AMC 10 (Grades 9-10)

249 core topics + 100 prerequisite topics taught as needed · approximately 113 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph - topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Competition Math: AMC 10/12 · 9 topics

Symmetric Systems [H]	Never solve for x and y - combine the symmetric pieces.
Polynomial Root Play [H]	Vieta for cubics: coefficients are symmetric functions of roots.
Counting Probability [H]	Count with combinations; flip to the complement.
AMC Number Theory [H]	Divisor counts from exponents; lcm from gcd.
AMC Geometry [H]	Areas scale with the square; coordinates give areas for free.
Logs & Exponents on the AMC [H]	Move exponents through logs; match bases.
Telescoping & Series [H]	Collapse the sum before computing it.
Functional Equations [H]	Plug in smart values; iterate carefully.
Complex Numbers on the AMC [H]	Powers of i cycle; moduli multiply.

AMC 10/12: Advanced Problem-Solving Techniques · 15 topics

AM-GM & Inequality Bounds [H]	Bound sums and products without calculus.
Vieta: Symmetric Functions of Roots [H]	Turn any symmetric expression in the roots into the coefficients.
Stars, Bars & Bijective Counting [H]	Count integer solutions by placing dividers.
Principle of Inclusion-Exclusion [H]	Add the singles, subtract the overlaps, add back the triple.
Modular Arithmetic & the CRT [H]	Stitch remainders modulo coprime bases into one answer.
Divisor Counting & Sum of Divisors [H]	Read divisor data straight off the prime factorization.
Units & Last Digits via Cycles [H]	Powers repeat their tail digits on a short cycle.
Power of a Point [H]	One product controls every line through a point.
Mass Points, Menelaus & Ceva [H]	Hang weights on the vertices and let ratios balance.
Pythagorean Triples & Diophantine Structure [H]	Generate every right triangle from two parameters.
Recursion & Characteristic Equations [H]	Linear recurrences are governed by a small polynomial.
Roots of Unity & Complex Geometry [H]	The nth roots of unity are a regular polygon on the unit circle.
Invariants & Coloring Arguments [H]	Find a quantity the moves can't change.
Generating Functions (Intro) [H]	Encode a counting problem as a coefficient.
Cyclic Quadrilaterals & Angle Chasing [H]	Concyclic points unlock angle and length relations.

AMC 10/12 - Deep II · 35 topics

Remainder Theorem [H]	Dividing by (x − a) leaves remainder p(a).
Factor Theorem [H]	(x − a) divides p(x) exactly when p(a) = 0.
Sum of Reciprocal Roots [H]	1/r + 1/s = (r+s)/(rs) via Vieta.
Cauchy-Schwarz Bound [H]	(px+qy)² ≤ (p²+q²)(x²+y²).
Rearrangement Inequality [H]	Same-order pairing maximizes the sum of products.
Simplifying Radicals [H]	Pull out the largest square factor.
Odd-Denominator Telescoping [H]	1/((2k−1)(2k+1)) = ½(1/(2k−1) − 1/(2k+1)).
Infinite Geometric Series [H]	Sum = a/(1 − r) for |r| < 1.
Reciprocal Functional Equation [H]	Substitute x → 1/x and solve the linear system.
Binomial Coefficients [H]	Term of xᵏ in (x+c)ⁿ is C(n,k)c^{n−k}.
Fermat's Little Theorem [H]	a^{p−1} ≡ 1 (mod p) for prime p ∤ a.
Wilson's Theorem [H]	(p−1)! ≡ −1 (mod p) for prime p.
Multiplicative Order [H]	Smallest k with aᵏ ≡ 1 (mod n).
Legendre's Formula [H]	Exponent of p in n! is Σ⌊n/pᵏ⌋.
Base Representation [H]	Repeatedly divide to change base.
Euler's Totient Function [H]	φ(pᵃqᵇ) = pᵃqᵇ(1−1/p)(1−1/q).
Linear Diophantine Counting [H]	Count positive (x,y) on ax + by = N.
Modular Inverses [H]	Solve ax ≡ 1 (mod m).
Squarefree Part [H]	k is the product of odd-exponent primes.
Complementary Counting [H]	Subtract the bad arrangements from all.
Integer-Sided Triangles [H]	Casework with the triangle inequality.
Hockey Stick Identity [H]	Σ_{i=k}^{n} C(i,k) = C(n+1,k+1).
Expected Value (Linearity) [H]	E[sum] = sum of E's, no independence needed.
Conditional Probability [H]	Condition shrinks the sample space.
Geometric Probability [H]	Probability = favorable length / total length.
Gambler's Ruin [H]	Fair walk: reach N before 0 with prob k/N.
Pigeonhole Guarantees [H]	Worst case then one more.
Circular Arrangements [H]	Fix one seat: (n−1)! around a circle.
Law of Cosines [H]	c² = a² + b² − 2ab cos C.
Law of Sines & Circumradius [H]	a/sin A = b/sin B = 2R.
Stewart / Median Length [H]	4m_a² = 2b² + 2c² − a².
Ptolemy's Theorem [H]	AC·BD = AB·CD + BC·DA.
Heron's Formula [H]	Area = √(s(s−a)(s−b)(s−c)).
Angle Bisector Theorem [H]	Bisector splits the opposite side in ratio b:c.
Section Formula [H]	P = (m·B + n·A)/(m+n) for ratio m:n.

