Week 5 of 28 · Toolkit · due 2026-06-27 · 26–30 hrs

Analysis: Differentiation + Algebra: Inequalities + LA: Eigenvalues + Comb: Bijections + NT: Divisibility + Hidden Tool: Convexity/Jensen/Smoothing

Analysis: Differentiation + Algebra: Inequalities + LA: Eigenvalues + Comb: Bijections + NT: Divisibility + Hidden Tool: Convexity/Jensen/Smoothing gate complete

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The derivative as witness: MVT, smoothing, and the arithmetic of divisibility

The Mean Value Theorem is the most quietly used theorem on the Putnam — it almost never stars in a problem, and it appears inside the solution of dozens. It converts information about values into information about derivatives and back, which is the entire business of analysis. Meanwhile the inequality track learns the two moves behind ninety percent of all contest inequalities, and number theory opens with the gcd — the most underrated object in the subject.

Every tool this week is a converter: MVT converts value gaps into derivative values, convexity converts midpoints into bounds, and Bézout converts gcd's into integer combinations. Learn what each converter eats and what it emits.

Key ideas — the week on one card

Rolle — continuous on $[a,b]$, differentiable on $(a,b)$, equal endpoint values — forces $f'(c) = 0$; MVT is Rolle applied to $f$ minus its chord.
Engel form (Cauchy–Schwarz): $\sum \frac{a_i^2}{b_i} \ge \frac{(\sum a_i)^2}{\sum b_i}$ for positive $b_i$ — squares upstairs is the cue.
$\operatorname{tr} A = \sum \lambda_i$, $\det A = \prod \lambda_i$, and $p(A)$ has eigenvalues $p(\lambda_i)$ — two invariants often pin the whole spectrum.
$\gcd(a,b)$ is the smallest positive integer combination of $a$ and $b$; Euclid reduces it in two-line steps.

1MVT and its favorite disguise: Rolle

Rolle's theorem — for $f$ continuous on $[a,b]$, differentiable on $(a,b)$, equal values at the endpoints force a zero of the derivative between them — is the engine (and those two hypotheses are graded: state them every time); the Mean Value Theorem $f(b) - f(a) = f'(c)(b-a)$ is Rolle applied to $f(x)$ minus its chord. The contest pattern: to prove SOMETHING about $f'$ (a zero, a large value, a repeated behavior), find two points where $f$ behaves identically and let Rolle produce the witness $c$. You will never locate $c$; like IVT, this is an existence factory.

Iterating Rolle climbs to higher derivatives: if $f$ is $n$ times differentiable and vanishes at $n+1$ distinct points, then $f'$ vanishes at $n$ points (one between each adjacent pair), $f''$ at $n-1$, ..., $f^{(n)}$ at some point. This is the standard weapon for 'show some derivative is zero' and for error terms in interpolation — a degree-$n$ polynomial agreeing with $f$ at $n+1$ points forces $f^{(n+1)}$ to appear in the error.

MVT on f(x) = x²: the secant on [1, 7] has slope 8, and the tangent matches it exactly at c = 4.

Worked example

Let $f$ be differentiable on $[0,1]$ with $f(0) = 0$ and $f(1) = 1$. Show there exist DISTINCT $a, b \in (0,1)$ with $f'(a)\, f'(b) = 1$.

Nudge 1
Choose the cut so the two slopes will MULTIPLY cleanly — cross $f$ with $1 - x$, not with a constant.
Reveal step 1
Cut where $f$ crosses the anti-diagonal: $g(x) = f(x) - (1 - x)$ is continuous (differentiable functions are continuous) with $g(0) = -1 < 0 < 1 = g(1)$, so IVT gives $t \in (0,1)$ with $f(t) = 1 - t$ — Week 4 supplying the anchor, chosen so the two slopes will multiply cleanly.
Nudge 2
MVT on $[0,t]$ and on $[t,1]$ separately: write both difference quotients.
Reveal step 2
MVT on each half: some $a \in (0, t)$ has $f'(a) = \frac{f(t) - 0}{t} = \frac{1-t}{t}$, and some $b \in (t, 1)$ has $f'(b) = \frac{1 - f(t)}{1 - t} = \frac{t}{1-t}$.
Nudge 3
Multiply the two slopes and watch everything cancel.
Reveal step 3
Multiply: $f'(a)f'(b) = \frac{1-t}{t} \cdot \frac{t}{1-t} = 1$ exactly, and $a < t < b$ guarantees distinctness. The skeleton — cut where the VALUE says to, harvest one MVT witness per piece, combine — is the reusable move; the art was choosing the cut ($f(t) = 1-t$) so the combination collapses.
Answer
Cut at $f(t) = 1 - t$ (IVT), MVT on each half: the slopes are $\frac{1-t}{t}$ and $\frac{t}{1-t}$, whose product is exactly 1.

