Full Mock #1: the dress rehearsal, and what to do with its wreckage
First full mock in the real 2026 format: four 90-minute sessions, three problems each, twelve problems total. Nothing new gets learned this week — the mock MEASURES, and the structured autopsy converts the measurement into your final month's agenda. Treat the mock day itself with exam-day seriousness (timing, table, no phone, printed problems), because half of what you are testing is not mathematics; it is you, under the actual conditions.
A mock is an instrument, not a verdict: its score is one noisy sample, but its FAILURE PATTERN is high-signal data you can still act on with five weeks left.
Key ideas — the week on one card
Mock fidelity is the experiment: real timing, formal triage with written tags, margin timeline with timestamps — and no post-mortems between sessions.
The checkpoint decides switches, the structural test breaks ties, and any switch is executed with a parking note and no oscillation.
Autopsy next-day and cold, in three passes: regrade hostile, classify every non-10 into five bins, pattern-match across all twelve problems.
Near-misses (5-7 points) are the highest-information attempts — one classification decides their entire repair.
Judge the trend against the last mock, not the raw score; two banked problems per session is the whole December target.
1Running the mock like it counts
Fidelity rules: real timing with a visible clock, sessions separated by real breaks, write-ups on paper in real time (not 'I basically had it' notes), and the triage ritual from Week 17 executed formally — five-minute read, tags written down, plan committed. The tags matter doubly on a mock: they let the autopsy compare your PREDICTED difficulty ordering against reality, which is the calibration statistic you have tracked since Week 13.
In-session policy is exactly the trained one: bank first with finish-clean, fight with a pre-set checkpoint, park with partial-credit hunting. The one mock-specific addition: keep a margin timeline — one-word notes with timestamps ('committed FE 0:07', 'parked 0:38', 'back, new idea 1:11'). Memory of session dynamics is fiction by evening; the margin notes are the only honest record the autopsy will have.
Between sessions, no post-mortems — the December rule, rehearsed now. Chewing on session one during the break taxes session two; the score is fixed, the next session is not. Practicing that emotional pivot is a genuine part of what this week trains, and it is harder than it sounds after a bad session.
Worked example
Session dynamics, worked: at minute 30 your FIGHT problem has a half-working idea; your PARK problem suddenly looks approachable. Stay or switch?
Nudge 1
Consult the checkpoint you wrote at triage, not the excitement of the half-idea.
Reveal step 1
Consult the checkpoint you set at triage, not your excitement. If the fight problem's checkpoint was 'park at 0:35 without a completable plan', you have five minutes to convert the half-idea into a named next step — the checkpoint question, verbatim, from Week 13.
Nudge 2
Test the half-idea structurally: reduction, symmetry, or invariant — or just a vibe?
Reveal step 2
Evaluate the half-idea structurally (Week 18's rule): does it expose symmetry, reduce the problem, or create an invariant? A half-idea that is a REDUCTION is usually worth the five minutes; one that is a vibe is not.
Nudge 3
If switching: parking note first, then full commitment — no oscillation.
Reveal step 3
If you switch: thirty seconds writing the parking note (current state, next thing to try), then full commitment to the new problem — no oscillating. Oscillation is the mock's most common self-inflicted wound, and the margin timeline will show you exactly how much it cost, in your own handwriting.
Answer
The checkpoint decides, the structural test breaks ties, and any switch is executed with a parking note and no oscillation.
Pitfall. Treating the mock as practice ('I'll just see how it goes'). Casual mocks measure casual performance — a quantity December does not feature. The conditions are the experiment; protect them.
2The structured autopsy
Run it the NEXT day, cold, in three passes. Pass one — regrade: score each attempt as a hostile grader (0-10, rubric mentality), and log the gap between evening-of self-scores and cold scores; that gap is your calibration error, and shrinking it is a named goal of the final month. Pass two — classify every non-10: knowledge gap (technique missing), recognition gap (technique owned, never surfaced), execution slip (arithmetic, case dropped), write-up leak (solved, under-communicated), or strategy error (time misallocated — the margin timeline testifies). Pass three — pattern-match ACROSS the twelve problems: three recognition gaps in one topic is a repair plan; three write-up leaks is a different one.
