Course contents document · Competition · generated 2026-09-01

Purple Comet! Math Meet (Grades 6-12)

260 core topics · approximately 87 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

Purple Comet - Middle School · 67 topics

Linear Word Equations [H]	Translate a one-line word condition into ax + b = c and solve.
Sum-and-Difference Systems [H]	From a sum and a difference recover the two numbers, then combine them.
Consecutive Integers [H]	The sum of consecutive integers equals count times the middle value.
Age Relationships [H]	Set up ages now and shift both by the same number of years.
Fraction of a Number [H]	If a fraction of N is known, divide by the fraction to recover N.
Percent Word Problems [H]	Percent means per hundred; multiply the whole by p/100.
Splitting by a Ratio [H]	A ratio a:b divides a total into a+b equal parts.
Hitting a Target Average [H]	The needed value equals target·count minus the running total.
Arithmetic Sequence Term [H]	The nth term is a1 + (n−1)d.
Arithmetic Series Sum [H]	A finite arithmetic series equals count times the average of its ends.
Evaluating Expressions [H]	Substitute the value and follow order of operations.
Mixture Percentages [H]	Total pure amount divided by total volume gives the blend's percentage.
Distance = Rate × Time [H]	Distance is the product of speed and elapsed time.
Combined Work Rates [H]	Rates add: together they finish in the reciprocal of the summed rates.
Digit Sums [H]	Add the individual digits of the number.
Reversed Digits [H]	A two-digit number plus its reversal equals 11 times the digit sum.
Counting Divisors [H]	Add one to each prime exponent and multiply.
GCD and LCM [H]	For any two numbers, gcd·lcm equals their product.
Remainders [H]	The remainder is what is left after taking out all full multiples.
Units Digit of a Power [H]	The last digit of a power repeats in a short cycle.
Base Conversion [H]	Each digit carries a power of the base, growing right to left.
Counting Multiples [H]	The count of multiples of d up to N is the floor of N/d.
Sum of Prime Factors [H]	Factor into primes, then add the distinct primes once each.
Triangular Numbers [H]	The nth triangular number is n(n+1)/2.
Simultaneous Events (LCM) [H]	Two periodic events realign after the least common multiple of their periods.
Choosing a Committee [H]	Order-free selections of k from n number C(n,k).
Ordered Awards [H]	Ordered selections of k from n number n·(n−1)···(n−k+1).
Basic Probability (m+n) [H]	Probability is favorable over total; report the reduced m+n.
Complementary Counting [H]	Count the complement: total minus the unwanted cases.
Arrangements with Repeats [H]	Divide the total factorial by the factorial of each repeated letter's count.
Lattice Paths [H]	Monotone corner-to-corner paths number C(a+b, a).
Distributing Identical Items [H]	Nonnegative solutions of x1+…+xk = n number C(n+k−1, k−1).