AMC 10/12 - Algebra, Functions and Sequences · 49 topics

Power Sums of x + 1/x [H]	The quantities x^n + 1/x^n obey s(n+1) = k·s(n) − s(n−1).
Evaluating the Polynomial Itself [H]	A product over the roots is the polynomial evaluated at one point.
Symmetric Sums for a Quartic [H]	Read e₁ through e₄ off the coefficients and rewrite the request in them.
Two Quadratics Sharing a Root [H]	Subtracting the two quadratics leaves a linear equation the shared root must satisfy.
Vieta a Second Time, in the Parameter [H]	A condition on the roots becomes a quadratic in the parameter, so Vieta applies again.
The Cubic With Transformed Roots [H]	Shifting every root replaces p(x) by p(x − h); inverting every root reverses the coefficients.
Symmetric Fractions of the Roots [H]	Put the symmetric fraction over one denominator and read both parts from Vieta.
Climbing the Power Sums [H]	Every x^n + y^n follows from x + y and xy, in either direction.
Three Pair Sums [H]	Adding the three pair sums gives twice the total, and each variable is the total minus one equation.
Systems Written in the Reciprocals [H]	xy/(x+y) = A means 1/x + 1/y = 1/A, which turns the system linear.
Systems Written in the Products [H]	Multiplying all three product equations gives (xyz)², and dividing recovers each variable.
Adding Every Equation at Once [H]	Summing a symmetric system produces the total, and each variable is one subtraction away.
Substituting the Partner Value [H]	Substitute the value that swaps the two unknown outputs, then solve the linear system.
Additive With a Correction Term [H]	f(x+y) = f(x) + f(y) + cxy builds f(n) one step at a time, adding c(n−1) each step.
Pairing f(x) With f(1 − x) [H]	When f(x) + f(1−x) is constant, a long sum collapses to a count of pairs.
Iterating a Map Into Its Cycle [H]	Compose the map a few times, find the period, then reduce the exponent modulo it.
Equations Mixing x With Its Floor [H]	Write x as n + f with n = ⌊x⌋ an integer and 0 ≤ f < 1, then read n off the size of the equation.
Counting With the Floor Function [H]	Group the index range into blocks on which the floor is constant.
The Fractional Part as Its Own Unknown [H]	Treat ⌊x⌋ = n and {x} = f as separate unknowns tied by 0 ≤ f < 1.
Counting With the Floor of a Square Root [H]	⌊√n⌋ = k exactly on the 2k+1 integers from k² to k²+2k.
Nested Absolute Values [H]	Peel one absolute value at a time and count how many branches survive.
Minimizing a Sum of Distances [H]	A sum of absolute values is piecewise linear, so its minimum sits at one of the marked points.
Regions Cut Out by Absolute Values [H]	Absolute-value inequalities bound polygons whose vertices are the axis intercepts.
An Arithmetic Sequence Pinned by Two Facts [H]	Two facts determine a₁ and d, and sums are linear in both.
Terms Two Progressions Share [H]	Common terms of two arithmetic progressions form a third, with difference lcm(d₁, d₂).
