Week 22 of 28 · Mastery · due 2026-10-24 · 28–32 hrs

USAMO Bridge: Nightmare Entry

USAMO Bridge: Nightmare Entry gate complete

This week's lesson · 9 min read

not started · read first, then train the gates

Nightmare entry: how to work a problem built to defeat you

A5, B5, A6, B6 — the slots where the national median is zero. This week you enter them deliberately, with a protocol, because nightmare problems are not harder versions of A3; they are a different activity. Nobody solves them by recognizing a technique. They are solved by sustained investigation — building the problem's private theory until a structure appears — and by studying published solutions the RIGHT way when the attempt falls short. This week teaches both halves.

At the nightmare tier the problem is a research project in miniature: you are not retrieving a method, you are discovering the problem's own local theorem — and that skill trains differently.

Key ideas — the week on one card

Nightmare problems are research projects in miniature: data, conjectures, local lemmas, assembly — in that order, returning to data when assembly jams.
Progress is measured in artifacts (tables, killed conjectures, proven lemmas), never in feeling close.
Vieta jumping: fix the value, view the symmetric condition as a quadratic in ONE variable, and the second root descends to a base case.
Study solutions by the interrupted read: borrow ONE move, re-derive the rest yourself — each solution yields a cue, a portable move-sentence, and a named jam diagnosis.

1The investigation protocol

Sustained attack has phases, and naming them prevents the drift that eats hours. PHASE ONE — data (15-20 min): compute every small case that exists, tabulate, and record EVERY pattern candidate, even silly ones. PHASE TWO — conjecture: promote patterns to precise statements, kill the false ones with more data. PHASE THREE — local theory: prove small lemmas about the object ('the sequence is eventually periodic', 'the quantity is invariant under the swap', 'extremal configurations have equal parts') without knowing which lemma is THE one — at this tier, lemmas are the currency. PHASE FOUR — assembly: try to chain lemmas into the claim, and when the chain jams, return to phase one with sharper questions.

The psychological calibration is different here: on an A1, ten minutes without progress means you missed the observation; on an A6, two hours of accumulating partial structure IS progress. Track advancement by artifacts — data tables, killed conjectures, proven lemmas — never by 'do I feel close'. A session that produced three true lemmas about the problem was a good session regardless of whether the problem fell.

Session economics for the exam itself: nightmare slots get attempted LAST and only with banked time, and the goal there is the 1-3 point artifact — a proven lemma, the key special case, a verified conjecture with partial proof. In training, however, you give them hours on purpose: the capacity for sustained investigation is trainable and decays if unused, and these weeks are where it gets built.

The investigation's fossil: Vieta jumping descends the k = 4 chain to a base case that answers the problem.

Worked example

Investigation, worked (the 1988 IMO problem 6 shape): $a, b$ positive integers with $ab + 1 \mid a^2 + b^2$. Investigate $k = \frac{a^2+b^2}{ab+1}$.

Nudge 1
Compute a table of small solutions first — the chains hiding in it ARE the discovery.
Reveal step 1
Data: $(1,1) \to k = 1$; $(2, 8) \to \frac{68}{17} = 4$; $(3, 27) \to \frac{738}{82} = 9$; scan finds $(8, 30) \to \frac{964}{241} = 4$ again. Conjecture: $k$ is always a perfect square — and the pairs for $k = 4$ chain: $2 \to 8 \to 30 \to 112$, each $\approx 4\times$ the last. A recurrence smells like Vieta.
Nudge 2
Freeze $k$ and read the condition as a quadratic in $a$ alone; Vieta hands you a second root.
Reveal step 2
Local theory: fix $k$ and view $a^2 - k a b + b^2 - k = 0$ as a QUADRATIC IN $a$. If $(a, b)$ is a solution with $a > b$, Vieta gives a second root $a' = kb - a = \frac{b^2 - k}{a}$ — an integer, and one can prove $0 \le a' < b$: the solution JUMPS DOWN to a smaller pair with the same $k$.
Nudge 3
Show the new root is a smaller nonnegative solution and ride the descent to its floor.
Reveal step 3
Assembly: descend until the smaller member hits 0; at $(a', 0)$ the equation reads $k = a'^2$ — a perfect square, and the descent is legal because each jump stays a nonnegative-integer solution. This is Vieta jumping, and the investigation FOUND it: data → chained-pairs pattern → quadratic-in-one-variable lemma → descent. The famous technique is just the fossil of this investigation.
Answer
$k$ is always a perfect square — proved by Vieta jumping, i.e. root-swapping descent to the $(a', 0)$ base case, discovered from the data table.