Committee with a Fixed Member [H]	Seat the required member first, then choose the rest.
Counting Dice Outcomes [H]	Independent choices multiply: f^n outcomes for n dice of f faces.
Counting Subsets [H]	A set of n elements has 2ⁿ subsets.
Rectangle Area and Perimeter [H]	Area is length times width; perimeter is twice their sum.
Triangle Area [H]	A triangle's area is half base times height.
Pythagorean Theorem [H]	In a right triangle the hypotenuse squared equals the sum of the legs squared.
Triangle Angle Sum [H]	The interior angles of a triangle sum to 180°.
Polygon Angle Sum [H]	An n-gon's interior angles total (n−2)·180°.
Circle Area Coefficient [H]	The area of a circle is πr²; the coefficient of π is r².
Volume of a Box [H]	Volume of a rectangular box is the product of its three edges.
Similar Triangles [H]	Corresponding sides of similar triangles share one scale factor.
Squared Distance [H]	The squared distance is Δx² + Δy² - no square root needed.
Shaded Region by Subtraction [H]	Remove the cut-out area from the whole figure's area.
Surface Area of a Cube [H]	A cube has six square faces, so surface area is 6s².
Clock Hand Angles [H]	Each hour mark is 30° apart; take the smaller of the two arcs.
Parallel Lines and a Transversal [H]	Corresponding angles are equal; co-interior angles are supplementary.
Scaling Area [H]	Scaling lengths by k scales area by k².
Perimeter of an L-Shape [H]	A corner notch does not change a rectangle's perimeter.
Average Speed [H]	Average speed is total distance over total time, not the mean of speeds.
Coin Totals [H]	Multiply each coin count by its value and add.
Polygon Diagonals [H]	A convex n-gon has n(n−3)/2 diagonals.
Handshakes [H]	Pairs among n people number C(n,2) = n(n−1)/2.
Sums of Evens and Odds [H]	The first n odds sum to n²; the first n evens sum to n(n+1).
Leftover Remainder Problems [H]	Find the least value in a range with a given remainder modulo d.
Reduce then Sum (m+n) [H]	Cancel the common factor, then add numerator and denominator.
Multiples in a Range [H]	Count multiples up to the top and subtract those below the bottom.
Square of Equal Area [H]	A square equal in area to a rectangle has side √(area).
Units Digit of a Sum (MC) [H]	Only the units digits of the addends determine the units digit of the sum.
Naming Polygons (MC) [H]	Polygon names encode their number of sides.
Packing Leftovers [H]	The leftover equals the total modulo the carton size.
Two Draws Without Replacement (m+n) [H]	Divide favorable pairs by total pairs; reduce and report m+n.
Missing Digit for Divisibility by 9 [H]	A number is divisible by 9 exactly when its digit sum is.
Fence-Post Counting [H]	A line of gaps has one more post than the number of gaps.
Pairs with an Even Sum [H]	A pair sums to an even number exactly when both are even or both odd.
Painted Cube (Two Faces) [H]	Two-face cubes lie along edges, away from the corners.