An Arithmetic Progression Turned Geometric [H]	The geometric condition (middle)² = (first)(third) reduces to d² = ac.
Telescoping by Partial Fractions [H]	Split 1/(k(k+m)) into (1/m)(1/k − 1/(k+m)) and cancel the interior.
Telescoping With Radicals [H]	Rationalizing a radical denominator turns each term into a difference that cancels.
Infinite Geometric Series in Disguise [H]	Identify the constant ratio hidden in a physical or geometric process, then sum a/(1−r).
A Geometric Series From Two Conditions [H]	Two sums involving the same a and r give two equations whose quotient isolates r.
Recursions That Cycle [H]	The recursion aₙ = aₙ₋₁ − aₙ₋₂ repeats with period 6 and any six consecutive terms sum to zero.
Chains of Change of Base [H]	log_a b · log_b c = log_a c, so a chain of logarithms collapses to its endpoints.
Systems Written in the Logarithms [H]	Set u = log x and v = log y; the system becomes ordinary algebra in u and v.
Quadratics in log x [H]	Substitute L = log x; Vieta on L gives the sum of the logs, hence the product of the roots.
One Value Shared by Several Exponentials [H]	Set the common value equal to k and take logarithms, so the exponents become reciprocals.
How Many Digits, and Which Leading Digit [H]	A positive integer n has ⌊log₁₀ n⌋ + 1 digits.
Nested Logarithms and Hidden Quadratics [H]	Work from the outside in, and name the repeated exponential as a single unknown.
Powers of 1 + i and the Sums They Compute [H]	(1+i)² = 2i, so every power of 1+i reduces to a power of 2 times one of 1, i, −1, −i.
Complex Roots of Real Quadratics [H]	For a real quadratic the roots are conjugates, so |z|² is the constant term and z + z̄ is minus the linear coefficient.
Cube Roots of Unity as an Algebraic Tool [H]	With ω³ = 1 and 1 + ω + ω² = 0, any expression in ω collapses to a short normal form.
Rotation by Multiplication [H]	Multiplying by i rotates a point 90° counterclockwise about the origin.
Splitting a Term to Make AM-GM Fit [H]	Break one term into equal pieces so the product of all pieces is free of x.
The Cauchy-Schwarz (Titu) Bound [H]	a²/x + b²/y ≥ (a+b)²/(x+y), with equality when x : y = a : b.
The Largest Product With a Fixed Sum [H]	Splitting a fixed sum into 3s (with at most two 2s) maximizes the product.
The Bound a Discriminant Forces [H]	Two reals with a known sum and product are the roots of a quadratic, and reality forces a discriminant bound.
Average Speed Is a Harmonic Mean [H]	Average speed is total distance over total time, so equal distances give the harmonic mean.
Mixtures and Repeated Replacement [H]	Track the amount of the pure substance, not the percentage.
Digit Problems and What Reversal Does [H]	Write the number as 10a + b, and reversal differences become multiples of 9 or 99.
Work Rates Add, Times Do Not [H]	Convert each time to a rate of one job per hour, add the rates, then invert.