Pitfall. Skipping phase one because the problem 'obviously' needs cleverness, not computation. The $k = 4$ chain in the data table IS the discovery; investigators who compute find the jump, spectators who stare do not.

2Solution study: extraction, not admiration

Most nightmare attempts end at the solution text, and HOW you read it decides whether the hours convert to skill. The interrupted-read from Week 18, hardened: (1) write your final attempt state first — conjectures, lemmas, where you jammed; (2) read ONLY to the solution's first move you did not have; (3) close the text and push your attempt with that one move as far as it goes; (4) repeat until done. Reading this way, one solution costs an evening — and that is correct; a solution skimmed in ten minutes teaches ten minutes of content.

After finishing, run the two extraction questions. RETROSPECTIVE: 'what in the problem STATEMENT could have pointed me to this?' — the answer becomes a recognition cue you file ('divisibility with a symmetric quadratic numerator → try the quadratic-in-one-variable view'). PORTABLE: 'state the reusable move in one sentence, stripped of this problem's specifics' — that sentence goes on the technique card; the problem-specific details do not, because they will never recur but the move will.

Miss autopsies at this tier center on the JAM POINT: the exact moment your attempt stopped generating new lemmas. Common diagnoses: stopped gathering data too early (phase-one starvation), fixated on one representation (never rewrote the divisibility as a quadratic), or proved lemmas but never attempted assembly (lemma hoarding). Each has a drill; naming the diagnosis is what turns a loss into training.

Worked example

Extraction, worked, from the Vieta-jumping solution above.

Nudge 1
Ask what in the STATEMENT could have routed you to the technique — that phrasing becomes the filed cue.
Reveal step 1
Retrospective cue: the statement offered a SYMMETRIC condition polynomial in two variables plus an integrality claim. Cue filed: 'symmetric Diophantine condition, quadratic in each variable alone → fix the value, treat as a quadratic in one variable, examine the second root.'
Nudge 2
Strip the problem's specifics and state the reusable move in one sentence.
Reveal step 2
Portable sentence: 'When an integer solution exists, Vieta manufactures a second solution from the first; if the new one is strictly smaller, descend to a base case.' Note what is NOT filed: the constants of IMO 1988/6. The move survives; the numbers never matter again.
Nudge 3
Name the exact moment your own attempt stopped producing lemmas — that is the diagnosis to drill.
Reveal step 3
Jam-point autopsy (typical): 'I conjectured $k$ square from data but never wrote the equation as a quadratic in $a$ — representation fixation.' Drill assigned: five problems where the winning move is re-viewing a symmetric relation as a polynomial in ONE of its variables. That is a complete study cycle: cue, move, diagnosis, drill.
Answer
One recognition cue, one portable move-sentence, one named jam diagnosis with its drill — the full yield of a studied solution, and the only parts worth keeping.

Pitfall. Reading a solution start-to-finish and nodding. Comprehension is not acquisition — if you did not re-derive the middle from the first borrowed move, next month you will remember THAT it was solved, not HOW.

Before you open the gates

Four phases, in order: data → conjectures → local lemmas → assembly; when assembly jams, return to data with sharper questions.
Judge sessions by artifacts produced, not proximity feelings — three true lemmas is a good day.
Exam policy: nightmares last, on banked time, hunting the 1-3 point artifact; training policy: hours on purpose.
Study protocol: attempt state written first, borrow one move at a time, re-derive between borrows.
Every studied solution yields exactly three keepers: a recognition cue, a portable move-sentence, a named jam diagnosis with its drill.

Check yourself

1. At nightmare tier, session progress is measured by:

2. In Vieta jumping on $a^2 - kab + b^2 - k = 0$, the second root $a' = kb - a$ matters because:

3. The interrupted-read protocol has you stop reading a solution:

Practice ladder — three rungs, rising

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

Rung 1 (investigation, generate the family). For the relation $a^2 + b^2 = 4(ab + 1)$ — that is, value $k = 4$ — verify $(0, 2)$ is a solution, then use the Vieta jump $x_{\text{next}} = 4x - x_{\text{prev}}$ to list the next three solution pairs, and confirm the base solution makes $k$ a perfect square.

One hint

Read the relation as a quadratic in the larger variable: from a solution the partner root gives the single chain $0, 2, 8, 30, \dots$ via $x_{\text{next}} = 4x - x_{\text{prev}}$. Pair consecutive terms.