Purple Comet - HS Algebra & Number Theory · 54 topics

Square of the Root Difference [H]	The squared difference of roots equals s² − 4p from Vieta's formulas.
Sum of Reciprocal Squares of Roots (m+n) [H]	1/r² + 1/s² = (r²+s²)/(rs)², all obtainable from the coefficients.
Sum of Squares of Cubic Roots [H]	a²+b²+c² = (a+b+c)² − 2(ab+bc+ca).
Fourth Powers of the Roots [H]	Use power sums: p₂ = S²−2P, then p₄ = (p₂)² − 2P².
Finding the Integer Root [H]	A rational root must divide the constant term; test divisors.
Sum of Coefficients [H]	Substituting x = 1 gives the sum of all coefficients.
Quadratic from Three Values [H]	A quadratic has constant second differences; extend the table.
Sum of Cubes from Sum and Product [H]	x³ + y³ = (x+y)³ − 3xy(x+y).
Product from Sum and Sum-of-Squares [H]	xy = ((x+y)² − (x²+y²)) / 2.
Finite Geometric Series (m+n) [H]	A finite geometric series sums to a(1−rⁿ⁺¹)/(1−r).
Infinite Geometric Series (m+n) [H]	An infinite geometric series with \|r\|<1 sums to a/(1−r).
Telescoping 1/(k(k+1)) (m+n) [H]	1/(k(k+1)) = 1/k − 1/(k+1), so the sum telescopes to 1 − 1/(n+1).
Telescoping 1/(k(k+2)) (m+n) [H]	1/(k(k+2)) = ½(1/k − 1/(k+2)); alternate terms cancel.
Sum of the First n Squares [H]	1² + … + n² = n(n+1)(2n+1)/6.
Sum of the First n Cubes [H]	1³ + … + n³ = (n(n+1)/2)², the square of the nth triangular number.
Solving a Logarithmic Equation [H]	log_b(x) = e means x = bᵉ.
Telescoping Chain of Logs [H]	By change of base, consecutive log factors telescope to log₂ N.
Combining Logarithms [H]	log_b(x) + log_b(y) = log_b(xy).
Exponential Equations [H]	Factor out the smallest power to collapse the sum.
Matching Prime Exponents [H]	Factor into primes and read off the exponents.
Reciprocal Functional Equation (m+n) [H]	Substitute x → 1/x to get a second equation, then solve the linear system.
Additive Functional Equation [H]	Cauchy's equation with f(1) known forces f(n) = n·f(1) on integers.
Fibonacci-Type Recurrence [H]	Each term is the sum of the two before it; iterate forward.
Infinite Nested Radical [H]	Set x equal to the radical; then x² = a + x.
Continued Fractions (m+n) [H]	Evaluate a continued fraction from the innermost layer outward.
Minimizing a Sum of Distances [H]	A sum of absolute deviations is smallest at the median.
Minimum by Completing the Square [H]	x² − 2hx + c = (x − h)² + (c − h²); the minimum is c − h².
Sum of Floor Quotients [H]	Add ⌊k/m⌋ over the range; the values step up by 1 each m terms.
Binomial Coefficients [H]	C(n,k) = n!/(k!(n−k)!).
Counting Divisors [H]	For n = p^a q^b, the divisor count is (a+1)(b+1).
Sum of Divisors [H]	σ is multiplicative; for pᵃ the divisor sum is 1+p+…+pᵃ.
Trailing Zeros of a Factorial [H]	Trailing zeros count the factors of 5: ⌊n/5⌋+⌊n/25⌋+….
Legendre's Formula [H]	The exponent of prime p in n! is ⌊n/p⌋+⌊n/p²⌋+….
Modular Exponentiation [H]	Reduce the base mod m and use the repeating cycle of powers.
Last Two Digits [H]	The last two digits are the value modulo 100.
Chinese Remainder Theorem [H]	Coprime moduli have a unique combined residue modulo their product.
Recovering a Number from gcd and lcm [H]	For any two numbers, product = gcd × lcm.
Counting Diophantine Solutions [H]	Step x through valid values and test whether y is a nonnegative integer.
Fermat's Little Theorem [H]	For prime p and a not divisible by p, aᵖ⁻¹ ≡ 1 (mod p).
Euler's Totient [H]	φ(n) = n·∏(1 − 1/p) over distinct primes p dividing n.
Multiplicative Order [H]	The order is the least k with aᵏ ≡ 1; it divides φ(n).
Frobenius (Chicken McNugget) [H]	For coprime a, b the largest non-representable value is ab − a − b.
Repeating Decimals (m+n) [H]	A length-L repeating block equals block/(10ᴸ − 1).
Digit Sum in Another Base [H]	Repeatedly divide by the base, adding the remainders.
Smallest Multiplier for a Square [H]	Multiply by the product of primes appearing to an odd power.
Digit Problem from Sum and Difference [H]	Reversal subtraction gives 9(t−u); combine with the digit sum.
Counting Coprime Integers in a Range [H]	Use inclusion-exclusion on the prime factors of m, or count directly.
Divisible by a or b (Inclusion-Exclusion) [H]	Count multiples of a plus multiples of b minus multiples of lcm(a,b).
Counting Perfect Squares [H]	Perfect squares up to N number ⌊√N⌋.
Sum of Multiples in a Range [H]	The multiples of d form an arithmetic series; sum with count × average.
Identifying a Prime (MC) [H]	A prime has no divisors other than 1 and itself; test small factors.
Divisor Count (MC) [H]	Factor into primes and multiply one-more-than each exponent.
Integer Geometric Mean [H]	The geometric mean of a and b is √(ab).
Difference of Squares [H]	a² − b² = (a − b)(a + b).