Pitfall. Using MVT witnesses as if you chose them. The theorem gives SOME $c$ — you may not also assume $c$ is rational, or in the left half, or anything else. Each extra property needs its own argument.

2Inequalities: AM–GM and Cauchy–Schwarz are 90% of the syllabus

The Arithmetic–Geometric Mean inequality — for nonnegative reals, $\frac{a_1 + \cdots + a_n}{n} \ge \sqrt[n]{a_1 \cdots a_n}$, equality iff all equal — is the tool for bounding SUMS below by PRODUCTS and vice versa. Its tactical form is 'split and balance': to bound $x + \frac{1}{x}$, or $a + a + \frac{1}{a^2}$, split terms so the product becomes constant, and the bound falls out with its equality case attached.

Cauchy–Schwarz — $(\sum a_i b_i)^2 \le (\sum a_i^2)(\sum b_i^2)$ — is the tool for decoupling a product of sums. Its most useful avatar on this exam is Engel form (Titu): $\sum \frac{x_i^2}{y_i} \ge \frac{(\sum x_i)^2}{\sum y_i}$ for positive $y_i$. Any sum of fractions with squares upstairs should trigger it on sight.

The meta-skill is reading the EQUALITY CASE as a diagnostic. AM–GM is tight when terms are equal; Cauchy–Schwarz when the vectors are proportional. If the problem's suspected extremum has all variables equal, AM–GM in some split will likely close it; if the extremum is lopsided, you probably need a different split — or a different inequality. Check equality cases BEFORE grinding.

Worked example

For positive reals with $a + b + c = 3$, prove $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \ge 3$.

Nudge 1
Squares over positives: write the Engel lower bound with $(a+b+c)^2$ upstairs.
Reveal step 1
Shape recognition: squares upstairs, positive things downstairs — Engel form. $\sum \frac{a^2}{b} \ge \frac{(a+b+c)^2}{a+b+c} = a+b+c$.
Nudge 2
The constraint hands you the denominator.
Reveal step 2
Substitute the constraint: $a + b + c = 3$, so the sum is at least 3.
Nudge 3
When is Engel tight? Chase the proportionality condition to its only solution.
Reveal step 3
Equality diagnostic: Engel is tight when $\frac{a}{b} = \frac{b}{c} = \frac{c}{a}$, forcing $a = b = c = 1$ — consistent with the constraint, so the bound is sharp and the write-up is two lines.
Answer
Engel/Titu: $\sum \frac{a^2}{b} \ge \frac{(a+b+c)^2}{a+b+c} = 3$, equality at $a=b=c=1$.

Pitfall. Applying AM–GM in a direction that makes the bound useless (bounding below when you need above), or ignoring the equality case entirely. State the equality condition in every inequality solution — it is both a correctness check and free credit.

3Eigenvalues: what a matrix cannot hide

Whatever a matrix does to most vectors is complicated; what it does to eigenvectors is multiplication by a scalar. The two conserved traces of that hidden simplicity are: the TRACE equals the sum of eigenvalues, and the DETERMINANT equals their product (both with multiplicity, over $\mathbb{C}$). Contest problems about $\operatorname{tr}(A^k)$ are eigenvalue problems in disguise: $\operatorname{tr}(A^k) = \sum \lambda_i^k$, the power sums of the spectrum — and power sums connect to coefficients by Newton's identities (Week 19).

The characteristic polynomial $\det(\lambda I - A)$ packages the spectrum, and Cayley–Hamilton says the matrix satisfies its own characteristic polynomial. For $2 \times 2$ this is the workhorse identity $A^2 = (\operatorname{tr} A)\,A - (\det A)\,I$: every power of a $2\times2$ matrix is a LINEAR combination of $A$ and $I$, so 'compute $A^{100}$' problems collapse to a two-term recurrence on coefficients.

Worked example

$A$ is a $2\times 2$ real matrix with $\operatorname{tr} A = 1$ and $\det A = -6$. Compute $\operatorname{tr}(A^2)$ and $\det(A^2)$.