Worked resolution

Check $(0, 2)$: $0^2 + 2^2 = 4 = 4(0\cdot 2 + 1)$, so $k = 4$ and the base value is $4 = 2^2$, a perfect square. Reading $x^2 - 4b\,x + (b^2 - 4) = 0$, the chain of single terms advances by $x_{\text{next}} = 4x - x_{\text{prev}}$: from $0, 2$ we get $4\cdot 2 - 0 = 8$, $4\cdot 8 - 2 = 30$, $4\cdot 30 - 8 = 112$. So after $(0, 2)$ the next three solution pairs are $(2, 8), (8, 30), (30, 112)$; for instance $\tfrac{8^2 + 30^2}{8\cdot 30 + 1} = \tfrac{964}{241} = 4$. The chain descends to the base $(0, 2)$, where $k = 2^2$ — exactly the perfect square the general theorem predicts. [Source: this week's investigation-protocol section — the $k = 4$ chain $2 \to 8 \to 30 \to 112$ in the worked IMO-1988/6 data table.]

Rung 2 (Vieta jumping). Let $a, b$ be positive integers such that $ab$ divides $a^2 + b^2 + 1$. Prove that $\dfrac{a^2 + b^2 + 1}{ab} = 3$.

One hint

Set $k = \tfrac{a^2 + b^2 + 1}{ab}$ and take a solution minimizing $a + b$. Read $x^2 - kbx + (b^2 + 1) = 0$ as a quadratic in $x$; the partner root descends unless $a = b$. Solve the $a = b$ base case.

Worked resolution

Let $k = \dfrac{a^2 + b^2 + 1}{ab}$ be a positive integer and take a solution $(a, b)$ of $a^2 + b^2 + 1 = k\,ab$ minimizing $a + b$, WLOG $a \ge b$. View it as $x^2 - (kb)x + (b^2 + 1) = 0$ with root $a$; the other root is $a' = kb - a = \tfrac{b^2 + 1}{a}$, a positive integer (the product of the roots is $b^2 + 1 > 0$, and $a' = kb - a \in \mathbb{Z}$). Then $(a', b)$ is another positive-integer solution with the same $k$. If $a > b$, then $a \ge b + 1$ gives $a^2 \ge b^2 + 2b + 1 > b^2 + 1$, so $a' = \tfrac{b^2 + 1}{a} < a$ and $(a', b)$ is strictly smaller — contradicting minimality. Hence the minimal solution has $a = b$, where $k = \tfrac{2a^2 + 1}{a^2} = 2 + \tfrac{1}{a^2}$ is an integer only for $a = 1$, giving $k = 3$. Every solution shares this value, so $\dfrac{a^2 + b^2 + 1}{ab} = 3$. [Source: this week's investigation-protocol section — Vieta jumping: fix the value and read the symmetric condition as a quadratic in one variable.]

Rung 3 (investigate, then prove). Prove that if $a, b$ are positive integers with $ab \mid a^2 + b^2$, then $a = b$.

One hint

Small cases suggest $a = b$. To prove it, divide out $d = \gcd(a, b)$ and use coprimality: reduce to a coprime pair and show it must be $(1, 1)$.

Worked resolution

Small solutions $(1,1), (2,2), (3,3)$ suggest the pattern $a = b$. Proof: write $d = \gcd(a, b)$, $a = dx$, $b = dy$ with $\gcd(x, y) = 1$. Then $\dfrac{a^2 + b^2}{ab} = \dfrac{x^2 + y^2}{xy}$, so $xy \mid x^2 + y^2$. Now $x \mid x^2 + y^2$ and $x \mid x^2$ give $x \mid y^2$; since $\gcd(x, y) = 1$ also $\gcd(x, y^2) = 1$, forcing $x = 1$. By the same argument $y = 1$. Hence $a = b = d$ (and the quotient equals $2$). [Source: this week's investigation-protocol section — the four-phase method (data $\to$ conjecture $a = b$ $\to$ coprime-reduction lemma $\to$ done) run on a fresh instance.]

Prove it — constructed response

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Prove that $\dfrac{a^2 + b^2}{ab + 1}$ is a perfect square. Use the Vieta-jumping / minimal-counterexample method from this week's worked investigation, and justify every step of the descent: that the jumped value is a nonnegative integer, that it strictly decreases, and what the base case forces.