Purple Comet - HS Geometry & Combinatorics · 51 topics

Heron's Formula [H]	Area = √(s(s−a)(s−b)(s−c)) with s the semiperimeter.
Shoelace (Twice the Area) [H]	The shoelace sum gives twice the area of a coordinate polygon.
Pick's Theorem [H]	Area = I + B/2 − 1 for a lattice polygon.
Trapezoid Area [H]	Area = ½(b₁ + b₂)·h, the average of the parallel sides times the height.
Rhombus Area from Diagonals [H]	A rhombus's area is half the product of its diagonals.
Power of a Point (Two Secants) [H]	For two secants from P, PA·PB = PC·PD.
Power of a Point (Tangent-Secant) [H]	A tangent and a secant from P satisfy t² = PA·PB.
Angle Bisector Theorem [H]	The bisector from A divides BC in the ratio AB : AC.
Median Length Formula [H]	The median to side c satisfies 4m² = 2a² + 2b² − c².
Circumdiameter of a Right Triangle [H]	A right triangle's hypotenuse is a diameter of its circumcircle.
Inradius of a Right Triangle [H]	For a right triangle, r = (a + b − c)/2.
Law of Cosines (60° / 120°) [H]	c² = a² + b² − 2ab·cos C; cos 60° = ½ and cos 120° = −½.
30-60-90 Triangle [H]	Sides are in ratio 1 : √3 : 2, so the longer leg is √3 times the shorter.
45-45-90 Triangle [H]	Legs are equal and the hypotenuse is leg·√2, so hypotenuse² = 2·leg².
Square Inscribed in a Circle [H]	A square inscribed in a radius-r circle has diagonal 2r and area 2r².
Area of an Annulus [H]	The ring area is π(R² − r²), the difference of the two disk areas.
Area of a Sector [H]	A sector's area is (θ/360)·πr².
Volume of a Cylinder [H]	Volume = πr²h.
Volume of a Cone [H]	Volume = (1/3)πr²h, one third of the enclosing cylinder.
Volume of a Sphere [H]	Volume = (4/3)πr³.
Surface Area of a Sphere [H]	Surface area = 4πr².
Space Diagonal of a Box [H]	The space diagonal satisfies d² = a² + b² + c².
Point-to-Line Distance [H]	Distance = \|ax₀+by₀−c\| / √(a²+b²).
Reflecting a Point [H]	Reflecting over y = x swaps coordinates; over x = c sends x to 2c − x.
Triangle Inequality Counting [H]	The third side lies strictly between \|a−b\| and a+b.
Tangent Circles [H]	Centers are r₁+r₂ apart (external) or \|r₁−r₂\| apart (internal).
Stars and Bars (No Empty Box) [H]	Positive solutions of x₁+…+x_k = n number C(n−1, k−1).
Inclusion-Exclusion (Three Sets) [H]	\|A∪B∪C\| = Σ\|A\| − Σ\|A∩B\| + \|A∩B∩C\|.
Derangements [H]	Derangements satisfy D_n = (n−1)(D_{n−1} + D_{n−2}).
Permutations of a Multiset [H]	Divide n! by the factorial of each repeated symbol's multiplicity.
Circular Arrangements [H]	n distinct objects around a circle arrange in (n−1)! ways.
Lattice Paths Through a Point [H]	Multiply paths to the waypoint by paths from the waypoint to the end.
Hockey Stick Identity [H]	Σ_{k=r}^{n} C(k,r) = C(n+1, r+1).
Catalan Numbers [H]	The nth Catalan number is C(2n,n)/(n+1).
Counting Surjections [H]	Surjections = Σ (−1)^i C(k,i)(k−i)ⁿ by inclusion-exclusion.
Proper Colorings of a Path [H]	Color the first freely, then each next differently: k(k−1)^{n−1}.
Strings With No Two Adjacent 1s [H]	The count follows the Fibonacci recurrence f(n) = f(n−1) + f(n−2).
Counting Rectangles in a Grid [H]	Choose 2 of the m+1 horizontal and 2 of the n+1 vertical lines.
Subsets With an Even Sum [H]	Exactly half of all subsets have an even sum: 2^{n−1}.
Dice Sum Probability (m+n) [H]	Count ordered pairs giving the sum, over 36; reduce and report m+n.
Expected Value of a Die (m+n) [H]	The mean of 1 through s is (s+1)/2.
Conditional Probability (m+n) [H]	Restrict to outcomes meeting the condition, then count favorable ones.
At Least One Success (m+n) [H]	P(at least one) = 1 − P(none).
One of Each Color (m+n) [H]	Favorable = (reds)(greens); total = C(a+b, 2).
Faces of a Polyhedron (MC) [H]	Recall the face counts of the common solids.
Hypotenuse Squared (MC) [H]	By the Pythagorean theorem the hypotenuse squared is a² + b².
Area of an Equilateral Triangle [H]	Area = (√3/4)s², so the coefficient of √3 is s²/4.
Area of a Regular Hexagon [H]	Area = (3√3/2)s²; the coefficient of √3 is 3s²/2.
Shoelace for a Quadrilateral [H]	The shoelace formula extends to any polygon, term by term around it.
Ptolemy's Theorem [H]	In a cyclic quadrilateral, AC·BD = AB·CD + BC·DA.
Interior Angle of a Regular Polygon [H]	Each interior angle of a regular n-gon is (n−2)·180°/n.