Nudge 1
Trace and determinant DETERMINE the characteristic polynomial of a $2\times 2$: $\lambda^2 - (\operatorname{tr}A)\lambda + \det A$ — factor it.
Reveal step 1
Spectrum from the two invariants: eigenvalues satisfy $\lambda^2 - (\operatorname{tr}A)\lambda + \det A = \lambda^2 - \lambda - 6 = 0$, so $\lambda = 3, -2$.
Nudge 2
Squaring the matrix squares each eigenvalue.
Reveal step 2
Powers act on the spectrum: eigenvalues of $A^2$ are $9$ and $4$.
Nudge 3
The trace of $A^2$ is the sum of the squared eigenvalues.
Reveal step 3
Read off: $\operatorname{tr}(A^2) = 9 + 4 = 13$, $\det(A^2) = 9 \cdot 4 = 36 = (\det A)^2$ — the determinant square-checks itself. Alternative route: Cayley–Hamilton gives $A^2 = A + 6I$, so $\operatorname{tr}(A^2) = \operatorname{tr}A + 12 = 13$. Two roads, same answer — a free verification.
Answer
$\operatorname{tr}(A^2) = 13$, $\det(A^2) = 36$; via eigenvalues $3, -2$ or via Cayley–Hamilton $A^2 = A + 6I$.

Pitfall. Assuming real matrices have real eigenvalues. A rotation matrix has none; trace and determinant identities survive because they run over COMPLEX eigenvalues, which always exist in full multiplicity.

4Divisibility: gcd is a linear combination

Bézout's identity — $\gcd(a,b) = ax + by$ for some integers $x, y$ — is the bridge between divisibility (a multiplicative idea) and linear arithmetic (an additive one). Its most-used corollary: if $d \mid ab$ and $\gcd(d, a) = 1$, then $d \mid b$. Write that on your hand; it is invoked silently in half of all number theory solutions and is exactly the step students skip and lose points on.

The Euclidean algorithm computes gcd's by $\gcd(a, b) = \gcd(b, a \bmod b)$, but its contest value is structural: $\gcd(a, b) = \gcd(a - kb, b)$ for ANY $k$. Aimed at polynomial expressions this becomes a weapon — $\gcd(n^2 + 1, n + 1)$ reduces by $n^2 + 1 - (n-1)(n+1) = 2$ to $\gcd(2, n+1)$, and the whole problem is now a parity question.

Keep the multiplicative worldview alongside: every $n$ factors uniquely into primes, and $\gcd$/$\operatorname{lcm}$ act coordinatewise on exponent vectors — $\gcd$ takes $\min$, $\operatorname{lcm}$ takes $\max$, and $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = ab$ is just $\min + \max = $ sum, one coordinate at a time.

Worked example

Show that $\frac{21n + 4}{14n + 3}$ is in lowest terms for every positive integer $n$ (IMO 1959, Problem 1).

Nudge 1
Run Euclid on the pair: subtract the right multiple of the denominator from the numerator.
Reveal step 1
Compute the gcd of numerator and denominator by Euclidean reduction: $\gcd(21n+4, 14n+3) = \gcd(21n + 4 - (14n + 3), 14n + 3) = \gcd(7n + 1, 14n + 3)$.
Nudge 2
One more reduction and the remainder collapses to a constant.
Reveal step 2
Reduce again: $\gcd(7n+1, 14n + 3 - 2(7n+1)) = \gcd(7n + 1, 1)$.
Nudge 3
$\gcd = 1$ means the fraction never reduces — say so explicitly.
Reveal step 3
$\gcd(\cdot, 1) = 1$ — numerator and denominator are coprime for every $n$, so the fraction never reduces. Note the shape: the algorithm ran on POLYNOMIALS in $n$, one linear elimination per line.
Answer
$\gcd(21n+4, 14n+3) = \gcd(7n+1, 14n+3) = \gcd(7n+1, 1) = 1$.

Pitfall. Cancelling a factor across a divisibility without checking coprimality. From $d \mid ab$ you may conclude $d \mid b$ ONLY when $\gcd(d, a) = 1$ — say it, cite it, then cancel.

Before you open the gates

To prove something about $f'$, hunt for two points where $f$ agrees — Rolle manufactures the witness.
Fractions with squares upstairs → Engel form on sight; then verify the equality case matches the problem's extremum.
State every inequality's equality condition — diagnostic first, free credit second.
$\operatorname{tr}$ and $\det$ are the spectrum's shadows; powers of matrices act on eigenvalues, and Cayley–Hamilton linearizes $2\times2$ powers.
Run the Euclidean algorithm on polynomial expressions to collapse gcd problems to small constants.