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

Fixes $k = \tfrac{a^2+b^2}{ab+1}$, assumes for contradiction that $k$ is not a perfect square, and among all positive-integer solutions of $a^2 + b^2 = k(ab+1)$ chooses one minimizing $a + b$, reducing to $a > b \ge 1$ (handling the $a=b$ case).
Views $x^2 - (kb)x + (b^2 - k) = 0$ as a quadratic in $x$ with known root $a$; identifies the second root $a' = kb - a = \tfrac{b^2 - k}{a}$, proves $a'$ is an integer and $a' \ge 0$ (ruling out $a' < 0$ from $a'^2 + b^2 = k(a'b + 1) > 0$), and shows $a' = 0$ would force $k = b^2$, a square.
Shows $a' < a$ (from $a' = \tfrac{b^2 - k}{a} < \tfrac{b^2}{a} < a$), so $(a', b)$ is a strictly smaller positive-integer solution with the same $k$, contradicting minimality; concludes $k$ must be a perfect square.

Model proof (compare AFTER grading yourself)

Let $k = \dfrac{a^2 + b^2}{ab + 1}$, a positive integer. Suppose, for contradiction, that $k$ is not a perfect square. Among all positive-integer pairs $(a,b)$ with $a^2 + b^2 = k(ab + 1)$, choose one with $a + b$ minimal, and assume WLOG $a \ge b$. First, $a \ne b$: if $a = b$ then $k = \tfrac{2a^2}{a^2 + 1}$, so $(a^2 + 1) \mid 2a^2 = 2(a^2 + 1) - 2$, forcing $a^2 + 1 \mid 2$; then $a = 1$ and $k = 1$, a square — excluded. So $a > b \ge 1$. Fix $b$ and read the relation as a quadratic in the first variable: $x^2 - (kb)x + (b^2 - k) = 0$, with $x = a$ a root. Let $a'$ be the other root. By Vieta, $a' = kb - a$ (an integer) and $a\,a' = b^2 - k$, so $a' = \tfrac{b^2 - k}{a}$. The pair $(a', b)$ again satisfies $a'^2 + b^2 = k(a'b + 1)$. Since $a'^2 + b^2 > 0$ and $k > 0$, we need $a'b + 1 > 0$; as $b \ge 1$ this forces $a' \ge 0$. If $a' = 0$, the relation gives $b^2 = k$, making $k$ a square — excluded; hence $a' \ge 1$ and $(a', b)$ is a pair of positive integers realizing the same $k$. Finally, using $k \ge 1$ and $b < a$, $a' = \dfrac{b^2 - k}{a} < \dfrac{b^2}{a} < \dfrac{a^2}{a} = a$, so $a' < a$ and therefore $a' + b < a + b$: a strictly smaller solution, contradicting minimality of $(a,b)$. No non-square $k$ can occur, so $\dfrac{a^2 + b^2}{ab + 1}$ is always a perfect square. $\blacksquare$

Confidence in your proof (0–5): 2

The gates

Technique Resources — USAMO-level proof craft (nightmare entry) (study before the reinforcing problems)

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Learning support for this week's reinforcing IMO/Putnam problems. Read/watch these before (and after) attempting the USAMO-level proof craft (nightmare entry) problems so each attempt has a source to learn the method and to check your write-up against.

sources & assignments (5)

Source Evan Chen — MOHS Hardness Scale — Evan Chen *MOHS Hardness Scale* — read fully; rate 3 past USAMO/Putnam A3 problems and predict the proof skeleton
Source Michael Penn — Proof Writing (playlist) — Michael Penn — olympiad-level proof walkthroughs Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Kedlaya Putnam archive — after each attempt, read the official write-up for that exact year/slot and re-write 1 proof to match
Source Po-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Po-Shen Loh — olympiad strategy & hard-problem walkthroughs
Source CMU 21-295 Putnam Seminar (full session recordings) — Seminar brainstorming on harder problems (companion to nightmare entry)

Nightmare Archive

Nightmare Archive accelerated only

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- Archive: 4 Nightmare · USAMO/hard Putnam A3–A4 · 35 min each - Archive: 3 Putnam A3/B3 · any topic · 30 min each

sources & assignments (6)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Writeup companion for “Nightmare Archive” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — Putnam 2024 A4 · Number Theory/Algebra · 35 min eachNightmare
Problems Archive Browser — Putnam 2005 A4 · Number Theory · 30 min eachNightmare

archive pull: 1 problems · Number Theory · A3/A4/B3 · Nightmare · 35 min →

A1/B1/A2/B2 Maintenance

A1/B1/A2/B2 Maintenance (do not drop these during nightmare weeks) accelerated only

