k=0}^{n-1}\omega^k = \tfrac{\omega^n - 1}{\omega - 1} = \tfrac{1 - 1}{\omega - 1} = 0$ (since $\omega^n = 1$). Writing $\omega^k = \cos\tfrac{2\pi k}{n} + i\sin\tfrac{2\pi k}{n}$ and taking the real part of $\sum \omega^k = 0$ gives $\sum_{k=0}^{n-1}\cos\tfrac{2\pi k}{n} = 0$ (the imaginary part gives the sine sum). [Source: this week's complex-numbers section — trigonometry's better notation.]

Rung 3 (finite differences). A polynomial $P$ of degree $\le 3$ satisfies $P(0) = 1,\ P(1) = 3,\ P(2) = 11,\ P(3) = 31$. Find $P(4)$ WITHOUT solving for the coefficients.

One hint

Build the difference table. For $\deg \le 3$ the third differences are constant — extend the table by repeating that constant and summing back up the diagonal.

Worked resolution

Values $1, 3, 11, 31$. First differences $2, 8, 20$; second differences $6, 12$; third difference $6$. For $\deg \le 3$ the third differences are constant, so the next one is again $6$: the next second difference is $12 + 6 = 18$, the next first difference is $20 + 18 = 38$, and $P(4) = 31 + 38 = 69$. (Check: the polynomial is $P(x) = x^3 + x + 1$, and $P(4) = 64 + 4 + 1 = 69$.) [Source: this week's hidden-tool section — finite differences.]

Prove it — constructed response

A hiker climbs a trail on Saturday, leaving the bottom at 6 a.m. and reaching the top at 6 p.m. On Sunday she descends the SAME trail, leaving the top at 6 a.m. and reaching the bottom at 6 p.m. Prove there is a time of day at which she stood at exactly the same point of the trail on both days. Name the theorem and be explicit about the function you apply it to.

The gates

Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load

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- Read: Rudin *PMA* Ch. 4 (~32 pp.) — limits of functions, continuity, EVT (compact → max attained), IVT (connected → intermediate values), uniform continuity - Watch: MIT 18.100A — continuity lecture (OCW, free) - Watch: Bright Side of Mathematics — continuity videos 9–10 - Do: Rudin Ch. 4 ex 1, 3, 5, 8, 10, 14, 17, 20 - Do: PnB section 3.2.3 probs 1–4 - Do: Stanford 05wk6 probs 1–4 - Do: MIT analysis.pdf probs 4–6 - Archive: 2 Putnam A1/B1 · Analysis · continuity/IVT · 20 min each

sources & assignments (5)

Source Rudin — Principles of Mathematical Analysis (PMA) — Rudin *PMA* Ch. 4 (~32 pp.) — limits of functions, continuity, EVT (compact → max attained), IVT (connected → intermediate values), uniform continuity
Source MIT 18.100A — MIT 18.100A — continuity lecture (OCW, free)
Source The Bright Side of Mathematics — Real Analysis (playlist) — Bright Side of Mathematics — continuity videos 9–10
Source Savchev & Andreescu — Mathematical Miniatures — Savchev–Andreescu *Mathematical Miniatures* Miniatures 1–4 (telescoping, invariants, extremal) — read + reconstruct each argument
Source MIT 18.100B Lecture 5 — Monotone Convergence Theorem (OCW) — Intuition companion for “Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Drill

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unlocks after: Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load

sources & assignments (8)

Source Rudin — Principles of Mathematical Analysis (PMA) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 4 (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. 4 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source The Bright Side of Mathematics — Real Analysis (playlist) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 4 (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 5 — Monotone Convergence Theorem (OCW) — Intuition companion for “Track A — Analysis: Rudin Ch. 4 (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. 4 ex 1, 3, 5, 8, 10, 14, 17, 20Putnam
Problems PnB — PnB section 3.2.3 probs 1–4Putnam
Problems Stanford 05wk6 — Stanford 05wk6 probs 1–4Putnam
Problems MIT analysis.pdf — MIT analysis.pdf probs 4–6Putnam

Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Archive bridge

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unlocks after: Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Drill

sources & assignments (5)

Source Rudin — Principles of Mathematical Analysis (PMA) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 4 (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. 4 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source The Bright Side of Mathematics — Real Analysis (playlist) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 4 (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 5 — Monotone Convergence Theorem (OCW) — Intuition companion for “Track A — Analysis: Rudin Ch. 4 (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 2004 A1, 1995 A1 · Analysis · 20 min eachPutnam

archive pull: 2 problems · Analysis · A1/B1 · Standard · 15 min →

Hidden Tool: Compactness + Extremal Existence: Load

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unlocks after: Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load

*Prerequisites: Rudin Ch. 2 (metric spaces) + Rudin Ch. 4 (EVT)* - Read: Berkeley *Problems in Mathematics* Ch. 4 Metric Spaces (pp. 59–63) — compactness and fixed points - Do: Berkeley Ch. 4 — 2 problems - Compactness: - Closed + bounded in ℝⁿ → compact (Heine-Borel) - Compact → EVT: max/min attained - Compact + sequence → convergent subsequence (Bolzano-Weierstrass) - First question: "Is the domain compact?" before claiming a maximum exists

why this gate: the lesson's §2 Compactness and EVT: why maxima exist is what it trains

sources & assignments (5)

Source Berkeley *Problems in Mathematics* — Berkeley *Problems in Mathematics* Ch. 4 Metric Spaces (pp. 59–63) — compactness and fixed points
Source Rudin — Principles of Mathematical Analysis (PMA) — Rudin Ch. 2 — compactness and Heine-Borel (revisit pp. 30–40)
Source The Bright Side of Mathematics — Real Analysis (playlist) — Bright Side of Mathematics — compactness / Heine-Borel videos
Source MathDoctorBob — Metric topology, connectedness, continuous maps on R^n — companion to compactness work
Source MIT 18.100B Lecture 5 — Monotone Convergence Theorem (OCW) — Intuition companion for “Hidden Tool: Compactness + Extremal Existence: Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Hidden Tool: Compactness + Extremal Existence: Drill

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unlocks after: Hidden Tool: Compactness + Extremal Existence: Load

why this gate: the lesson's §2 Compactness and EVT: why maxima exist is what it trains

sources & assignments (5)

Source Berkeley *Problems in Mathematics* — Drill reference — this gate trains the material of “Hidden Tool: Compactness + Extremal Existence: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source The Bright Side of Mathematics — Real Analysis (playlist) — Drill reference — this gate trains the material of “Hidden Tool: Compactness + Extremal Existence: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MathDoctorBob — Drill reference — this gate trains the material of “Hidden Tool: Compactness + Extremal Existence: Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.100B Lecture 5 — Monotone Convergence Theorem (OCW) — Intuition companion for “Hidden Tool: Compactness + Extremal Existence: Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Berkeley — Berkeley Ch. 4 — 2 problemsPutnam

Track B — Algebra

Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load

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- Read: Andreescu *Complex Numbers from A to Z* Ch. 1 — Euler form, modulus, argument, conjugate, rotation - Read: Stanford Polya 07wk4 — roots of unity, complex-plane geometry, trig via complex - Read: PnB section 2.2.4 (roots of unity pp. 57–60) - Watch: 3Blue1Brown — complex numbers and e^{iπ} video - Do: 100 Polynomials Problems #35, 40, 45 - Do: MIT polynomials.pdf probs 7–12 - Do: PnB section 2.2.4 probs 1–4 - Archive: 3 Putnam A1/A2 · Algebra · any slot · 15 min each

why this gate: the lesson's §3 Complex numbers: trigonometry's better notation is what it trains

sources & assignments (6)