Check yourself

1. $f$ is TWICE differentiable with $f(0) = f(1) = f(2) = 0$. What is guaranteed?

2. For positive reals, $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} \ge 3$ follows fastest from:

3. A $2\times2$ matrix has $\operatorname{tr} A = 0$ and $\det A = -1$. Then $A^2$ equals:

Practice ladder — three rungs, rising

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

Rung 1 (Rolle, direct). Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$ with $f(a) = f(b)$. A student claims 'then $f$ is constant.' Give an explicit counterexample, then state what Rolle's theorem actually guarantees and prove it in one line from the MVT.

One hint

For the counterexample, try a parabola on a symmetric interval. Rolle is the MVT in the case $f(a) = f(b)$.

Worked resolution

Counterexample: $f(x) = x^2$ on $[-1, 1]$ has $f(-1) = f(1) = 1$ but is not constant. What Rolle guarantees: some $c \in (a,b)$ with $f'(c) = 0$. From the MVT: there is $c \in (a,b)$ with $f'(c) = \tfrac{f(b) - f(a)}{b - a}$; when $f(a) = f(b)$ the numerator is $0$, so $f'(c) = 0$. (For $x^2$ on $[-1,1]$, $c = 0$.) [Source: this week's MVT section — Rolle as the MVT's favorite disguise.]

Rung 2 (AM-GM / Cauchy). For positive reals $a, b, c$ prove $\tfrac{a}{b} + \tfrac{b}{c} + \tfrac{c}{a} \ge 3$, and separately $(a + b + c)\left(\tfrac1a + \tfrac1b + \tfrac1c\right) \ge 9$.

One hint

First: AM-GM on the three terms — their product is $1$. Second: Cauchy-Schwarz, or AM-GM on each factor.

Worked resolution

First: by AM-GM on the three positive terms, $\tfrac{a}{b} + \tfrac{b}{c} + \tfrac{c}{a} \ge 3\sqrt[3]{\tfrac{a}{b}\cdot\tfrac{b}{c}\cdot\tfrac{c}{a}} = 3\sqrt[3]{1} = 3$, equality iff $a = b = c$. Second: by Cauchy-Schwarz, $(a + b + c)\left(\tfrac1a + \tfrac1b + \tfrac1c\right) \ge \left(\sqrt{a}\cdot\tfrac{1}{\sqrt a} + \sqrt{b}\cdot\tfrac{1}{\sqrt b} + \sqrt{c}\cdot\tfrac{1}{\sqrt c}\right)^2 = (1 + 1 + 1)^2 = 9$. [Source: this week's inequalities section — AM-GM and Cauchy-Schwarz.]

Rung 3 (gcd, a step up). Prove that $\gcd(n! + 1,\ (n+1)! + 1) = 1$ for every positive integer $n$.

One hint

Let $d$ divide both. Use $(n+1)! + 1 = (n+1)\,n! + 1$ to eliminate $n!$: form $(n+1)! + 1 - (n+1)(n! + 1)$.

Worked resolution

Let $d = \gcd(n! + 1,\ (n+1)! + 1)$. Since $(n+1)! + 1 = (n+1)\,n! + 1$, compute $(n+1)! + 1 - (n+1)(n! + 1) = (n+1)n! + 1 - (n+1)n! - (n+1) = 1 - (n+1) = -n$, so $d \mid n$. Then $d \mid n$ and $n \mid n!$ give $d \mid n!$, and with $d \mid (n! + 1)$ we get $d \mid (n! + 1) - n! = 1$, so $d = 1$. [Source: this week's divisibility section — gcd as a linear combination, one step past the IMO warm-up.]

Prove it — constructed response

Prove that for every positive integer $n$, the fraction $\dfrac{21n + 4}{14n + 3}$ is in lowest terms (IMO 1959, Problem 1). Make the gcd argument explicit — exhibit an integer combination, don't just assert coprimality.