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- Archive: 6 mixed A1/A2/B1/B2 · all topics · 20 min each

sources & assignments (5)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Writeup companion for “A1/B1/A2/B2 Maintenance (do not drop these during nightmare weeks)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — Putnam 1989 A1 · Number Theory/Algebra · 20 min eachPutnam

archive pull: 1 problems · Number Theory/Algebra · A1/A2/B1/B2 · Challenge · 20 min →

Certified Lemma

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- On any unsolved problem: write 1 clean sub-lemma

Ritual: On any unsolved problem, write one clean, fully-proved sub-lemma — a self-contained claim you *can* certify, even if the full problem stays open.

sources & assignments (4)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Writeup companion for “Certified Lemma” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Miss Autopsy

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- Classify each miss: content gap / wrong first move / proof-conversion failure / execution error

Ritual: - Classify each miss: content gap / wrong first move / proof-conversion failure / execution error

sources & assignments (4)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Writeup companion for “Miss Autopsy” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

SPIRAL

SPIRAL: mixed A1/A2/B1/B2 maintenance

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Added from AGENTS.md Phase 3 SPIRAL rule: Weeks 22–24 keep mixed prior-topic reinforcement while nightmare work runs.

sources & assignments (5)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Writeup companion for “SPIRAL: mixed A1/A2/B1/B2 maintenance” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — Putnam 1988 B1 · Analysis · 25 min eachPutnam

archive pull: 1 problems · Analysis · A1/A2/B1/B2 · Challenge · 25 min →

Deep Study — Solution-study protocol on real nightmare problems

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Mastery-phase learning gate: at the nightmare tier, studying solutions WITH a protocol is learning — passive reading is not.

Ritual: Output: 3 first-move notes + 4 miniature cards.

why this gate: the lesson's §1 The investigation protocol is what it trains

sources & assignments (7)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — 2021–2023, slots A3/A4/B3: attempt 30 min each, then study the official solution and write a 'how would I have found this first move' note for each (3 notes total).
Source Matousek — Thirty-three Miniatures (full free PDF) — Miniatures 12–15: the heavier linear-algebra weapons — summarize each in 5 lines.
Source Putnam and Beyond — Pick 3 starred problems from your weakest section; study the printed solutions with the same 'find the first move' protocol.
Source CMU 21-295 Putnam Seminar (full session recordings) — Pick any ONE full session (this is deliberately your choice — match it to your weakest hard family); pause at each solution start and pre-commit your own first move before watching theirs.
Source Michael Penn — One of my favorites from the Putnam — A full hard-Putnam walkthrough — run the same pre-commit-the-first-move protocol on it.
Source Supplemental — Lupu TTU MATH 4000, Lecture 9 (direct) — Putnam seminar spine, followed in course order (Lecture 9 of 11 assigned). Livestream pace — watch at 1.5x; pause before each solution and commit to a first move yourself before he reveals his.
Source Supplemental — Lupu TTU MATH 4000, Recitation 9 (direct) — Problem session paired with Lecture 9. Solutions are presented and defended live — grade each one against your own writeup standard: would it earn 10/10?

USAMO/IMO Bridge — IMO 1988/6

USAMO/IMO Bridge — IMO 1988/6 (solution study) (stretch) accelerated only

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Ritual: Solution-study debrief: write the one sentence you would have needed to find the key move yourself. File it as a pattern note.

why this gate: the lesson's §2 Solution study: extraction, not admiration is what it trains

sources & assignments (5)

Source IMO 1988/6 (solution study) — AoPS wiki (statement + solutions + discussion link) — ab+1 | a²+b² ⇒ (a²+b²)/(ab+1) is a perfect square — the Vieta-jumping legend. Run this week's solution-study protocol on the most famous hard problem in olympiad history. Attempt 25 min, then STUDY the Vieta-jumping solution with the protocol; tomorrow reconstruct it cold. Time cap 50 min — do not scroll to the Solutions section until the attempt is over.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Strategy companion for “USAMO/IMO Bridge — IMO 1988/6 (solution study) (stretch)” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
Source Cezar Lupu — TTU MATH 4000, Lecture 1 — Pacing companion for “USAMO/IMO Bridge — IMO 1988/6 (solution study) (stretch)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Michael Penn — Putnam Exam Solutions (playlist) — Writeup companion for “USAMO/IMO Bridge — IMO 1988/6 (solution study) (stretch)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems IMO 1988/6 — Attempt 25 min cold, then run the solution-study protocol on the official Vieta-jumping solution: copy the key move, name WHY it works, close the page, reconstruct the full proof from memory. Tomorrow: one cold reconstruction, no notes.USAMO bridge

