W2 · Boot— gates27–31 hrs

Week 2 of 28 · Boot · due 2026-06-06 · 27–31 hrs

Methods + AIME Fluency + Analysis: Metric Spaces

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This week's lesson · 8 min read

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Speed is a proof skill: AIME reflexes and the geometry of distance

Ninety minutes, three problems. The 2026 format is not generous with time, and the minutes you burn re-deriving how to shift a sum or complete a square are minutes stolen from the argument that actually scores. This week does two things that look unrelated and are not: it rebuilds contest-speed algebra until it is reflex, and it introduces metric spaces — because 'distance' is the first definition you will meet where your intuition needs formal rails.

Fluency below, rigor above: automate the manipulations so your conscious attention is free for the one step in every problem that is actually new.

Key ideas — the week on one card

  1. Telescoping: rewrite the term as $f(k) - f(k+j)$ and the sum collapses to $j$ leading terms minus $j$ trailing ones.
  2. A metric needs positivity, symmetry, and the triangle inequality — and the triangle inequality does all the work.
  3. Limits in any metric space are unique: route $d(x,y)$ through $x_n$ and squeeze it below every $\varepsilon$.
  4. The 90-minute session is a trained skill: read all three problems first, bank the routine one clean, pre-commit your exits.

1The AIME layer: manipulations you should never think about again

There is a short list of moves that appear inside virtually every Putnam A1/B1, and none of them deserve conscious thought by December: telescoping a sum or product, completing the square in two variables, substituting $x \mapsto 1/x$ or $x \mapsto 1-x$ to exploit symmetry, and reading off Vieta's relations without expanding. The test of fluency is not whether you can do them — it is whether you can do them while thinking about something else.

Telescoping is the deepest of the four because it is really a statement about structure: a sum collapses when its terms are differences $a_k = f(k+1) - f(k)$. The skill is manufacturing $f$, usually by partial fractions or by recognizing a product pattern. When you see $\sum \frac{1}{k(k+1)}$, the answer $1 - \frac{1}{n+1}$ should arrive before you finish reading the summand.

Symmetric substitutions earn their keep on functional equations and definite integrals alike. The identity $\int_0^1 f(x)\,dx = \int_0^1 f(1-x)\,dx$ costs one line and cracks integrals that resist every direct technique — you will meet it again in Week 6 as 'the reflection trick', and it is worth owning early.

1/1−1/21/2−1/31/3−1/41/4−1/51/5−1/6every interior pair cancels;only the endpoints survive:Σ = 1 − 1/6 = 5/6
The telescope: write the term as a difference and the sum collapses to its two ends.

Worked example

Evaluate $\sum_{k=1}^{n} \frac{1}{k(k+2)}$ in closed form.

  1. Nudge 1

    The factors of $k(k+2)$ differ by 2 — partial fractions will carry a $\frac12$ out front.

    Reveal step 1

    Partial fractions with the gap in mind: $\frac{1}{k(k+2)} = \frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)$. The gap of 2 means the telescope collapses with TWO surviving terms at each end, not one.

  2. Nudge 2

    Write four leading terms and four trailing ones; watch which survive the cancellation.

    Reveal step 2

    Write out the first and last few terms: $\frac{1}{2}\left[\left(1 - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \cdots\right]$. Everything cancels except $1, \frac{1}{2}$ at the front and $-\frac{1}{n+1}, -\frac{1}{n+2}$ at the back.

  3. Nudge 3

    Two heads survive and two tails vanish in the limit — assemble, then sanity-check at $n = 1$.

    Reveal step 3

    Collect: $\frac{1}{2}\left(1 + \frac{1}{2} - \frac{1}{n+1} - \frac{1}{n+2}\right) = \frac{3}{4} - \frac{2n+3}{2(n+1)(n+2)}$. Sanity check at $n=1$: formula gives $\frac{3}{4} - \frac{5}{12} = \frac{1}{3}$, and directly $\frac{1}{1 \cdot 3} = \frac{1}{3}$. It matches — always spend the ten seconds.