The gates

Track A — Analysis: Rudin Ch. 5 (1.5 hrs/day): Load

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- Read: Rudin *PMA* Ch. 5 (~26 pp.) — derivative definition, chain rule, Rolle, MVT, generalized MVT, Taylor's theorem, L'Hôpital - Watch: MIT 18.100A — differentiation lecture (OCW) - Do: Rudin Ch. 5 ex 1, 3, 5, 7, 9, 13, 15, 18, 21, 24 - Do: PnB section 3.2.5–3.2.6 probs 1–4 - Do: CMU 04-Calculus exercise sheet probs 3–4 - Note (MAT 260 background): Leibniz rule only — d/dt ∫f(x,t)dx = ∫∂f/∂t dx. Do 1 Feynman-technique problem from MIT sum_integrals.pdf to preview Week 8.

sources & assignments (5)

Source Rudin — Principles of Mathematical Analysis (PMA) — Rudin *PMA* Ch. 5 (~26 pp.) — derivative definition, chain rule, Rolle, MVT, generalized MVT, Taylor's theorem, L'Hôpital
Source MIT 18.100A — MIT 18.100A — differentiation lecture (OCW)
Source MIT 18.01SC — Single Variable Calculus (OCW) — Use four 18.01SC units only: Mean Value Theorem, Taylor Series, Fundamental Theorem of Calculus, and Approximating Sums by Integrals. Watch the first lecture video in each unit; output one theorem card per unit.
Source Spivak — Calculus — Spivak *Calculus* Ch. 11 (MVT, probs 1–6) + Ch. 20 (Taylor's theorem with remainder, probs 1–3)
Source 3Blue1Brown — Essence of Calculus — Visual calculus intuition (limits, derivatives, integrals, Taylor)

Track A — Analysis: Rudin Ch. 5 (1.5 hrs/day): Drill

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unlocks after: Track A — Analysis: Rudin Ch. 5 (1.5 hrs/day): Load

sources & assignments (7)

Source Rudin — Principles of Mathematical Analysis (PMA) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 5 (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. 5 (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.01SC — Single Variable Calculus (OCW) — Drill reference — this gate trains the material of “Track A — Analysis: Rudin Ch. 5 (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. 5 (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. 5 ex 1, 3, 5, 7, 9, 13, 15, 18, 21, 24Putnam
Problems PnB — PnB section 3.2.5–3.2.6 probs 1–4Putnam
Problems CMU 04-Calculus exercise sheet — CMU 04-Calculus exercise sheet probs 3–4Putnam

Track B — Algebra

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

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- Read: PnB section 2.1.3–2.1.5 (AM-GM pp. 39–45, Cauchy-Schwarz pp. 32–38, Jensen pp. 45–50) - Read: Engel Ch. 12 — intro + probs 1–15 - Watch: little fermat Inequalities Tutorial — lessons 1–3 - Watch: Chan Lye Lee Math Olympiad Inequalities — 2–3 videos - Do: MIT ineq.pdf probs 1–6 (write equality case for every problem) - Do: PnB section 2.1.5 probs 1–5 - Do: MIT ss3.pdf probs 28–30 - Do: CMU 06-Inequalities exercise sheet probs 1–2 - Do: Stanford 06wk6 probs 1–3 - Do: Manfrino *Inequalities* section 1.3 problem 1 + section 1.4 problem 1 - Archive: 3 Putnam A1/A2 · Algebra/Inequalities · any slot · 18 min each Inequality technique ladder (try in order, always state the equality case): AM-GM → Cauchy-Schwarz (both forms) → Jensen (check f″≥0) → tangent-line trick → SOS → uvw/Schur → smoothing → normalization (set ∑=1 or ∏=1) → homogenization.

why this gate: the lesson's §2 Inequalities: AM–GM and Cauchy–Schwarz are 90% of the syllabus is what it trains

sources & assignments (5)

Source Putnam and Beyond — PnB section 2.1.3–2.1.5 (AM-GM pp. 39–45, Cauchy-Schwarz pp. 32–38, Jensen pp. 45–50)
Source Engel — Problem-Solving Strategies — Engel Ch. 12 — intro + probs 1–15
Source little fermat — Olympiad Inequalities Tutorial (playlist) — little fermat Inequalities Tutorial — lessons 1–3
Source Chan Lye Lee Math Olympiad Inequalities — Chan Lye Lee Math Olympiad Inequalities — 2–3 videos
Source MIT 18.06 Strang — Intuition companion for “Track B — Algebra: Inequalities (1.5 hrs/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