The repair block consumes the rest of the week, driven by classification counts, not feelings. Knowledge gaps at this date get triaged honestly: a hole in a core topic gets patched (targeted drill, technique card, three problems); a hole in a fringe topic gets ACCEPTED and routed around — five weeks out, breadth additions cost more than they pay. Recognition gaps get the five-problem burst treatment; execution slips get verification-ritual reps; strategy errors get their scenario re-run on paper.
Score interpretation, once, honestly: this mock minus your Week 13 mini-mock is the trend; the trend, not the level, predicts December. And the placement arithmetic is steadying: top-200 historically needs roughly 30-40 of 120 — two banked problems per session with honest partial credit elsewhere is the entire requirement. The mock's job is telling you which two per session are reachable and what is stealing them.
Worked example
Autopsy, worked: your mock produced 26 points — solves on two A1-tier and one B2-tier problem, near-misses (5-7 pts) on two more, zeros elsewhere. Classify and build the week's repair plan.
Nudge 1
Start with the near-misses; classify each one's single fixable defect.
Reveal step 1
Regrade the near-misses first — they are the highest-information attempts. A 6/10 that was 'solved with a dropped case' is an execution slip (repair: verification reps); a 6/10 that stalled at the second move is a recognition gap on the transformed problem (repair: five-problem burst on that seam). Suppose one of each.
Nudge 2
Look for topic-level patterns across the zeros before treating them as diagnoses.
Reveal step 2
Pattern-check the zeros: if both fight-tier zeros were probability-flavored, that is ONE topic-level recognition gap, and a five-problem burst on probability setups is the repair. If the zeros share no pattern, they are the exam being the exam — A5/B6 zeros are the national norm, not a diagnosis.
Nudge 3
Let the bin COUNTS write the week's repair plan — one targeted intervention per bin.
Reveal step 3
Assemble the week: two repair drills (verification reps; probability burst), one write-up rehab if any solve leaked points, and one strategy note for the timeline ('spent 20 min post-checkpoint on the parked A4'). That is a full repair block — four targeted interventions from one structured reading, each cheap, each aimed at a demonstrated leak.
Answer
Regrade near-misses first, classify each miss into the five bins, pattern-match across problems, and let the bin COUNTS write the repair block.
Pitfall. Autopsying the same evening. Evening-you regrades with the solver's memory and the solver's ego; the classification bins only get filled honestly by next-day-you, working from ink.
Before you open the gates
Full fidelity: real timing, real write-ups, formal triage with written tags, margin timeline with timestamps.
Between sessions: no post-mortems — the pivot is a trained skill, rehearsed now.
Autopsy next-day and cold: regrade, classify into the five bins, pattern-match across all twelve.
2. Five weeks out, a knowledge gap in a fringe topic gets:
Late-stage hours go where points are: core-topic repairs and craft. A conscious routing decision is not the same as ignoring it — triage is honest about the cost.Repair: revisit §2 The structured autopsy ↑
3. The most information-dense attempts to regrade are:
A near-miss is one fixable defect away from ten points; whether that defect is execution, recognition, or write-up determines the entire repair, and misreading it wastes the week.Repair: revisit §2 The structured autopsy ↑
Each rung: attempt cold, one hint if stuck, worked resolution only after a real try.
Rung 1 (Complex keep-warm, direct). For an integer $n \ge 2$, evaluate $\sum_{k=0}^{n-1}\cos\frac{2\pi k}{n}$.
One hint
These cosines are the real parts of the $n$-th roots of unity. What is the sum of all $n$-th roots of unity?
Worked resolution
The $n$-th roots of unity are $e^{2\pi i k/n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}$, and their sum is $0$ (the coefficient of $z^{n-1}$ in $z^n - 1$ vanishes for $n \ge 2$). Taking real parts of the sum, $\sum_{k=0}^{n-1}\cos\frac{2\pi k}{n} = \operatorname{Re}(0) = 0$. [Source: this week's endgame keep-warm — the closed-book Complex recall (De Moivre + the $n$-th-roots-of-unity sum).]