  4. Answer

    $\frac{3}{4} - \frac{2n+3}{2(n+1)(n+2)}$, hence the infinite sum is $\frac{3}{4}$.

Pitfall. Telescoping with a gap and keeping only one boundary term per end. A gap of $g$ leaves $g$ survivors at each end. Write the first three and last three terms explicitly every time — the pattern is cheaper to see than to remember.

2Metric spaces: distance with the training wheels off

A metric space is a set $X$ with a function $d(x,y)$ that behaves like distance: positive between distinct points, symmetric, and obeying the triangle inequality $d(x,z) \le d(x,y) + d(y,z)$. That is the whole definition, and its power is what it does NOT say — nothing about numbers, coordinates, or dimensions. The set of continuous functions on $[0,1]$ with $d(f,g) = \max_x |f(x)-g(x)|$ is a metric space, and suddenly every theorem you prove about 'points' applies to functions.

The triangle inequality is the only axiom with content, and ninety percent of its uses on this exam are one two-step pattern: to show two things are close, route through a third. $d(x,z) \le d(x,y) + d(y,z)$ — pick the midpoint $y$ wisely and both terms on the right are things you control. The reverse triangle inequality $|d(x,z) - d(y,z)| \le d(x,y)$ is the same fact worn inside out, and it is how you prove distance itself is continuous.

Open balls $B(x,r) = \{y : d(x,y) < r\}$ turn the $\varepsilon$-$N$ language of Week 1 into geometry: convergence means the sequence eventually enters every ball around the limit. Keep both dialects fluent — some proofs are shorter in epsilons, others in balls, and the exam does not tell you which.

Worked example

In any metric space, prove: if $x_n \to x$ and $x_n \to y$, then $x = y$ (limits are unique).

  1. Nudge 1

    You cannot compare $x$ and $y$ directly — route the distance through $x_n$ with the triangle inequality.

    Reveal step 1

    Route the distance $d(x,y)$ through the sequence: for every $n$, the triangle inequality gives $d(x,y) \le d(x, x_n) + d(x_n, y)$.

  2. Nudge 2

    Split $\varepsilon$ between the two convergences; each supplies its own $N$.

    Reveal step 2

    Fix $\varepsilon > 0$. Convergence to $x$ gives $N_1$ with $d(x, x_n) < \varepsilon/2$ for $n \ge N_1$; convergence to $y$ gives $N_2$ likewise. Take $n = \max(N_1, N_2)$ — one concrete index making both terms small.

  3. Nudge 3

    $d(x,y)$ is a constant sitting below every $\varepsilon$ — Week 1's lemma finishes it.

    Reveal step 3

    Then $d(x,y) < \varepsilon/2 + \varepsilon/2 = \varepsilon$ for EVERY $\varepsilon > 0$. By Week 1's micro-lemma ('$\le b + \varepsilon$ for all $\varepsilon$ implies $\le b$', here with $b = 0$), $d(x,y) = 0$, so $x = y$ by the metric axioms.

  4. Answer

    Triangle through $x_n$, split $\varepsilon$, conclude $d(x,y) < \varepsilon$ for all $\varepsilon > 0$, hence $d(x,y) = 0$ and $x = y$.

Pitfall. Using $\max(N_1, N_2)$ without saying so — or worse, using the same $N$ for two different convergence statements as if they came synchronized. Every limit statement issues its own $N$; you must reconcile them explicitly.

3The 90-minute session as a trained skill

This week you also run your first timed blocks in the real 2026 shape: ninety minutes, three problems. The optimal strategy is known and boring — read all three first (five minutes), rank them, bank the easiest completely before touching the hardest. A written, checked solution to one problem beats three half-solutions every time the grader is honest, and Putnam graders are ruthless.

Train the exit reflex now: when an approach has produced nothing on paper for eight minutes, you switch — not because the approach is wrong, but because you have two other problems and a banked partial credit beats a heroic zero. Write one line about where you stopped ('tried induction, step fails at the inequality') so re-entry is cheap if time remains. This is the move that separates trained contestants from strong students.