Purple Comet (Middle School) - Team Problems with Integer Answers · 44 topics

Digit Characters in a Run of Numbers [H]	Count digits one width at a time: nine one-digit numbers, ninety two-digit numbers, nine hundred three-digit numbers.
Counting One Digit's Appearances [H]	Count a digit place by place, and keep occurrences of the digit separate from the count of numbers containing it.
The Alternating Test for Eleven [H]	A number is a multiple of eleven exactly when the alternating sum of its digits is.
Palindromes With a Condition [H]	Build palindromes from their free digits instead of testing every number in the range.
The Twos and Fives Inside a Product [H]	A trailing zero is a factor of ten, so count the factors of two and of five separately and take the smaller count.
Sums of Proper Divisors [H]	Pair each divisor with its cofactor to list divisors quickly, then drop the number itself.
Divisors That Are Perfect Squares [H]	A divisor is a perfect square exactly when every exponent in it is even, so halve each exponent and count.
Runs of Consecutive Integers [H]	A run of k consecutive integers starting at a sums to ka plus k(k-1)/2, so k must divide what is left.
One Remainder, Several Divisors [H]	Numbers with the same remainder under several divisors are exactly the multiples of the least common multiple, shifted.
Three Repeating Cycles [H]	Cycles coincide at the multiples of their least common multiple, and pairs coincide at pairwise least common multiples.
Primes Inside a Range [H]	Test a candidate only against primes up to its square root, and sweep a range once rather than testing numbers at random.
Squares Hidden in a Grid [H]	Count squares one size at a time: a k by k square is fixed by the position of its top left corner.
Lattice Paths Around a Closure [H]	Count all the paths and subtract the ones that use the forbidden point, or build the count corner by corner.
Where the Diagonals Cross [H]	Every interior crossing of two diagonals comes from one set of four vertices, so crossings are counted by choosing four.
Three-Digit Numbers Under Two Conditions [H]	Organize the count by one digit at a time, and impose the second condition on the digits rather than on the number.
Committees With Floors on Two Groups [H]	Split the count by how many come from the first group, or subtract the forbidden committees from all of them.
Round-Robin Schedules [H]	Each unordered pair of teams contributes a fixed number of games, so count the pairs and multiply.
Choosing With Forced Gaps [H]	Set aside the chairs that must sit between the chosen ones, then choose freely from what is left.
Distributing With Floors and Caps [H]	Hand out the required minimums first, then use inclusion and exclusion to remove the distributions that break a cap.
Spinner Sums Reported as m + n [H]	List every equally likely pair of outcomes, count the favourable ones, reduce, and add numerator to denominator.
Matching Picks Reported as m + n [H]	Fix the first pick and count the choices left to the others, then split the outcomes into all different, exactly two alike, and all alike.
Adjusting a Jar to Hit a Target Probability [H]	Write the new probability as new reds over new total and cross multiply; the change hits the numerator and the denominator differently.
Expected Values Scaled to an Integer [H]	An expected value is a weighted average of the outcomes; multiplying by the number of outcomes turns it into a plain sum.
Dice Products Reported as m + n [H]	A product carries a prime exactly when some factor does, so count the rolls that avoid the prime and subtract.
The Area of a Border [H]	A uniform border adds twice its width to each dimension, so its area is the outer rectangle minus the inner one.
Perimeter and Area of a Staircase [H]	A staircase has exactly the perimeter of its bounding rectangle, but its area is a triangular number of step blocks.
Two Rectangles That Overlap [H]	The area covered by two shapes is the sum of their areas minus the area counted twice.
Semicircles on a Divided Diameter [H]	Subtract the small semicircles from the large one and let the square of the whole diameter expand.
Cutting Corners Off a Rectangle [H]	An inscribed triangle is the rectangle minus the three right triangles that sit in its corners.
Squares and Circles Nested Inside Each Other [H]	An inscribed circle has the square's side as its diameter, while an inscribed square has the circle's diameter as its diagonal.
Tiling a Rectangular Floor [H]	Count tiles along each dimension, not by dividing areas, because a leftover strip cannot hold a whole tile.
Shaded Fractions Reported as p + q [H]	Write the shaded area over the total area, reduce, and add the numerator to the denominator.
Chaining Two Ratios Together [H]	Rescale the two ratios so the shared quantity has the same size in both, then read the three-part ratio off directly.
Combining Two Groups With Different Ratios [H]	Convert each ratio into actual counts before merging, because ratios themselves never add.
Reading and Changing a Map Scale [H]	A scale is a rate of ground distance per map distance, so convert through the ground distance whenever the map changes.
Sharing in a Three-Part Ratio [H]	Name the part size once; every share, every difference and the equal split are all multiples of it.
Reversing the Digits and Subtracting [H]	Reversing a two-digit number changes it by nine times the difference of its digits, and a three-digit number by ninety-nine times the outer difference.
Digit Sums of Structured Numbers [H]	A power of ten minus a small number is a string of nines with a fixed tail, so its digits can be described without computing it.
Two-Digit Numbers by Digit Product [H]	List the factor pairs of the target that fit in one digit each, and every pair gives at most two numbers.
An Arithmetic Sequence From Two Facts [H]	The gap between two known terms is the common difference times the gap between their positions.
Sums With Alternating Signs [H]	Group the terms in pairs or in blocks so that each group collapses to a constant, then count the groups.
Running a Recursion Forwards and Backwards [H]	Apply the rule step by step to move forwards, and invert the rule step by step to move back to the start.
Symmetric Sums From a Sum and a Product [H]	Every symmetric expression in two numbers rebuilds from their sum and their product, so the numbers themselves are never needed.
Reading a Value Straight Off a System [H]	Subtract two equations to cancel the shared variable, and add all three to get twice the grand total.