AMC 10/12 - Number Theory and Counting · 46 topics

Smallest Integer with a Given Divisor Count [H]	Match the exponent pattern to the factorization of the divisor count, then assign the largest exponents to the smallest primes.
Divisors That Are Perfect Powers [H]	A divisor of n is a perfect k-th power exactly when every exponent it uses is a multiple of k, so the count is the product of floor(e/k) + 1.
Writing n as a Product of Two Factors [H]	Factorizations n = ab pair each divisor with its cofactor, and coprime factorizations assign each prime power wholly to one side.
Divisors That Are Multiples of m [H]	The divisors of n that are multiples of m are exactly m times the divisors of n/m, so there are tau(n/m) of them.
Divisors Common to Two Numbers [H]	A number divides both a and b exactly when it divides gcd(a, b), so the common divisors number tau(gcd(a, b)).
Recovering a Number from gcd and lcm [H]	For any two positive integers, gcd times lcm equals the product, and each prime is split so that one number carries the smaller exponent and the other the larger.
Counting Pairs with a Given lcm [H]	For lcm(a, b) = p^e the pair of exponents must have maximum e, which happens in 2e + 1 ways, and the primes multiply independently.
gcd of Two Linear Forms [H]	Any common divisor of two linear expressions in n divides every integer combination of them, and eliminating n exposes the largest value it can reach.
Coincident Cycles [H]	Events repeating every a and every b units coincide exactly at multiples of lcm(a, b), and three cycles need inclusion and exclusion to separate double from triple coincidences.
Reducible Fractions [H]	A fraction (n+a)/(n+b) reduces exactly when some prime factor of b - a divides n + a, so only the primes of the difference matter.
The Last Two Digits of a Power [H]	The last two digits of a power are the value modulo 100, and the powers of a fixed base repeat with a period that divides 20 for bases coprime to 10.
A Far-Out Digit of a Repeating Decimal [H]	The decimal expansion of a fraction repeats with period equal to the multiplicative order of 10 modulo the reduced denominator, so the position of a digit is read modulo that period.
A Long Power Sum Modulo m [H]	The residues of i^k modulo m repeat with period m, so a long power sum reduces to whole blocks plus a short leftover.
A Factorial Sum Modulo m [H]	Every factorial from m! onward is divisible by m, so a factorial sum modulo m collapses to its first m terms.
Two Congruences at Once [H]	With coprime moduli there is exactly one residue class modulo the product satisfying both congruences, found by walking one congruence until the other holds.
Counting Solutions of a Congruence System [H]	Two coprime congruences combine into a single congruence modulo the product, so the solutions in a range form an arithmetic sequence with that common difference.
Three Simultaneous Remainders [H]	Three pairwise coprime congruences pin down one residue class modulo the product of the moduli, best found by merging two at a time.
Solving for the Base [H]	A base-b numeral is a polynomial in b with the digits as coefficients, so an equation in base ten becomes a polynomial equation in the base.
Digit Length in Two Bases [H]	Having exactly k digits in base b means lying in the interval from b^(k-1) to b^k - 1, so a two-base condition is an intersection of intervals.
Palindromes in Another Base [H]	Palindromes of a fixed digit length in base b are counted by choosing the free half of the digits, with the leading digit nonzero.
Counting Numbers by Digit Sum [H]	Counting numbers with a prescribed digit sum is a stars-and-bars count on the digits, corrected by inclusion and exclusion for the cap of nine and the nonzero leading digit.
Reversals and Multiples of Nine [H]	A number minus its reversal is a fixed multiple of nine times the difference of the outer digits, and the sum of a number and its reversal depends only on the digit sums by place.
Finding the Missing Digit [H]	Replacing an unknown digit changes the number by a fixed power of ten times the digit, so divisibility becomes a single linear congruence in that digit.