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

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

why this gate: the lesson's §2 Inequalities: AM–GM and Cauchy–Schwarz are 90% of the syllabus is what it trains

sources & assignments (10)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track B — Algebra: Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Chan Lye Lee Math Olympiad Inequalities — Drill reference — this gate trains the material of “Track B — Algebra: Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.06 Strang — Intuition companion for “Track B — Algebra: Inequalities (1.5 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems MIT ineq.pdf — MIT ineq.pdf probs 1–6 (write equality case for every problem)Putnam
Problems PnB — PnB section 2.1.5 probs 1–5Putnam
Problems MIT ss3.pdf — MIT ss3.pdf probs 28–30Putnam
Problems CMU 06-Inequalities exercise sheet — CMU 06-Inequalities exercise sheet probs 1–2Putnam
Problems Stanford 06wk6 — Stanford 06wk6 probs 1–3Putnam
Problems Manfrino *Inequalities* — Manfrino *Inequalities* §1.3 problem 1 + §1.4 problem 1Putnam

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

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

why this gate: the lesson's §2 Inequalities: AM–GM and Cauchy–Schwarz are 90% of the syllabus is what it trains

sources & assignments (5)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track B — Algebra: Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source little fermat — Olympiad Inequalities Tutorial (playlist) — Drill reference — this gate trains the material of “Track B — Algebra: Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Chan Lye Lee Math Olympiad Inequalities — Drill reference — this gate trains the material of “Track B — Algebra: Inequalities (1.5 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source MIT 18.06 Strang — Intuition companion for “Track B — Algebra: Inequalities (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 2024 A1, 2007 A1, 2006 A1, 2002 A1 · Algebra · 18 min eachPutnam

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

Hidden Tool

Hidden Tool: Convexity / Jensen / Tangent Line / Smoothing

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*Prerequisites: inequalities reading (Track B above) + Rudin Ch. 4 (continuity)* - Convexity: - f convex ↔ f'' ≥ 0 for smooth f — check this first - Jensen: f convex → f(Σλᵢxᵢ) ≤ Σλᵢf(xᵢ), weights λᵢ ≥ 0, Σλᵢ = 1 - Tangent line: f convex → f(x) ≥ f(a) + f'(a)(x−a) for all x - Smoothing: replace two unequal variables with their average; check if inequality improves; repeat to equality - Homogenization: equalize degree on both sides → eliminate one constraint - SOS: is LHS − RHS a sum of squares? - Equality case: identify before closing any inequality proof

sources & assignments (7)

Source Engel — Problem-Solving Strategies — Engel Ch. 12 — convexity, Jensen, SOS, smoothing
Source Manfrino — Inequalities — Manfrino *Inequalities* §1.3 (Cauchy–Schwarz) probs 1–4 + §2.2 (convexity/Jensen) probs 1–4
Source little fermat — Olympiad Inequalities Tutorial (playlist) — little fermat Inequalities Tutorial — Jensen / convexity lesson
Source Michael Penn — Putnam Exam Solutions (playlist) — Lecture series — direct playlist companion to this gate's reading
Source MIT 18.100A — Intuition companion for “Hidden Tool: Convexity / Jensen / Smoothing” — 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. 16 Inequalities. Use as a contest-style inequality warmup before Jensen/smoothing; output one equality-case check.

Problems Convexity drill (standard examples) — (1) Prove AM-GM for three variables by applying Jensen to -ln x; (2) tangent-line method (displaces the old min-of-x+1/x warm-up, which it strictly contains): for positive a, b, c with a + b + c = 3, prove 1/(1+a²) + 1/(1+b²) + 1/(1+c²) ≥ 3/2 — show f(x) = 1/(1+x²) ≥ (2 − x)/2 for x > 0 by reducing to x(x−1)² ≥ 0, then sum the three tangent-line bounds; name the move: tangent line at the equality point x = 1; (3) prove that among positive reals with fixed sum, the product is maximized when all are equal, via a smoothing/replacement argument.AMC

Track D — Linear Algebra

Track D — Linear Algebra: Axler Ch. 4–5B (1 hr/day): Load

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- Read: Axler Ch. 4 (~15 pp.) — polynomials applied to operators - Read: Axler Ch. 5A–5B (~40 pp.) — invariant subspaces, eigenvalues over ℂ, upper triangular matrices - Watch: Sheldon Axler LADR lectures — lectures 8–10 - Do: Axler 4: 1, 3, 5, 7 · 5A: 1, 3, 5, 9, 12, 15 · 5B: 1, 5, 9, 13 - Do: MIT linalg.pdf probs 1–2

why this gate: the lesson's §4 Divisibility: gcd is a linear combination is what it trains

sources & assignments (4)