Source Andreescu *Complex Numbers from A to Z* — Andreescu *Complex Numbers from A to Z* Ch. 1 — Euler form, modulus, argument, conjugate, rotation
Source Stanford Polya 07wk4 — Stanford Polya 07wk4 — roots of unity, complex-plane geometry, trig via complex
Source Putnam and Beyond — PnB section 2.2.4 (roots of unity pp. 57–60)
Source Maths 505 — Complex Analysis Lectures (playlist) — Maths 505 — complex-analysis lectures (rotation, modulus, complex methods) — direct playlist
Source Anulus Smaragdinus — Polynomials (playlist) — Polynomials lecture series — direct playlist companion to this gate's reading Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Intuition companion for “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Drill

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unlocks after: Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load

why this gate: the lesson's §3 Complex numbers: trigonometry's better notation is what it trains

sources & assignments (7)

Source Andreescu *Complex Numbers from A to Z* — Drill reference — this gate trains the material of “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Maths 505 — Complex Analysis Lectures (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Anulus Smaragdinus — Polynomials (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Intuition companion for “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems 100 Polynomials — 100 Polynomials Problems #35, 40, 45Putnam
Problems MIT polynomials.pdf — MIT polynomials.pdf probs 7–12Putnam
Problems PnB — PnB section 2.2.4 probs 1–4Putnam

Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Archive bridge

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unlocks after: Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Drill

why this gate: the lesson's §3 Complex numbers: trigonometry's better notation is what it trains

sources & assignments (5)

Source Andreescu *Complex Numbers from A to Z* — Drill reference — this gate trains the material of “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Maths 505 — Complex Analysis Lectures (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Anulus Smaragdinus — Polynomials (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Intuition companion for “Track B — Algebra: Complex Numbers + Polynomials II (1 hr/day): Archive bridge” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Archive Browser — Putnam 2003 A1, 2020 A1 · Algebra · 15 min eachPutnam

archive pull: 2 problems · Algebra · A1/A2 · Challenge · 15 min →

Hidden Tool

Hidden Tool: Finite Differences: Load

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*Prerequisites: polynomials (Track B Week 3 + above)* - Read: CMU 11-Integer-Polynomials — finite difference / integer-valued polynomial section (pp. 1–4) - Do — 3 targeted problems: 1. Compute Δf, Δ²f, Δ³f for f(n) = n³. Verify Δ³f is constant. State why: for a degree-d polynomial, Δᵈf is constant. 2. Newton forward difference formula: f(n) = Σₖ C(n,k)·Δᵏf(0). Find the polynomial with f(0)=1, f(1)=3, f(2)=7, f(3)=13. 3. Sum ∑ₙ₌₀ᴺ n² using finite differences: express n² in falling-factorial basis C(n,1)+2·C(n,2), sum term-by-term using ΣC(n,k)=C(N+1,k+1), get closed form. - Finite Difference: - Δf(n) = f(n+1)−f(n) - Δᵈ[degree-d poly] = constant - Newton: f(n) = Σ C(n,k)Δᵏf(0) - Sum ∑f via falling-factorial antidifference: Σ C(n,k) = C(N+1,k+1) - Connection to derivatives: Δ ↔ d/dx in discrete calculus

why this gate: the lesson's §4 Hidden tool: finite differences is what it trains

sources & assignments (5)

Source Yufei Zhao — Integer Polynomials (intpoly.pdf) — CMU 11-Integer-Polynomials — finite difference / integer-valued polynomial section (pp. 1–4)
Source Concrete Mathematics (Graham, Knuth, Patashnik) — Concrete Mathematics Ch. 2 — finite/discrete calculus, falling factorials
Source Michael Penn — Abstract Algebra (playlist) — Michael Penn — finite differences & summation video (companion to the CMU handout) 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 Anulus Smaragdinus — Polynomials (playlist) — Watch the lecture matching this gate's polynomial topic BEFORE the problem set; write the 3 key moves it demonstrates as trigger lines. Watch-for: olympiad polynomial-problem walkthroughs (titled by problem number, not topic) — attempt each problem cold before watching the solution.
Source MIT 18.100B Lecture 5 — Monotone Convergence Theorem (OCW) — Intuition companion for “Hidden Tool: Finite Differences: Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Hidden Tool: Finite Differences: Drill