Purple Comet (High School) - Algebra & Geometry with Integer Answers · 44 topics

Shifting the Roots of a Monic Polynomial [H]	Translating every root by k rewrites the polynomial as P(x - k), so each new coefficient is an elementary symmetric function of the shifted roots.
Products of Root Differences [H]	A product of the form (m - r)(m - s)(m - t) over all roots of a monic polynomial is just P evaluated at m.
The Linear Remainder Modulo a Quadratic [H]	Dividing by a quadratic leaves a remainder ax + b, and two evaluations of the dividend pin that line down.
Repeated Roots and the Derivative [H]	A repeated root is a common root of the polynomial and its derivative, which turns a hidden double root into an ordinary equation.
Iterating a Linear Function [H]	Composing f(x) = ax + b with itself n times multiplies the slope and accumulates the intercept as a geometric series.
The Substitution x to 1 - x [H]	Replacing x by 1 - x pairs each input with its partner, which either sums a whole list at once or produces a second equation to solve.
Systems in Two Logarithms [H]	Name the two logarithms as new unknowns, solve the ordinary system in them, and only then exponentiate.
Equations Quadratic in the Logarithm [H]	Let L be the logarithm of the unknown; the equation becomes a quadratic in L, and each root of L gives one value of x.
Logarithms Between Powers of One Base [H]	A logarithm whose base and argument are powers of the same number is the exact ratio of the two exponents.
Symmetric Expressions in a^x and a^-x [H]	Setting t = a^x makes a^x + a^-x a sum with product 1, so every higher power follows from the Newton identities.
Exponential Equations by Substitution [H]	Substituting y = 2^x turns an equation in 4^x and 2^x into a quadratic whose roots must themselves be powers of 2.
Terms Shared by Arithmetic Sequences [H]	Numbers in two arithmetic progressions form a third progression whose common difference is the lcm of the two.
Arithmetico-Geometric Sums [H]	Multiply the series by its ratio and subtract; what is left is an ordinary geometric series.
A Recurrence That Repeats [H]	The rule a(n+2) = a(n+1) - a(n) cycles with period six, so a huge index reduces to one of six values.
Alternating Sums in Blocks [H]	Group an alternating sum into blocks of equal length; each block contributes the same simple amount.
Inserting Arithmetic Means [H]	Inserting k means between two numbers splits their gap into k + 1 equal steps.
Recovering Terms from a Partial Sum [H]	A term is the difference of two consecutive partial sums, so a formula for S(n) hands over every term at once.
Exact Sines of Doubled and Summed Angles [H]	A rational sine comes from a Pythagorean triple, so the cosine is rational too and every addition formula stays exact.
Tangent Addition as an Exact Fraction [H]	The tangent addition formula turns two rational tangents into a third rational tangent with no radicals anywhere.
Half-Angle Values from a Rational Cosine [H]	The half-angle identities turn a rational cosine into a rational square, and reading them backwards recovers the whole angle.
Counting Solutions of a Multiple-Angle Equation [H]	Solve for the multiple angle first, list every value it may take in the stretched interval, and divide back down.
Area from Two Sides and a Special Angle [H]	Half the product of two sides times the sine of the angle between them gives the area, and the special angles keep that sine exact.