Continuation — Abstract Algebra

Continuation — Abstract Algebra: rings, fields & modular rings (≈2 hrs)

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Ritual: Thread: Abstract Algebra. Required ~2-hr continuation swapped from the Matoušek miniature — the ring/field layer behind modular arithmetic.

sources & assignments (6)

Source Matousek — Thirty-three Miniatures (full free PDF) — Continuation-lane companion for: LA in combinatorics: one Matoušek miniature (≈2 hrs). Reused from the verified pool — watch/read with this week's lens.
Source Microsoft Research — The Wonders of the Probabilistic Method — Continuation-lane companion for: LA in combinatorics: one Matoušek miniature (≈2 hrs). Reused from the verified pool — watch/read with this week's lens.
Source Dr. Ebrahimian — Putnam Math Competition (playlist) — Pacing companion for “Continuation — LA in combinatorics: one Matoušek miniature (≈2 hrs)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Cezar Lupu — TTU MATH 4000, Lecture 1 — Writeup companion for “Continuation — LA in combinatorics: one Matoušek miniature (≈2 hrs)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Source Michael Penn — Abstract Algebra (playlist) — Companion lecture series for this week's abstract-algebra rep — watch the relevant 1–2 videos, then do the written rep below. Watch-for: this playlist opens with set-theory / proof-writing videos — skip to the 'Abstract Algebra |' entries for the group/ring/field content this gate needs.

Problems Continuation lane — Abstract Algebra — (1) In the ring ℤ/nℤ, find all units and all zero-divisors for n = 12. (2) Prove ℤ/pℤ is a field iff p is prime (no zero divisors ⟺ every nonzero element is invertible ⟺ p prime). (3) Show ℤ/6ℤ has zero divisors, hence is not a field. (4) State the definition of a field (commutative ring in which every nonzero element is invertible) and give two examples and two non-examples. Displaces the Matoušek rank-argument continuation — rank arguments remain in the W20 algebraic-method deep-study.Standard

Keep-Warm Rotation — polynomials · complex / roots of unity · extremal / graph

Keep-Warm Rotation — polynomials · complex / roots of unity · extremal / graph (30 min)

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Ritual: If any micro-rep took over 10 minutes or failed, that family goes on this week's repair list — it was rusting.

sources & assignments (2)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Pull keep-warm reps here when the archive button's pick doesn't match the rotation topic.

Problems Keep-warm rotation — Three 10-minute micro-reps, no notes: (1) polynomials — one polynomial rep (Vieta or interpolation), or re-derive the finite-difference degree test; (2) complex / roots of unity — one roots-of-unity filter rep, or re-derive the sum of the n-th roots of unity; (3) extremal / graph — one extremal rep: state the extremal configuration before proving it. The dormancy rule: no tool family sleeps longer than 2 weeks — these are the three most overdue right now.Standard

archive pull: 1 problems · Algebra/Combinatorics · A1/B1 · Putnam · 10 min →

Track A∗ — Analysis

Track A∗ — Analysis (parallel mastery track): Convexity & integral inequalities

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Runs EVERY mastery week so Analysis never goes cold — the Toolkit parallel-track model carried into Mastery. One depth problem + one A3–A6 reach problem each week.

sources & assignments (4)

Source Pólya & Szegő — Problems and Theorems in Analysis I — This week: Convexity & integral inequalities. Read the matching section, then do the two problems below.
Source Radulescu — Problems in Real Analysis — Companion reference for the same material.

Problems Analysis (depth) — Putnam 1995 B2 — work it with the reading open; one clean write-up.Putnam
Problems Analysis (reach) — Putnam 1992 A4 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track D∗ — Linear Algebra

Track D∗ — Linear Algebra (parallel mastery track): Matrix identities, commutators, Cayley–Hamilton powers

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Runs EVERY mastery week so Linear Algebra never goes cold — the Toolkit parallel-track model carried into Mastery. One depth problem + one A3–A6 reach problem each week.

sources & assignments (4)

Source Prasolov — Problems and Theorems in Linear Algebra — This week: Matrix identities, commutators, Cayley–Hamilton powers. Read the matching section, then do the two problems below.
Source Axler — Linear Algebra Done Right (Ch. 8 & 10) — Companion reference for the same material.