Source Axler — Linear Algebra Done Right — Axler Ch. 5A–5B (~40 pp.) — invariant subspaces, eigenvalues over ℂ, upper triangular matrices
Source Sheldon Axler — Linear Algebra Done Right (lecture videos) — Sheldon Axler LADR lectures — lectures 8–10
Source 3Blue1Brown — Essence of Linear Algebra — Essence of Linear Algebra — visual companion to this gate's LA reading.
Source Chan Lye Lee Math Olympiad Inequalities — Intuition companion for “Track D — Linear Algebra: Axler Ch. 4–5A (1 hr/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track D — Linear Algebra: Axler Ch. 4–5B (1 hr/day): Drill

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unlocks after: Track D — Linear Algebra: Axler Ch. 4–5B (1 hr/day): Load

why this gate: the lesson's §4 Divisibility: gcd is a linear combination is what it trains

sources & assignments (6)

Source Axler — Linear Algebra Done Right — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 4–5B (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Sheldon Axler — Linear Algebra Done Right (lecture videos) — Drill reference — this gate trains the material of “Track D — Linear Algebra: Axler Ch. 4–5B (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. 4–5B (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 — Intuition companion for “Track D — Linear Algebra: Axler Ch. 4–5A (1 hr/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems Axler 4 — Axler 4: 1, 3, 5, 7 · 5A: 1, 3, 5, 9, 12, 15 · 5B: 1, 5, 9, 13Putnam
Problems MIT linalg.pdf — MIT linalg.pdf probs 1–2Putnam

Track C — Combinatorics

Track C — Combinatorics: Bijections + Recursion Intro (1 hr/day): Load

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- Read: PnB section 6.2.2 (GF intro pp. 298–302) + Engel Ch. 8 — recurrence sections - Watch: Shahriar Shahriari — Generating Functions intro - Do: 102 Combinatorial Problems #38, 43, 47, 52 - Do: MIT comb.pdf probs 5–8 - Do: MIT indep.pdf probs 1–2

sources & assignments (5)

Source Putnam and Beyond — PnB section 6.2.2 (GF intro pp. 298–302) + Engel Ch. 8 — recurrence sections
Source Shahriar Shahriari — Shahriar Shahriari — Generating Functions intro
Source Shahriar Shahriari — Combinatorics, An Invitation (playlist) — Combinatorics lecture series — direct playlist companion to this gate's reading.
Source Brualdi — Introductory Combinatorics — Brualdi *Introductory Combinatorics* Ch. 2 (permutations & combinations) + Ch. 6 (inclusion–exclusion) — read + do 5 exercises each
Source Generating-functions lecture series — Intuition companion for “Track C — Combinatorics: Bijections + Recursion Intro (1 hr/day): Load” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Track C — Combinatorics: Bijections + Recursion Intro (1 hr/day): Drill

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unlocks after: Track C — Combinatorics: Bijections + Recursion Intro (1 hr/day): Load

sources & assignments (7)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track C — Combinatorics: Bijections + Recursion Intro (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Shahriar Shahriari — Drill reference — this gate trains the material of “Track C — Combinatorics: Bijections + Recursion Intro (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: Bijections + Recursion Intro (1 hr/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Generating-functions lecture series — Intuition companion for “Track C — Combinatorics: Bijections + Recursion Intro (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 #38, 43, 47, 52Putnam
Problems MIT comb.pdf — MIT comb.pdf probs 5–8Putnam
Problems MIT indep.pdf — MIT indep.pdf probs 1–2Putnam

Track E — Number Theory

Track E — Number Theory: Divisibility + Foundations (0.75 hrs/day): Load

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- Read: PnB section 5.1 (pp. 247–257) — divisibility, gcd, Bezout, primes - Read: Engel Ch. 13 — intro + probs 1–8 - Watch: Michael Penn Number Theory v2 — videos 1–3 - Do: 104 NT Problems #1, 2, 4, 6, 8, 10 - Do: PnB section 5.1 probs 1–3

why this gate: the lesson's §4 Divisibility: gcd is a linear combination is what it trains

sources & assignments (6)

Source Putnam and Beyond — PnB section 5.1 (pp. 247–257) — divisibility, gcd, Bezout, primes
Source Engel — Problem-Solving Strategies — Engel Ch. 13 — intro + probs 1–8
Source Michael Penn — Number Theory v2 (playlist) — Number-theory lecture series — direct playlist companion to this gate's reading.
Source Niven, Zuckerman & Montgomery — Introduction to the Theory of Numbers — Niven–Zuckerman–Montgomery *Intro to the Theory of Numbers* §2.1 problems 1–2, §2.2 problems 1–2, §2.3 problems 1–2
Source CMU 03-Number Theory (Putnam seminar) — Intuition companion for “Track E — Number Theory: Divisibility + Foundations (0.75 hrs/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. 3 Divisibility and Remainders. Use before divisibility/archive reps; output one modular reduction trigger.