Counting Numbers by Digit Product [H]	A prescribed digit product forces a factorization of that product into digits, and each factorization contributes its number of distinct orderings.
The Multiplication Principle Under Restriction [H]	When digits or slots are chosen with restrictions, fill the most constrained slot first and multiply the remaining independent choices.
Casework and the Addition Principle [H]	Splitting a count by the value of one variable turns a hard problem into a sum of easy independent counts.
Arrangements with a Block Together [H]	People who must stay together are glued into a single block, and the block's internal orders multiply the arrangements of the reduced list.
Arrangements That Keep People Apart [H]	Separation constraints are handled either by subtracting the glued arrangements or by seating the unrestricted people first and dropping the others into the gaps.
Fixing a Relative Order [H]	Requiring some items to appear in one prescribed relative order divides the total arrangements by the number of orders those items could take.
Committees with Required and Barred Members [H]	Forcing a person onto a committee fixes one seat and reduces both the pool and the number of seats, while barring a person only shrinks the pool.
At Least k of a Type [H]	An at-least condition is a sum over the exact counts, because reserving seats for a required type and filling the rest freely double counts.
Complementary Counting on Sequences [H]	Counting sequences that contain a pattern is easiest as the total minus the sequences that avoid it, and the avoiding sequences satisfy a short recursion.
Inclusion and Exclusion on Two Conditions [H]	Counting multiples of a or b requires subtracting the multiples of their least common multiple, and exactly one of the two requires subtracting it twice.
Inclusion and Exclusion on Three Conditions [H]	With three conditions the alternating sum adds singles, subtracts pairs and adds the triple, and exactly-one counts subtract the pairs twice and restore the triple three times.
Venn Counts from Overlap Totals [H]	Survey totals count overlapping regions repeatedly, so peel the triple overlap out of each pair count before reading off the exactly-one and exactly-two regions.
Arrangements of Repeated Letters [H]	Arrangements of a multiset equal the factorial of the length divided by the factorial of each repeated letter's multiplicity.
Repeated Letters Together or Apart [H]	Identical letters forced together become one block with no internal orderings, while forcing them apart means arranging the others first and choosing gaps.
Distributions with a Minimum [H]	Giving every variable its required minimum first reduces the problem to a nonnegative stars-and-bars count on the remaining total.
Distributions with a Cap [H]	An upper bound on each variable is enforced by subtracting the solutions where one variable exceeds the cap and adding back the double violations.
Solutions of an Inequality [H]	An inequality becomes an equation by adding a slack variable, so counting solutions below a bound uses one extra box in stars and bars.
Lattice Paths Through a Checkpoint [H]	The number of monotone lattice paths to a point is a binomial coefficient, and paths through a required point multiply the counts of the two legs.
Lattice Paths Around a Blockage [H]	Paths avoiding a closed intersection are the total minus the paths through it, and two closures need inclusion and exclusion for the paths using both.
Paths That Stay Below the Diagonal [H]	Monotone paths from the origin to (n, n) that never rise above the line y = x are counted by the Catalan number C(2n, n)/(n + 1).
Forcing a Forbidden Difference [H]	Grouping the integers by residue modulo d turns a forbidden difference of d into a no-two-adjacent condition on separate chains, each of which admits half its length rounded up.
Worst-Case Draws [H]	The guarantee threshold is one more than the largest draw that still avoids the goal, and a color with fewer socks than the target caps its own contribution.
Forcing a Pair with a Given Sum [H]	Pairing each integer with its complement to the target sum creates disjoint pigeonholes, and a set avoiding the sum can use at most one integer from each pair.