Problems Linear Algebra (depth) — Putnam 1992 B5 — work it with the reading open; one clean write-up.Putnam
Problems Linear Algebra (reach) — Putnam 1992 B6 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track E∗ — Number Theory

Track E∗ — Number Theory (parallel mastery track): Sums of two squares, Gaussian integers

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Runs EVERY mastery week so Number Theory never goes cold — the Toolkit parallel-track model carried into Mastery. One depth problem + one A3–A6 reach problem each week.

sources & assignments (4)

Source 104 Number Theory Problems (Andreescu, Andrica, Feng) — Advanced — This week: Sums of two squares, Gaussian integers. Read the matching section, then do the two problems below.
Source Niven, Zuckerman & Montgomery — Theory of Numbers — Companion reference for the same material.

Problems Number Theory (depth) — Putnam 2017 B2 — work it with the reading open; one clean write-up.Putnam
Problems Number Theory (reach) — Putnam 1966 A4 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track C∗ — Combinatorics

Track C∗ — Combinatorics (parallel mastery track): Probabilistic method — expectation & existence

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Continuous Combinatorics mastery spine (Track C∗). Combinatorics is the #1 Putnam genre (~20%); A1–B2 combinatorics is already drilled hard in the slots, so this spine adds the continuous theory ladder plus a weekly A3–A6 reach problem (the real gap). One depth re-solve + one fresh reach each week.

sources & assignments (4)

Source Combinatorics spine — Probability text — This week: Probabilistic method — expectation & existence. Expectation existence arguments; first/second moment; alteration. Read the matching section, then do the two problems below.
Source Combinatorics continuous spine — Runs every Mastery week (parallel-track model) so Combinatorics — the single most frequent Putnam genre (~20%) — never goes cold. 1 depth re-solve + 1 fresh reach problem.

Problems Combinatorics (depth) — Putnam 2003 A1 — cold re-solve with the reading open; one clean write-up.Putnam
Problems Combinatorics (reach) — Putnam 1996 B5 — 45 min; extract one rigorous lemma / reach-point (2–4 of 10), do not force a full solve.Nightmare

Track F∗ — Geometry · Combinatorics · Probability

Track F∗ — Geometry · Combinatorics · Probability (parallel maintenance): Probability — Variance & tail bounds

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Bundled parallel maintenance track so Geometry, Combinatorics, and Probability each get a fixed weekly cadence in Mastery (probability previously had multi-week gaps). Rotates focus; ~1.5 hrs.

sources & assignments (3)

Source Blitzstein & Hwang — Introduction to Probability (free) — This week’s focus: Variance & tail bounds. Variance/covariance, Markov & Chebyshev, the union bound, second-moment method.
Source Probability keep-warm — Lower-cadence topic on a guaranteed weekly rotation (Toolkit Track F model). 1 problem this week; Probability comes round again every 3rd week.

Problems Probability (F∗) — Putnam 1989 B1 — variance & tail bounds; full attempt + one clean write-up.Putnam

Reach Mock — 3 hard problems, lemma goal

Reach Mock — 3 hard problems, lemma goal (105 min)

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Turns nightmare work into measurable partial-credit practice instead of open-ended suffering.

Ritual: For each problem: first move, lemma, obstruction, and whether continuing would be rational on exam day.

sources & assignments (2)

Source Hard-problem partial-credit protocol — Three hard problems, 35 minutes each. The score target is one certified lemma per problem, not full solves.

Problems Reach mock — Putnam 2024 A4, Putnam 2021 A3, Putnam 1999 A5 — 105 minutes total; one certified lemma or obstruction per problem.Nightmare

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. Slot reinforcement supplement: A2/B2 are deliberately overtrained so they become bankable, not occasional exposure. Slot reinforcement supplement: A3/B3 now get repeated lemma-hunt reps so slot 3 is trained as a strong reach tier. Slot reinforcement supplement: A4-B6 remain a good reach tier without displacing A1-B3 scoring practice.

sources & assignments (17)