Worked example

You have 90 minutes and three problems: one looks routine (telescoping sum), one is a functional equation you half-recognize, one is unfamiliar geometry. Plan the session.

  1. Nudge 1

    Five minutes reading all three beats twenty-five on the wrong one — one first-move line per problem.

    Reveal step 1

    Minutes 0-5: read all three fully. Write one first-move line under each — 'partial fractions', 'try $f(x)=cx$, then plug $x=y=0$', 'no idea: draw it big'. Ranking is now evidence, not mood.

  2. Nudge 2

    Bank means grader-proof: the $n=1$ check and the write-up happen NOW, not at minute 85.

    Reveal step 2

    Minutes 5-30: bank the telescoping sum COMPLETELY — including the $n=1$ sanity check and clean write-up. A polished 10 in hand changes how calmly you attack the rest.

  3. Nudge 3

    Pre-committed exits survive adrenaline; improvised ones do not.

    Reveal step 3

    Minutes 30-65: the functional equation, with the exit reflex armed at minute 55 if the page is blank. Minutes 65-88: geometry, aiming for a lemma worth partial credit. Final 2 minutes: reread the banked solution for strict-vs-non-strict slips.

  4. Answer

    Read-rank-bank: 5 minute survey, complete the easiest to a checked 10, time-box the middle, mine the hardest for partial credit, reserve 2 minutes for verification.

Pitfall. Spending the first twenty minutes on the most interesting problem instead of the easiest. Interest is not a scoring category. Bank first, be brave later.

Before you open the gates

  • Drill the four reflex moves (telescope, complete the square, symmetric substitution, Vieta) until they run without conscious attention.
  • Sanity-check every closed form at $n = 1$ — ten seconds, catches sign and boundary errors that graders will not forgive.
  • When two things must be close, route the triangle inequality through a midpoint you control; split $\varepsilon$ deliberately.
  • In timed blocks: read all three problems before solving any; bank the easiest completely; arm the eight-minute exit reflex.
  • Keep both dialects — epsilons and balls — and translate freely between them.

Check yourself

1. $\sum_{k=1}^{\infty} \frac{1}{k(k+3)}$ equals which value?

2. Which function on $\mathbb{R}^2$ is NOT a metric?

3. Twenty minutes into a 90-minute session your first-choice problem has produced nothing on paper. The trained move is:

Practice ladder — three rungs, rising

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

Rung 1 (reflex check). Evaluate $\sum_{k=1}^{n} \dfrac{1}{k(k+1)}$ in closed form.

One hint

Partial fractions: $\tfrac{1}{k(k+1)} = \tfrac{1}{k} - \tfrac{1}{k+1}$. Then watch which terms survive.

Worked resolution

$\sum_{k=1}^{n} \left(\tfrac{1}{k} - \tfrac{1}{k+1}\right)$ telescopes: every interior term cancels, leaving $1 - \tfrac{1}{n+1} = \tfrac{n}{n+1}$. Sanity check $n = 1$: $\tfrac{1}{2}$ both ways. [Source: The telescoping reflex from this week's AIME layer, in its simplest honest form.]

Rung 2 (telescoping in disguise). Evaluate $\prod_{k=2}^{n} \left(1 - \dfrac{1}{k^2}\right)$ in closed form.

One hint

Factor each term as $\tfrac{(k-1)(k+1)}{k^2}$ and let the product telescope in two directions at once.

Worked resolution

$\prod_{k=2}^{n} \tfrac{(k-1)(k+1)}{k \cdot k} = \left(\prod_{k=2}^{n} \tfrac{k-1}{k}\right)\left(\prod_{k=2}^{n} \tfrac{k+1}{k}\right) = \tfrac{1}{n} \cdot \tfrac{n+1}{2} = \tfrac{n+1}{2n}$. Sanity check $n = 2$: $1 - \tfrac14 = \tfrac34$ and $\tfrac{3}{4}$ from the formula. [Source: The same reflex one disguise deeper — a standard product-telescope drill.]