AMC 10/12 - Geometry · 47 topics

Shoelace on a Lattice Polygon [H]	The shoelace sum gives the area of any simple polygon from its vertices.
Pick's Theorem [H]	A lattice polygon has area I + B/2 - 1, with I interior and B boundary points.
Heron Run Backwards into an Altitude [H]	Heron gives the area, and the area divided by half a side gives the altitude to it.
Two Sides and the Included Angle [H]	The area is one half the product of two sides times the sine of the angle between them.
Areas Around the Diagonal Point [H]	Around the meet of the diagonals, opposite triangle areas have equal products.
The Triangle Cut from a Square [H]	Subtract the three corner right triangles from the square to get the inner triangle.
A Parallel Cut Inside a Triangle [H]	A line parallel to a side cuts off a similar triangle whose areas scale as the square of the ratio.
Geometric Means at the Right Angle [H]	The altitude to the hypotenuse creates two triangles similar to the original and to each other.
Crossing Cables Between Two Poles [H]	Two crossed wires between poles of heights a and b meet at height ab/(a+b), whatever the gap.
The Two Squares Inside a Right Triangle [H]	A square at the right angle has side ab/(a+b); a square on the hypotenuse has side abc/(ab+c²).
Trapezoid Diagonals and Similar Pieces [H]	The diagonals of a trapezoid cut areas p², pq, pq, q² when the parallel sides are p and q.
Interior and Exterior Angles from Arcs [H]	An inscribed angle is half its arc, two chords give half the sum, two secants give half the difference.
The Tangent-Chord Angle [H]	The angle between a tangent and a chord equals half the arc the chord cuts off.
Star Polygon Point Angles [H]	The point angles of an {n/k} star sum to 180(n - 2k) degrees.
Inscribed Angles in a Regular Polygon [H]	Every arc of a regular n-gon is a multiple of 360/n, so every angle in it is a multiple of 180/n.
The Isosceles Ladder [H]	Each new equal-length segment repeats the base angle, so the chase steps down by a fixed amount.
Two Secants and a Tangent from One Point [H]	Every line through P meets a circle in two points whose signed distances have the same product.
A Chord Split by an Interior Point [H]	For a point inside a circle, every chord through it splits into pieces with product R² - d².
Brahmagupta's Area Formula [H]	A cyclic quadrilateral with sides a, b, c, d has area the square root of (s-a)(s-b)(s-c)(s-d).
Opposite Angles and the Diagonal of a Cyclic Quad [H]	Opposite angles of a cyclic quadrilateral are supplementary, so their cosines are negatives.
Ptolemy on a Regular Polygon [H]	Ptolemy applied to a regular polygon plus a point on its circumcircle gives a linear relation.
The Inradius as Area Over Semiperimeter [H]	Joining the incenter to the vertices splits a triangle into three pieces of total area rs.
The Circumradius from abc Over 4K [H]	A triangle with sides a, b, c and area K has circumradius abc divided by 4K.
Incircle Tangent Lengths [H]	The two tangent segments from a vertex to the incircle are equal and have length s minus the opposite side.
The Right Triangle Inradius Identities [H]	A right triangle has inradius (a+b-c)/2, and its semiperimeter equals r plus the hypotenuse.
Euler's Distance Between the Centers [H]	The circumcenter and incenter satisfy OI² = R(R - 2r), so R is at least 2r.
The British Flag Theorem [H]	For any point and any rectangle, the squared distances to opposite corners have equal sums.
Chord Length from the Distance to the Center [H]	A radius perpendicular to a chord bisects it, so half the chord, the distance, and R form a right triangle.
Tangency Conditions in Coordinates [H]	The squared tangent length from a point equals the power of that point with respect to the circle.
Centroids and Midpoint Triangles [H]	The centroid is the average of the three vertices, and the midpoint triangle has one quarter the area.
A Line That Splits a Region by Area [H]	A line bisects a rectangle exactly when it passes through the rectangle's center.
Shared Altitudes and Base Ratios [H]	Two triangles with the same altitude have areas in the ratio of their bases.
Nested Cevian Area Ratios [H]	Cutting two sides of a triangle at given ratios multiplies the two fractions in the area.
Areas Around Two Intersecting Cevians [H]	Two cevians cut a triangle into three triangles and a quadrilateral whose areas follow from one ratio.
The Six Triangles Made by the Medians [H]	The medians cut a triangle into six triangles of equal area, and the median triangle has three quarters of it.
A Point Inside a Parallelogram [H]	For P inside a parallelogram, the two triangles on opposite sides together make half its area.
Space Diagonals and Face Diagonals [H]	A box with edges l, w, h has space diagonal the square root of l² + w² + h².
The Shortest Path Across a Box Surface [H]	Unfold the box flat: the shortest surface path is a straight line in the unfolding.
Heights Inside a Pyramid [H]	The apothem gives the slant height and the half-diagonal gives the lateral edge.
Spheres, Cubes and Tight Fits [H]	An inscribed sphere matches the edge; a circumscribed sphere matches the space diagonal.
Cross Sections of a Cube [H]	Slicing a cube gives triangles, rectangles or a regular hexagon depending on the plane.
Similar Solids and the Cut Cone [H]	Similar solids scale lengths by k, areas by k squared and volumes by k cubed.
Cutting a Corner off a Solid [H]	A corner tetrahedron with mutually perpendicular legs has volume pqr/6 and obeys de Gua's theorem.
Slanted Slices of a Prism [H]	A prism cut by a slanted plane has volume the base area times the average of the edge heights.
Reflection and the Shortest Path [H]	Reflecting one endpoint across a line turns a bent shortest path into a straight segment.
Rotations About a Lattice Point [H]	Rotating 90 degrees about a center swaps the offsets and flips one sign.
Two Mirrors Make a Rotation [H]	Reflecting across two lines through a point is a rotation through twice the angle between them.