Chords Cut from a Circle [H]	Drop the perpendicular from the center to the line: the half chord, that distance and the radius form a right triangle.
Vertices from Centroids and Midpoints [H]	The centroid averages the three vertices and cuts each median in the ratio 2 to 1, so any one missing point can be solved for.
Shortest Paths by Reflection [H]	Reflect one endpoint across the mirror line; the bent path straightens into a single segment whose length is the minimum.
Lattice Points on Segments [H]	A segment between two lattice points carries gcd of the coordinate differences equal steps, so counting the points is a gcd computation.
Inscribed Angles and Their Arcs [H]	An inscribed angle is half its intercepted arc, so angles of an inscribed triangle and the three arcs determine one another.
The Angle Between Two Chords [H]	An angle formed inside a circle by two chords is half the SUM of the two arcs it and its vertical angle cut off.
Angles from a Point Outside a Circle [H]	An angle with its vertex outside a circle is half the DIFFERENCE of the far and near arcs it intercepts.
Common Tangents of Two Circles [H]	Sliding one radius along the other turns a common tangent into the leg of a right triangle on the line of centers.
Tangent Lengths from an Incircle [H]	The two tangent segments from a vertex to the incircle are equal, and each one equals the semiperimeter minus the opposite side.
Cevian Ratios by Mass Points [H]	Hang weights at the vertices so that every stated ratio balances; the intersection point then reads off every other ratio at once.
Necklaces and Bracelets by Burnside [H]	Average the number of colorings fixed by each symmetry; a rotation by d positions fixes exactly the colorings constant on its cycles.
Coloring a Square and a Cube [H]	List the rotations of the figure, count the colorings each one fixes, and average.
Permutations Counted by Fixed Points [H]	Choose which elements stay put, then derange the rest; nothing else is left to decide.
Choosing Without Neighbours [H]	Reserve one blank after each chosen element except the last; the leftovers can then be placed freely.
Round Tables with Forbidden Neighbours [H]	Fix the rotation by seating everyone else first, then drop the restricted people into the gaps between them.
Counting Integers Free of Square Factors [H]	Subtract the multiples of each prime square, add back the overlaps, and keep alternating.
Divisors That Are Perfect Powers [H]	A divisor is a perfect square exactly when every exponent in it is even, so the count is a product of independent choices.
Divisors Inside a Multiple Class [H]	Divisors of n that are multiples of k are exactly k times the divisors of n/k.
Solving a Linear Congruence [H]	Divide out the common factor of the coefficient and the modulus, invert what is left, and combine two congruences by stepping through one of them.
Counting Square Residues in a Range [H]	The condition on n depends only on n modulo m, so count the good residues once and multiply by the number of complete blocks.
Towers of Exponents Modulo Small Numbers [H]	Powers repeat with a fixed cycle length, so reduce the exponent modulo that length before computing anything.
Pairs and Triples with a Given LCM [H]	Handle one prime at a time: the lcm fixes the maximum exponent, so each prime contributes an independent count.

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