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unlocks after: Hidden Tool: Finite Differences: Load

why this gate: the lesson's §4 Hidden tool: finite differences is what it trains

sources & assignments (4)

Source Michael Penn — Abstract Algebra (playlist) — Re-watch the worked-solution segment only AFTER attempting this gate's drill problems; note where your route diverged. 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 MIT 18.100B Lecture 5 — Monotone Convergence Theorem (OCW) — Problem-session companion for “Hidden Tool: Finite Differences: Drill” — watch one worked problem, stop, finish it yourself on paper, then compare.
Source MIT 18.100A — Intuition companion for “Hidden Tool: Finite Differences: 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) Compute Δf, Δ²f, Δ³f for f(n) = n³. Verify Δ³f is constant. State why: for a degree-d polynomial, Δᵈf is constant. (2) Newton forward difference formula: f(n) = Σₖ C(n,k)·Δᵏf(0). Find the polynomial with f(0)=1, f(1)=3, f(2)=7, f(3)=13. (3) Sum ∑ₙ₌₀ᴺ n² using finite differences: express n² in falling-factorial basis C(n,1)+2·C(n,2), sum term-by-term using ΣC(n,k)=C(N+1,k+1), get closed form.Drill

Track D — Linear Algebra

Track D — Linear Algebra: Axler Ch. 3 (1 hr/day): Load

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- Read: Axler *LADR* Ch. 3 (~50 pp.) — linear maps, null space, range, rank-nullity theorem, matrix representation - Watch: MIT 18.06 Strang — column space and nullspace lecture (OCW) - Watch: 3Blue1Brown — rank and nullspace video - Watch: Sheldon Axler LADR lectures — lectures 5–7 - Do: Axler 3A: 1, 5, 8, 11, 14 · 3B: 1, 5, 10, 15 · 3C: 1, 5, 9 · 3D: 1, 3, 7, 11 - Do: PnB section 2.3.2 probs 1–3

sources & assignments (4)

Source Axler — Linear Algebra Done Right — Axler *LADR* Ch. 3 (~50 pp.) — linear maps, null space, range, rank-nullity theorem, matrix representation
Source MIT 18.06 Strang — MIT 18.06 Strang — column space and nullspace lecture (OCW)
Source 3Blue1Brown — Essence of Linear Algebra — 3Blue1Brown — rank and nullspace video (opens the official Essence of Linear Algebra series)
Source Sheldon Axler — Linear Algebra Done Right (lecture videos) — Sheldon Axler LADR lectures — lectures 5–7

Track D — Linear Algebra: Axler Ch. 3 (1 hr/day): Drill

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unlocks after: Track D — Linear Algebra: Axler Ch. 3 (1 hr/day): Load

sources & assignments (6)

Source Axler — Linear Algebra Done Right — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 3 (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. 3 (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source 3Blue1Brown — Essence of Linear Algebra — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 3 (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. 3 (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Axler 3A — Axler 3A: 1, 5, 8, 11, 14 · 3B: 1, 5, 10, 15 · 3C: 1, 5, 9 · 3D: 1, 3, 7, 11Putnam
Problems PnB — PnB section 2.3.2 probs 1–3Putnam

Track C — Combinatorics

Track C — Combinatorics: Inclusion-Exclusion + Pigeonhole (1 hr/day): Load

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- Read: PnB section 6.2.3–6.2.4 (pp. 302–318) — IE, double counting - Watch: Shahriar Shahriari — Inclusion-Exclusion lesson - Watch: Art of Problem Solving counting playlist — videos 1–3 - Do: 102 Combinatorial Problems #13, 17, 20, 23, 28, 33 - Do: PnB section 6.2.4 probs 1–5 - Do: MIT comb.pdf probs 1–4 - Do: Engel Ch. 4 probs 16–25

why this gate: the lesson's §5 Inclusion-exclusion: counting by overcorrection is what it trains

sources & assignments (4)

Source Putnam and Beyond — PnB section 6.2.3–6.2.4 (pp. 302–318) — IE, double counting
Source Shahriar Shahriari — Combinatorics, An Invitation (playlist) — Art of Problem Solving counting playlist — videos 1–3
Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Intuition companion for “Track C — Combinatorics: Inclusion-Exclusion + Pigeonhole (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. 4 Pigeonhole Principle. Use before inclusion-exclusion reps; output one pigeonhole model choice note.