AMC 10/12 - Probability and Advanced Topics · 48 topics

Choosing the Sample Space [H]	Fix the sample space first, then count only the outcomes it contains.
Relative Order by Symmetry [H]	All k! relative orders of k chosen items are equally likely.
All Choices Different [H]	Independent choices give m^n outcomes; distinct ones give m(m-1)···.
Keeping a Block Together [H]	Glue a required block into one object; for separation, place gaps first.
Drawing Without Replacement [H]	Choose the reds and the blues separately, then divide by all C(n,k) hands.
The Largest Value Shown [H]	Count max at most m, then subtract max at most m-1.
Bayes on Two Boxes [H]	Posterior odds equal prior odds times the likelihood of the evidence.
What the Information Really Says [H]	Conditioning on an existence statement is not the same as naming a child.
Which Events Are Independent [H]	Events are independent exactly when P(A and B) equals P(A)·P(B).
Transfer, Then Draw [H]	Split on the hidden first stage and weight each branch by its probability.
The Alternating Game [H]	Sum the geometric series over full rounds, or use one self-referential equation.
Indicators on a Permutation [H]	Add the probabilities of the individual events; linearity ignores dependence.
Expected Adjacent Pairs [H]	Each adjacent slot is a uniformly random unordered pair of the people.
Collecting Every Type [H]	Sum the expected waits for each new type: n/n + n/(n-1) + ... + n/1.
Expected Wait for a Pattern [H]	Set up one equation per state and solve; a single success waits 1/p flips.
Expectation of a Maximum [H]	For a non-negative integer variable, E[X] equals the sum of P(X >= k).
Two Random Points on a Segment [H]	Two uniform points are one uniform point in a square; probability is area.
The Meeting Problem [H]	The two arrival times form a uniform point in a square; meeting is a band.
Breaking a Stick [H]	Cut points (x,y) fill a unit square; each piece condition is a triangle.
Uniform Points on a Disk [H]	On a disk, probability scales with area, so it scales with radius squared.
Random Coefficients, Real Roots [H]	Real roots means the discriminant is non-negative; count lattice pairs.
A Bug on the Vertices [H]	Collapse the vertices into two states and iterate one recursion.
Biased Walk to Absorption [H]	For a biased walk, the hitting probability is a geometric ratio, not linear.
Forbidden Runs by Recursion [H]	Count admissible strings by their last letter; Fibonacci counts no-two-heads.
Expected Steps to Absorption [H]	Expected absorption time for a fair walk on 0..N from k is k(N-k).
Returning on a Cycle [H]	Track the full distribution over the cycle one step at a time.
The General Term [H]	Write the general term, set its exponent to the target, and solve for k.
Weighted Binomial Sums [H]	Differentiate or integrate the binomial theorem, or use k·C(n,k) = n·C(n-1,k-1).
Alternating Binomial Sums [H]	A partial alternating row sum collapses to a single entry one row up.
Vandermonde's Identity [H]	Splitting a committee across two groups gives C(m,k)C(n,r-k) summing to C(m+n,r).
Equal Heads and Tails [H]	Split the outcomes into more heads, more tails, and the tie, then use symmetry.
Roots of Unity Filter [H]	Averaging a generating function over the d-th roots of unity keeps one residue class.
Sums Over Roots of Unity [H]	The n-th roots of unity are the roots of z^n - 1, so their symmetric sums are known.
Distances in a Regular Polygon [H]	Place the vertices at R times the n-th roots of unity and read the algebra off.
Pairing Terms by Symmetry [H]	When f(x) + f(1-x) is constant, pair the terms from the two ends.
Telescoping Products [H]	Factor each term so consecutive numerators and denominators cancel.
Partial Fractions That Telescope [H]	Split each term into differences so all but the boundary terms cancel.
Telescoping with Radicals [H]	Rationalize each term so it becomes a difference of consecutive roots.
Telescoping Logarithms [H]	A sum of logs is the log of a product, and the product telescopes.
The Blackboard Invariant [H]	Find a quantity every move preserves; it decides the final state.
Reachability by Invariant [H]	Differences of the counts modulo 3 are unchanged by every meeting.
Splitting Piles and Monovariants [H]	A quantity that changes by a fixed amount each move settles the outcome.
Maximum Product, Fixed Sum [H]	Split a fixed sum into 3s, replacing one 3 by two 2s when the remainder is 1.
The Largest Admissible Subset [H]	Split the set into chains, then take alternate elements of each chain.
Clock Hands [H]	The minute hand gains 5.5 degrees per minute on the hour hand.
Calendar Arithmetic [H]	Days of the week repeat with period 7, so count elapsed days modulo 7.
Combined Work Rates [H]	Rates add; times do not, so convert every time to a rate first.
Average Speed [H]	Average speed is total distance over total time, a harmonic mean for equal distances.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 6 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators)	Same-denominator addition.
Adding Fractions (Unlike Denominators)	Rewrite over a common denominator first.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Multi-Step Equations	Equations needing distribution or variables on both sides.
Linear Inequalities	Solving with <, >, ≤, ≥.