✓ rung 1 done
Rung 2 (number-theory keep-warm). Prove that $30 \mid n^5 - n$ for every integer $n$.
One hint
Factor $n^5 - n$, then show divisibility by $2$, $3$, and $5$ separately; $30 = 2\cdot 3\cdot 5$.
Worked resolution
Factor $n^5 - n = n(n^2-1)(n^2+1) = (n-1)n(n+1)(n^2+1)$. The three consecutive integers $(n-1)n(n+1)$ make the product divisible by both $2$ and $3$. For $5$: by Fermat's little theorem $n^5 \equiv n \pmod 5$, so $5 \mid n^5 - n$ (or check the five residues mod $5$ directly). Since $2, 3, 5$ are pairwise coprime, $30 \mid n^5 - n$. [Source: this week's endgame keep-warm — the number-theory conversion rep.]
✓ rung 2 done
Rung 3 (linear-algebra keep-warm). Let $A$ be an $n \times n$ real matrix with $A^2 = 0$. Prove that $\operatorname{rank}(A) \le n/2$.
One hint
$A^2 = 0$ says the image of $A$ lies inside the kernel of $A$. Combine that containment with rank-nullity.
Worked resolution
From $A^2 = 0$ we get $A(Av) = 0$ for every $v$, i.e. $\operatorname{im}(A) \subseteq \ker(A)$, so $\operatorname{rank}(A) = \dim\operatorname{im}(A) \le \dim\ker(A)$. By rank-nullity $\dim\ker(A) = n - \operatorname{rank}(A)$, hence $\operatorname{rank}(A) \le n - \operatorname{rank}(A)$, giving $2\operatorname{rank}(A) \le n$, i.e. $\operatorname{rank}(A) \le n/2$. [Source: this week's endgame keep-warm — the linear-algebra conversion rep.]
✓ rung 3 done
Prove it — constructed response
A closed-book Complex keep-warm, promoted to a proof. Let $n \ge 2$. Prove that the sum of all $n$-th roots of unity is $0$. Give TWO independent justifications: (a) an algebraic one (Vieta on $z^n - 1$, or the finite geometric series), and (b) a rotation-symmetry one. Name De Moivre's theorem at the point where you use it to identify the roots.
Self-grade against the rubric — completion requires the judgment, not the text
Model proof (compare AFTER grading yourself)
Write $\zeta_k = e^{2\pi i k/n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}$ for $k = 0, 1, \dots, n-1$. By De Moivre, $\zeta_k^n = \cos 2\pi k + i\sin 2\pi k = 1$, so each $\zeta_k$ is a root of $z^n - 1$; the $\zeta_k$ are distinct, so they are all $n$ of its roots. Let $S = \sum_{k=0}^{n-1}\zeta_k$. (a) Algebraic: in $z^n - 1 = z^n + 0\cdot z^{n-1} + \dots + 0\cdot z - 1$ the coefficient of $z^{n-1}$ is $0$, so by Vieta $S = 0$; equivalently, with $\omega = \zeta_1 \ne 1$, the geometric series gives $S = \frac{\omega^n - 1}{\omega - 1} = \frac{0}{\omega - 1} = 0$. (b) Symmetry: multiplication by $\omega = \zeta_1$ sends $\zeta_k \mapsto \zeta_{k+1 \bmod n}$, permuting the roots, so $\omega S = S$; thus $(\omega - 1)S = 0$, and since $\omega \ne 1$ (because $n \ge 2$) we conclude $S = 0$. $\blacksquare$
The gates
Technique Resources — full mock #1 — review resources (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 full mock #1 — review resources problems so each attempt has a source to learn the method and to check your write-up against.