Rung 3 (a metric you have not met). Show that $d(x,y) = \dfrac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb{R}$ — the triangle inequality is the whole fight — and explain why it is bounded yet has exactly the same convergent sequences as the usual distance.

One hint

Let $f(t) = \tfrac{t}{1+t}$ on $t \ge 0$. Show $f$ is increasing and subadditive ($f(a+b) \le f(a) + f(b)$); then chain $d(x,y) = f(|x-y|) \le f(|x-z| + |z-y|)$.

Worked resolution

Positivity and symmetry are inherited from $|x-y|$ ($f(t) = 0$ iff $t = 0$). For the triangle inequality: $f$ is increasing on $[0,\infty)$ since $f'(t) = \tfrac{1}{(1+t)^2} > 0$, and subadditive because $f(a+b) = \tfrac{a}{1+a+b} + \tfrac{b}{1+a+b} \le \tfrac{a}{1+a} + \tfrac{b}{1+b} = f(a) + f(b)$. Then $d(x,y) = f(|x-y|) \le f(|x-z| + |z-y|) \le f(|x-z|) + f(|z-y|) = d(x,z) + d(z,y)$, using the ordinary triangle inequality inside the increasing $f$. Boundedness: $f(t) < 1$ always, so no two points are ever at distance $\ge 1$. Same convergence: $d(x_n, x) \to 0$ iff $|x_n - x| \to 0$ because for $\varepsilon \in (0,1)$, $f(t) < \varepsilon$ exactly when $t < \tfrac{\varepsilon}{1 - \varepsilon}$ — the two distances squeeze each other near $0$. Distance can be reshaped radically and still tell the same stories about limits: that is the metric-space idea in one example. [Source: The standard bounded-metric example; pairs with this week's metric-space section and foreshadows Rudin Ch. 2.]

Prove it — constructed response

In any metric space $(X, d)$, prove the reverse triangle inequality: $|d(x,z) - d(y,z)| \le d(x,y)$ for all $x, y, z \in X$. Cite the axiom you use at every step.

The gates

Track — Proof Methods + Problem Strategies (2 hrs/day)

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- Read: PnB section 1.2–1.5 (induction pp. 14–18, pigeonhole pp. 18–20, extremal pp. 20–22, invariants pp. 22–24) + section 6.2.1 (bijection counting intro pp. 291–300) - Read: Putnam Guide (Ebrahimian) — General Strategies chapter (~20 pp.) - Read: CMU 01-Intro — worked examples scan - Watch: Eddie Woo induction (4 videos) + Dr. Ebrahimian Putnam Competition playlist — first 2 videos + Po-Shen Loh CMU 295 intro lecture - Do: PnB section 1.4 probs 1–6 · section 1.5 probs 1–4 · section 6.2.1 probs 1–4 - Do: Engel Ch. 4 probs 1–15 · Ch. 5 probs 1–12 - Do: Stanford 07wk5 probs 1–3 · Stanford 07wk1 probs 1–7 - Do: 102 Combinatorial Problems #2, 5, 7, 10, 13, 16, 20, 23 - Do: MIT ss1.pdf probs 1–3 · MIT ss4.pdf probs 40–42 - Archive: 5 AMC-band problems · any topic · any slot · 12 min each - Archive: 5 AIME-band problems · any topic · any slot · 12 min each

sources & assignments (13)
  • Problems PnB — PnB section 1.4 probs 1–6 · section 1.5 probs 1–4 · section 6.2.1 probs 1–4Putnam
  • Problems Engel — Engel Ch. 4 probs 1–15 · Ch. 5 probs 1–12Putnam
  • Problems Stanford 07wk5 — Stanford 07wk5 probs 1–3 · Stanford 07wk1 probs 1–7Putnam
  • Problems 102 Combinatorial — 102 Combinatorial Problems #2, 5, 7, 10, 13, 16, 20, 23Putnam
  • Problems MIT ss1.pdf — MIT ss1.pdf probs 1–3 · MIT ss4.pdf probs 40–42Putnam
  • Problems Archive (hand-picked) — 3 AMC/AIME combinatorics speed reps (open each below) · ~8 min eachAMC · archive: 2021 AMC 10 B2021 AMC 10 BAMC 12 A
  • Problems Archive (hand-picked) — 3 AIME combinatorics speed reps (open each below) · ~8 min eachAIME · archive: AIME II (Standard)2022 AIME I (Standard)AIME II (Standard)