Problems Engel/PnB — Engel Ch. 6 probs 17–19 · PnB Ch. 6 probs 11–12 (maintenance, mixed)Putnam
Problems Problems from the Book — Look at the Exponent chapter, probs 1–2Nightmare
Problems Kedlaya — Putnam 1999 A5,A6 · 2000 B5,B6Nightmare
Problems Putnam Archive — Putnam 2023 A2 - 18 min clean-solve; write the final proof and fatal-slip check.Putnam
Problems Putnam Archive — Putnam 1998 B2 - 22 min clean-solve; write the final proof and fatal-slip check.Putnam
Problems Putnam Archive — Putnam 2013 A3 - 35 min lemma hunt; record one rigorous lemma even if the full proof does not close.Nightmare
Problems Putnam Archive — Putnam 2012 B3 - 35 min lemma hunt; compare to official solution after the attempt.Nightmare
Problems Putnam Archive — Putnam 2001 A4 - 45 min reach; extract the first invariant/structure and stop only after a written lemma.Nightmare
Problems Putnam Archive — Putnam 2001 B4 - 45 min reach; extract the first invariant/structure and stop only after a written lemma.Nightmare
Problems Putnam Archive — Putnam 2001 A5 - 50 min reach; official-solution autopsy required after the attempt.Nightmare
Problems Putnam Archive — Putnam 2018 B5 - 50 min reach; official-solution autopsy required after the attempt.Nightmare
Problems Putnam Slot 2 Equalizer — Slot-2 Equalizer Set 9: Putnam 2015 B2, Putnam 2018 B2, Putnam 2019 B2 - clean-solve focus; write final proofs and fatal-slip checks.Putnam
Problems Putnam Slot 4 Equalizer — Slot-4 Equalizer Set 9: Putnam 2000 B4 - reach/lemma focus; extract the first invariant or structure before reading solutions.Nightmare
Problems AMC/AIME Fluency Opener — Optional 5-minute opener: solve the three linked AMC/AIME reps before Putnam work; stop at 15 minutes total.AMC/AIME · archive: AMC integer algebra AMC floor function AMC systems/geometry
Problems Slot 3 Pair Balancer — Putnam 2022 A3 - one certified lemma plus official-solution reconstruction; balances slot-3/slot-4 hard-contact reps.Nightmare
Problems Putnam Full Past Exam — Full Past Exam 21: Putnam 2015 A1, Putnam 2015 A2, Putnam 2015 A3, Putnam 2015 A4, Putnam 2015 A5, Putnam 2015 A6, Putnam 2015 B1, Putnam 2015 B2, Putnam 2015 B3, Putnam 2015 B4, Putnam 2015 B5, Putnam 2015 B6 - Primary paper protocol: A-session A1-A6 in one 180-minute block; B-session B1-B6 in one 180-minute block within 48 hours; next day score all 12 slots, tag every miss, and choose three repair pulls.Putnam
Problems Putnam Full Past Exam — Full Past Exam 22: Putnam 2016 A1, Putnam 2016 A2, Putnam 2016 A3, Putnam 2016 A4, Putnam 2016 A5, Putnam 2016 A6, Putnam 2016 B1, Putnam 2016 B2, Putnam 2016 B3, Putnam 2016 B4, Putnam 2016 B5, Putnam 2016 B6 - Secondary paper protocol: Day 1 bankable sweep A1-A2-B1-B2 in 100 minutes; Day 2 medium/hard sweep A3-A4-B3-B4 in 100 minutes with certified-lemma goal; Day 3 reach scan A5-A6-B5-B6 in 50 minutes; Day 4 score all 12 slots and re-solve one same-family miss.Putnam

Reflect

Reflect: reconstruct and log the pattern

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- Standard ---

Ritual: Pick your best nightmare-tier partial or solve. Close all notes and the solution. Rewrite the full solution from memory, then compare with the original — what did you miss or reorder? Write one pattern note: "When I see [pattern], do [first move]."

sources & assignments (4)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — Catalan Numbers: generating function and closed form — 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

- Standard ---

Ritual: Pick one solution. Re-check for: (1) a false claim, (2) a missing case, (3) a wrong bound. Extra check at nightmare tier: is the certified lemma stated precisely enough to earn its partial credit alone?

sources & assignments (4)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source CMU 21-295 Putnam Seminar (full session recordings) — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — One of my favorites from the Putnam — Week reference — this gate applies this week's lead material (“Deep Study — Solution-study protocol on real nightmare problems”). If an attempt stalls, the repair source is here.
Source Michael Penn — Catalan Numbers: generating function and closed form — Intuition companion for “Verify: audit one proof before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Exit contract — Week 22

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W23 · Nightmare Ramp I
the exam follows the final taper week

Train Week 22 in the trainer →