Track C — Combinatorics: Inclusion-Exclusion + Pigeonhole (1 hr/day): Drill

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unlocks after: Track C — Combinatorics: Inclusion-Exclusion + Pigeonhole (1 hr/day): Load

why this gate: the lesson's §5 Inclusion-exclusion: counting by overcorrection is what it trains

sources & assignments (7)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track C — Combinatorics: Inclusion-Exclusion + Pigeonhole (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: Inclusion-Exclusion + Pigeonhole (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) — Intuition companion for “Track C — Combinatorics: Inclusion-Exclusion + Pigeonhole (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 #13, 17, 20, 23, 28, 33Putnam
Problems PnB — PnB section 6.2.4 probs 1–5Putnam
Problems MIT comb.pdf — MIT comb.pdf probs 1–4Putnam
Problems Engel — Engel Ch. 4 probs 16–25Putnam

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. 4 (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. 4 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source The Bright Side of Mathematics — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.100B Lecture 5 — Monotone Convergence Theorem (OCW) — 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.

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

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 (4)

Problems Engel — Engel Ch. 10 probs 1–8 (polynomials/complex roots)Putnam
Problems 102 Comb — 102 Combinatorial Problems — inclusion–exclusion, Introductory probs 1–4Putnam
Problems MAA archive — Exposure: 2014 A1 solved + 2013 A1 attemptPutnam
Problems Probability Warmup (hand-picked) — Tiny probability warmup: solve the three linked reps before the main week gates; identify sample space, random variable, and expectation/conditioning move.AMC · archive: Putnam 2002 B1 AMC probability AMC expectation

Reflect

Reflect: reconstruct and log the pattern

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- Reconstruct: 1 Rudin Ch. 4 exercise → close → rewrite → pattern note - Reflect: IVT/EVT trigger exact hypotheses from memory - Reflect: complex numbers card (Euler form / roots of unity / rotation = multiplication) - Reflect: rank-nullity statement from memory - Verify: on one continuity proof — every ε-δ step justified? ---

Ritual: 1 Rudin Ch. 4 exercise → close → rewrite → pattern note | IVT/EVT trigger exact hypotheses from memory | complex numbers card (Euler form / roots of unity / rotation = multiplication) | rank-nullity statement 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. 4 (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. 4 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source The Bright Side of Mathematics — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source Michael Penn — Putnam Exam Solutions (playlist) — Intuition companion for “Reflect: reconstruct and log the pattern” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Verify: audit one proof before closing

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unlocks after: Reflect: reconstruct and log the pattern

Ritual: on one continuity proof — every ε-δ step justified?

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. 4 (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. 4 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source The Bright Side of Mathematics — Real Analysis (playlist) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 4 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source Michael Penn — Putnam Exam Solutions (playlist) — Intuition companion for “Verify: audit one proof before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Lerma Training

Lerma Training: Number Theory 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 (2)

Source Lerma Putnam Training 2023 (Northwestern) — NU Putnam-team number-theory section — congruences, divisibility, orders, Diophantine.

Problems Lerma Training 2023 §3 Number Theory — Work problems 3.1–3.37 (4–6 per session). Modular reduction, CRT, orders/LTE, Diophantine descent; reduce mod a clever modulus early. Self-check against the PDF's hints/solutions parts.Putnam

Exit contract — Week 4

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W5 · Analysis: Differentiation + Algebra: Inequalities + LA: Eigenvalues + Comb: Bijections + NT: Divisibility + Hidden Tool: Convexity/Jensen/Smoothing
the exam follows the final taper week

Train Week 4 in the trainer →