Hidden Tool: Invariants + Monovariants

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*Prerequisites: proof methods (above)* - Read: Engel Ch. 1 (Invariance Principle) — intro + probs 1–8 - Do: Engel Ch. 1 probs 1–8 - Do: Stanford 07wk5 probs 1–10 — complete the full invariants/monovariants handout - Invariant: - Coloring/parity invariant - Weight function - Monovariant (quantity that strictly increases/decreases) - Potential function - First question: "What quantity is preserved or monotone?"

sources & assignments (8)
  • Source Engel — Problem-Solving Strategies — Engel Ch. 1 (Invariance Principle) — intro + probs 1–8
  • Source Putnam and Beyond — PnB section 1.5 — invariants and monovariants (pp. 22–24)
  • Source Michael Penn — Abstract Algebra (playlist) — Michael Penn — invariants & monovariants problem-solving videos 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 Michael Penn — Proof Writing (playlist) — Watch ONE video matching this gate's proof type (direct/contrapositive/contradiction/induction), then redo one of this gate's exercises in that video's style. Output: the rewritten proof. Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
  • Source Cezar Lupu — TTU MATH 4000, Lecture 1 — Intuition companion for “Hidden Tool: Invariants + Monovariants” — 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. 1 Parity + Ch. 12 Invariants. Use as first-move practice before the invariant/monovariant drill; output one reusable invariant trigger.
  • Problems Engel — Engel Ch. 1 probs 1–8Putnam
  • Problems Stanford 07wk5 — Stanford 07wk5 probs 1–10 — complete the full invariants/monovariants handoutPutnam

Hidden Tool: Construction + Counterexample

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*Prerequisites: proof methods (above)* - Construction: - Small cases (n = 1, 2, 3, 4 first) - Recursive construction (prove termination) - Perturb a known example - Impossibility: - Parity obstruction - Modular obstruction - Invariant impossibility - After each problem: write "This was [construction/impossibility]; I used [method] because ___."

sources & assignments (7)
  • Problems Construction drill (standard examples) — Construct, with full proof: (1) a sequence of reals whose set of subsequential limits is exactly [0,1]; (2) a function R->R continuous at every irrational and discontinuous at every rational (Thomae-style); (3) an explicit bijection N->Q. Each construction must come with the one-line reason it works.AMC

Track A — Analysis: Rudin Ch. 2 (1.5 hrs/day)

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- Read: Rudin *PMA* Ch. 2 (~46 pp.) — metric spaces, open/closed sets, compactness, Heine-Borel, Bolzano-Weierstrass, connected sets - Watch: Bright Side of Mathematics Real Analysis — videos 3–5 - Do: Rudin Ch. 2 ex 2, 5, 9, 11, 14, 16, 20, 23, 26 - Do: CMU 07-Convergence — read lecture notes alongside Ch. 2

sources & assignments (6)
  • Problems Rudin — Rudin Ch. 2 ex 2, 5, 9, 11, 14, 16, 20, 23, 26Putnam
  • Problems CMU 07-Convergence — CMU 07-Convergence — read lecture notes alongside Ch. 2Putnam

Milestone Gate

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- Score ≥ 4/5 timed AIME problems. If not: complete one Archive AMC session before proceeding to Week 3.

Ritual: - Score ≥ 4/5 timed AIME problems. If not: complete one Archive AMC session before proceeding to Week 3.

sources & assignments (5)
  • Problems ≥ 4/5 timed AIME — ≥ 4/5 timed AIME problems. If not: complete one Archive AMC session before proceeding to Week 3.AMC

Putnam Contact — first clean A1 attempt loop

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This is still A1 contact, not full archive volume; serious archive work begins Week 3.