SourceMichael Penn — Interesting Integrals (playlist) — Writeup companion for “Technique Resources — full mock #1 — review resources (study before the reinforcing problems)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Mock Exam
Mock Exam
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- Do: 6-hour timed mock (A-session 3 hr + B-session 3 hr, all 12 problems; use a real full exam year)
- Score: 0–2 (stuck/false start) or 8–10 (clean/near-clean) — no in-betweens
Ritual: - Do: 6-hour timed mock (A-session 3 hr + B-session 3 hr, all 12 problems; use a real full exam year)
- Score: 0–2 (stuck/false start) or 8–10 (clean/near-clean) — no in-betweens
SourceMichael Penn — Interesting Integrals (playlist) — Strategy companion for “Mock Exam” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
SourceSilver — Integration Bee Training (playlist) — Pacing companion for “Mock Exam” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourcePo-Shen Loh — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Interesting Integrals (playlist) — Writeup companion for “Structured Miss Autopsy” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Repair Block
Repair Block
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- Do: 3 repairs from routing table
sources & assignments (5)
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourcePo-Shen Loh — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Interesting Integrals (playlist) — Writeup companion for “Repair Block” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Problems3 repairs from routing table — 3 repairs from routing tablePutnam
Spiral Reinforcement
Spiral Reinforcement
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- Archive: Default: Putnam 2014 A1, Putnam 2015 B1, Putnam 2016 A1. If routing table has a weaker A1/B1 topic, swap only within that same slot band.
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sources & assignments (5)
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourcePo-Shen Loh — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Interesting Integrals (playlist) — Writeup companion for “Spiral Reinforcement” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
ProblemsArchive Browser — Default: Putnam 2014 A1, Putnam 2015 B1, Putnam 2009 B2. If routing table has a weaker A1/B1 topic, swap only within that same slot band.Putnam
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 (5)
ProblemsPnB — PnB — technique on tap; reference during the mock’s gap-repairPutnam
ProblemsMAA archive — One full recent paper from a year you have NOT drilled this program (least-touched recent years are cleanest — e.g. 2020 or 2006), 6 hrs timedPutnam
ProblemsAlexanderson — 4 problems on the mock’s missed slotsPutnam
ProblemsPutnam Full Past Exam — Full Past Exam 27: Putnam 2021 A1, Putnam 2021 A2, Putnam 2021 A3, Putnam 2021 A4, Putnam 2021 A5, Putnam 2021 A6, Putnam 2021 B1, Putnam 2021 B2, Putnam 2021 B3, Putnam 2021 B4, Putnam 2021 B5, Putnam 2021 B6 - Calibration mock protocol: strict A-session A1-A6 for 180 minutes and strict B-session B1-B6 for 180 minutes under exam rules; next day score all 12 slots, write a mock autopsy, tag the three largest point leaks, and complete one repair pull before any new paper.Putnam
ProblemsPutnam Full Past Exam — Full Past Exam 28: Putnam 2022 A1, Putnam 2022 A2, Putnam 2022 A3, Putnam 2022 A4, Putnam 2022 A5, Putnam 2022 A6, Putnam 2022 B1, Putnam 2022 B2, Putnam 2022 B3, Putnam 2022 B4, Putnam 2022 B5, Putnam 2022 B6 - Calibration mock protocol: strict A-session A1-A6 for 180 minutes and strict B-session B1-B6 for 180 minutes under exam rules; next day score all 12 slots, write a mock autopsy, tag the three largest point leaks, and complete one repair pull before any new paper.Putnam
Reflect
Reflect: reconstruct and log the pattern
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Mandatory Reflect gate from AGENTS.md for every week 3–28.
Ritual: Pick one problem you solved this week. Close all notes and the solution. Rewrite the full solution from memory. Then compare and write: When I see [pattern], do [first move].
sources & assignments (4)
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourcePo-Shen Loh — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMaths 505 — This integral taught me Feynman's technique — 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: fatal-slip audit before closing
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unlocks after: Reflect: reconstruct and log the pattern
Mandatory Verify gate from AGENTS.md for every week 3–28.
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourcePo-Shen Loh — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #1 — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMaths 505 — This integral taught me Feynman's technique — Intuition companion for “Verify: fatal-slip audit before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Endgame keep-warm
Endgame keep-warm: linear algebra · number theory · geometry · probability · complex numbers (30 min)
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Retention keep-warm (2026-07-02 audit): the mock endgame left linear algebra, number theory, geometry, probability and complex numbers untouched for 3-4 weeks before the exam. 25-30 minutes: one conversion-speed rep per topic; log any miss into repair.
Ritual: Recall (closed book): Complex: state De Moivre + n-th roots of unity sum, closed book.