Track E — Number Theory: Divisibility + Foundations (0.75 hrs/day): Drill

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unlocks after: Track E — Number Theory: Divisibility + Foundations (0.75 hrs/day): Load

why this gate: the lesson's §4 Divisibility: gcd is a linear combination is what it trains

sources & assignments (5)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Track E — Number Theory: Divisibility + Foundations (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Number Theory v2 (playlist) — Drill reference — this gate trains the material of “Track E — Number Theory: Divisibility + Foundations (0.75 hrs/day): Load”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source CMU 03-Number Theory (Putnam seminar) — Intuition companion for “Track E — Number Theory: Divisibility + Foundations (0.75 hrs/day): Drill” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Problems 104 NT — 104 NT Problems #1, 2, 4, 6, 8, 10Putnam
Problems PnB — PnB section 5.1 probs 1–3Putnam

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. 5 (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. 5 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.01SC — Single Variable Calculus (OCW) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 5 (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 number theory speed reps (open each below) · ~8 min eachAMC · archive: AMC 12 A AMC 12 B 2021 AMC 10 B

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 Larson — Larson Ch. 7 probs 1–8 (inequalities)Putnam
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 2021 B1 Putnam 2001 A2 AMC probability/NT

Reflect

Reflect: reconstruct and log the pattern

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- Reconstruct: 1 Rudin Ch. 5 exercise → close → rewrite → pattern note - Reflect: MVT trigger hypotheses from memory - Reflect: inequality named move all 7 moves from memory - Reflect: eigenvalue card stub (trace = Σλ, det = Πλ) - Verify: one inequality — equality case identified and tight? ---

Ritual: 1 Rudin Ch. 5 exercise → close → rewrite → pattern note | MVT trigger hypotheses from memory | inequality named move all 7 moves from memory | eigenvalue card stub (trace = Σλ, det = Πλ)

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. 5 (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. 5 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.01SC — Single Variable Calculus (OCW) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 5 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source The Bright Side of Mathematics — Real Analysis (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: one inequality — equality case identified and tight?

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. 5 (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. 5 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source MIT 18.01SC — Single Variable Calculus (OCW) — Week reference — this gate applies this week's lead material (“Track A — Analysis: Rudin Ch. 5 (1.5 hrs/day): Load”). If an attempt stalls, the repair source is here.
Source The Bright Side of Mathematics — Real Analysis (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: Algebra 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 (6)

Source Lerma Putnam Training 2023 (Northwestern) — NU Putnam-team algebra sections — inequalities, polynomials, complex numbers, recurrences, symmetries.

Problems Lerma Training 2023 §2 Inequalities — Work problems 2.1–2.38 (4–6 per session). AM–GM, Cauchy–Schwarz, convexity/Jensen; always name the equality case first. Self-check against the PDF's hints/solutions parts.Putnam
Problems Lerma Training 2023 §4 Polynomials — Work problems 4.1–4.33 (4–6 per session). Roots/coefficients (Vieta), factor theorem, irreducibility, symmetric functions. Self-check against the PDF's hints/solutions parts.Putnam
Problems Lerma Training 2023 §5 Complex Numbers — Work problems 5.1–5.10 (4–6 per session). Roots of unity, rotation, |z| arguments; translate geometry to complex when it simplifies. Self-check against the PDF's hints/solutions parts.Putnam
Problems Lerma Training 2023 §7 Recurrences — Work problems 7.1–7.9 (4–6 per session). Characteristic equations, telescoping, generating-function setup. Self-check against the PDF's hints/solutions parts.Putnam
Problems Lerma Training 2023 §11 Symmetries — Work problems 11.1–11.4 (4–6 per session). Exploit symmetry/substitution to collapse the problem. Self-check against the PDF's hints/solutions parts.Putnam

Exit contract — Week 5

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W6 · Analysis: Integration + FTC + Algebra: Functional Equations + LA: Inner Products + Comb: Generating Functions + NT: Congruences + Hidden Tools: ODE Recognition + Generating Functions
the exam follows the final taper week

Train Week 5 in the trainer →