Ritual: One rewritten solution is required. Watching/reading without reconstruction does not complete this gate.

sources & assignments (4)
  • Problems Putnam archive timed contact — Putnam 2001 B1 — 20-minute attempt; if stuck, write two failed first moves before checking.Putnam
  • Problems Putnam archive rewrite — Rewrite one complete solution from memory after reading the official solution.Putnam
  • Problems Putnam archive timed contact — Putnam 1976 B1 — 20-minute attempt; tag the mistake or the first move used.Putnam

Problem-Solving Reps — weekly homework

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Standing weekly homework — the problem-solving book stack, every week. Solve ALL listed; write ONE full clean solution (the rest may stay scratch). Up to 2 due re-solves from your review queue surface first.

sources & assignments (2)
  • Problems Newman — Newman, A Problem Seminar — problems 1–8Putnam
  • Problems MAA archive — Exposure: read solved Putnam 2017 A1, then attempt 2018 A1 (15 min) + autopsy noteAMC

Reflect: reconstruct and log the pattern

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- Reconstruct: 1 Rudin exercise → close → rewrite → pattern note - Reflect: write Heine-Borel and Bolzano-Weierstrass from memory (one line each) - Verify: on one proof — every "⊂" claim proved in both directions? --- ## Phase 2 — Toolkit (Weeks 3–13) Six parallel tracks run simultaneously. Within each track: gates lock sequentially (Read before Do). Across tracks in the same week: all open simultaneously. ---

Ritual: 1 Rudin exercise → close → rewrite → pattern note | write Heine-Borel and Bolzano-Weierstrass from memory (one line each)

sources & assignments (4)
  • Source Putnam and Beyond — Week reference — this gate applies this week's lead material (“Track — Proof Methods + Problem Strategies (2 hrs/day)”). If an attempt stalls, the repair source is here.
  • Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Week reference — this gate applies this week's lead material (“Track — Proof Methods + Problem Strategies (2 hrs/day)”). If an attempt stalls, the repair source is here.
  • Source Michael Penn — Abstract Algebra (playlist) — Week reference — this gate applies this week's lead material (“Track — Proof Methods + Problem Strategies (2 hrs/day)”). If an attempt stalls, the repair source is here. 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 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.

Problem Attack Protocol (use on every hard problem + every lecture)

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Pause-and-Preempt (for the lecture playlists): when the instructor writes a problem, pause before their first step; spend 5–10 min finding a foothold yourself; resume and compare their FIRST move to yours (vector-space framing? complex plane? a substitution?); if different, pause again and find the structural clue that revealed their path. Active preemption is what converts watching into pattern recognition.

Ritual: 30-minute scratch-pad: (0–5 min) Radical simplification — drop to n=1,2,3, set variables to 0/1, test boundary cases. (5–15 min) Systematic data — write the small cases out; look for known sequences, repetition mod a prime, bounds by powers of 2. (15–25 min) Invariant hunt — what stays constant while everything else changes (parity, total sum, a field, a dimension)? (25–30 min) Formalize — translate the pattern into a rigorous proof: induction, a dimension/linear-independence argument, or an ε–δ bounding chain, justifying every step.

sources & assignments (4)
  • Source MIT 18.A34 — Putnam Seminar (OCW) — MIT problem-solving seminar: the canonical Putnam seminar problem sets + structure
  • Source Stanford Polya Seminar (handouts) — Stanford Polya 07wk1 (induction/pigeonhole) probs 1–7 + 07wk5 (invariants/monovariants) probs 1–3
  • Source Michael Penn — Putnam Exam Solutions (playlist) — Putnam walkthrough playlist — worked-solution companion
  • Source Michael Penn — Abstract Algebra (playlist) — Intuition companion for “Problem Attack Protocol (use on every hard problem + every lecture)” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises. 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.

Exit contract — Week 2

Verification remaining

  • reading progress…

Carry-forward repairs

  • reading queue…

Next week opens with

W3 · Analysis: Sequences + Algebra: Polynomials I + LA: Vector Spaces + Comb: Counting Intro
the exam follows the final taper week

Train Week 2 in the trainer →