Linear Functions · 4 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.
Systems of Equations (Substitution)	Two equations, two unknowns.

Quadratics & Polynomials · 9 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k	Taking square roots of both sides - remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.

Radicals & Exponentials · 6 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Rational Exponents	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Geometry · 8 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Parallel Lines & Transversals	Angle pairs formed when a transversal crosses parallel lines.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Similar Triangles	Same shape, different size: corresponding sides are proportional.
Perimeter & Area	Measuring around and inside basic shapes.
Circles: Area & Circumference	C = 2πr and A = πr².
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

Functions & Algebra II · 16 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Function Composition	Feeding one function's output into another: f(g(x)).
Piecewise Functions	Functions defined by different rules on different intervals.
Nonlinear Systems	Where a line meets a parabola: set the two formulas equal.
Complex Numbers	The imaginary unit i = √(−1) and numbers of the form a + bi.
Operations with Complex Numbers	Multiplying complex numbers with FOIL and i² = −1.
Polynomial Division	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials	Far from the origin, only the leading term matters.
Simplifying Rational Expressions	Factor top and bottom, then cancel the common factor.
Logarithms	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms	Logs turn products into sums, quotients into differences, powers into multiples.
Exponential & Log Equations	Matching bases and rewriting between exponential and log form.
Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

Trigonometry · 2 topics

Degrees & Radians	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle	Exact sine, cosine, and tangent values at the special angles.

Precalculus · 1 topics

Asymptotes of Rational Functions

The vertical asymptotes and the end behavior (horizontal or slant asymptote) of a rational function.

Limits & Continuity · 4 topics

Limits: Graphical & Numerical	What value a function approaches - which need not be the value it takes.
One-Sided Limits	Approaching from the left or right - and when the two disagree.
Infinite Limits & Vertical Asymptotes	A limit of +∞ or −∞ at a vertical asymptote; the sign of the shrinking denominator decides which.
Limits at Infinity	The value a function approaches as x goes to +∞ or −∞; for a rational function, compare the degrees of the top and bottom.

Infinite Series · 2 topics

Convergence of Sequences (BC)	A sequence converges if aₙ approaches a limit.
Geometric Series (BC)	Σarⁿ = a/(1−r) when |r| < 1.

Math Olympiad: MOEMS & AMC 8 · 7 topics

Modular Arithmetic	Working with remainders directly.
Counting Divisors	Divisor count from the prime factorization.
Permutations	Ordered arrangements.
Combinations	Unordered selections: n choose k.
Counting & Probability	Favorable outcomes over total outcomes.
Vieta's Formulas	Relating roots to coefficients without solving.
Telescoping Sums	A sum that collapses to its endpoints.

Mathematical Methods for the Physical Sciences I · 2 topics

Complex Arithmetic	Multiply out, use i² = −1.
Polar Form & Euler's Formula	z = re^{iθ}: modulus and angle instead of real and imaginary parts.

Math Kangaroo: Junior & Student · 5 topics

Quadratic Tricks	Complete the square; work with sums and products of roots.
Angle Chasing	Push known angles through triangles and polygons.
Grid Paths & Selections	Shortest paths are just choices in disguise.
Divisibility Duels	Factor counts, trailing zeros, digit sums.
Logarithm Puzzles	Chain rule for logs: bases cancel.

Precalculus: Exponential & Logarithmic Functions · 9 topics

Evaluating Exponential Functions	Plug integer inputs into f(x) = a·bˣ - including 0 and negatives.
Graphs: y-Intercept & Asymptote	y-intercept a, growth when b > 1, decay when 0 < b < 1, floor at y = 0.
Evaluating Logarithms	log_b(x) asks: to what power must b be raised to get x?
Log Laws in Computation	Logs turn products into sums, quotients into differences, powers into multiples.
Exponential Equations: Same Base	Match the bases, then set the exponents equal.
Exponential Equations with Logs	Take a logarithm of both sides to bring the exponent down.
Exponential Growth & Decay Models	Model y = a·bᵗ: multiply the start by the factor once per time step.
Doubling Time	Doubling every T means y = a·2^(t/T) - count the doublings first.
Solving Logarithmic Equations	Rewrite log_b(expr) = k as expr = b^k, then solve.

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