Math Olympiad: MOEMS Division M (Grades 7-8)

268 core topics + 34 prerequisite topics taught as needed · approximately 96 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

Math Olympiad: MOEMS & AMC 8 · 14 topics

Modular Arithmetic [M]	Working with remainders directly.
Counting Divisors [M]	Divisor count from the prime factorization.
GCD & LCM Relationships [M]	The identity gcd(a,b) · lcm(a,b) = a · b.
Digit Problems [M]	Reasoning about the digits of a number.
Units Digit of Powers [M]	Combine last-digit cycles, then add.
Permutations [M]	Ordered arrangements.
Combinations [M]	Unordered selections: n choose k.
The Pigeonhole Principle [M]	Guaranteeing a repeat.
Complementary Counting [M]	Count the opposite, then subtract.
Counting & Probability [H]	Favorable outcomes over total outcomes.
Vieta's Formulas [H]	Relating roots to coefficients without solving.
Telescoping Sums [H]	A sum that collapses to its endpoints.
Sums, Parity & Invariants [M]	Elegant closed forms and even/odd reasoning.
Recursion & Sequences [M]	Each term built from earlier ones.

MOEMS Depth: Divisions E & M · 55 topics

Divisibility Rules [H]	Test divisibility without doing the division.
Remainders [H]	The amount left over after fair sharing.
Digit Sums [H]	Add up the digits of a number.
Reversing Two-Digit Numbers [H]	The gap between a number and its reversal is a multiple of 9.
Counting Divisors [H]	Read the divisor count off the prime factorization.
GCF and LCM [H]	Common factors and common multiples.
Units Digit of a Power [H]	Last digits of powers repeat in a short cycle.
Sums of Consecutive Integers [H]	Pair the ends to add a run quickly.
Factors and Factor Pairs [H]	Factors come in pairs that multiply to the number.
Common Multiples in a Range [H]	Numbers divisible by both a and b are multiples of their LCM.
Gauss Sums [H]	Add 1 to n by pairing the ends.
Arithmetic Sequence Sums [H]	Average the first and last term, then multiply by the count.
Finding the nth Term [H]	Count the steps from the start.
Averages and a Missing Value [H]	The total is the average times the count.
A Fraction of a Quantity [H]	Divide into equal parts, then take some of them.
Successive Fractions (What's Left) [H]	Each fraction acts on the amount remaining, not the original.
Sharing in a Ratio [H]	Split into equal 'shares', then hand them out.
Proportions and Scaling [H]	Scale both quantities by the same factor.
Unit Rates [H]	How much for exactly one.
Distance, Speed, and Time [H]	Distance equals speed times time.
Perimeter of Rectangles [H]	Add up the distance around the outside.
Area of Rectangles [H]	Rows times columns of unit squares.
Area of Triangles [H]	Half of the base times the height.
Area of Composite Shapes [H]	Break the figure into rectangles, then add or subtract.
Angles in a Triangle [H]	The three angles always add to 180 degrees.
Angles on a Straight Line [H]	Angles along a straight line add to 180 degrees.
Counting Squares in a Grid [H]	Count squares of each size separately, then add.
The Painted Cube [H]	Where a small cube sits decides how many faces are painted.
Volume of a Box [H]	Stack layers of unit cubes.
The Multiplication Principle [H]	Multiply the number of independent choices.
Arrangements in a Row [H]	Multiply the shrinking number of choices for each spot.
Handshakes and Choosing Pairs [H]	Each handshake is a chosen pair.
Simple Probability [H]	Favorable outcomes over total outcomes.
Counting Grid Paths [H]	Every shortest path is a sequence of the same moves.
Two-Set Venn Counting [H]	Do not double-count the overlap.
The Pigeonhole Principle [H]	Plan for the worst case, then add one.
Age Problems [H]	Both people age by the same amount.
Chickens and Cows [H]	Assume all of one kind, then fix the leg count.
Coin Problems [H]	Assume the cheaper coin, then account for the extra value.
Work Rate (Inverse Proportion) [H]	More workers means proportionally less time.
Working Backwards [H]	Undo each step in reverse order.
Fence Posts (Off-by-One) [H]	A straight row of posts has one more post than gaps.
Days of the Week [H]	The weekday pattern repeats every 7 days.
Clock Angles [H]	Each hour mark is 30 degrees apart.
Magic Squares [H]	The magic sum is the total divided by the number of rows.
Even and Odd (Parity) [H]	Even/odd follows simple rules under addition and multiplication.
Remainders that Cycle [H]	Adding to a number cycles its remainder.
Perfect Squares [H]	Perfect squares come from squaring whole numbers.
Counting Multiples [H]	Divide to count multiples up to a limit.
Multiplying Patterns [H]	Find the constant ratio, then apply it again.
Total Value Problems [H]	Multiply each value by its count, then add.
Differences of Squares [H]	Consecutive squares differ by an odd number.
Grouping and Leftovers [H]	The leftover is the remainder after grouping.
Average Speed of a Round Trip [H]	Average speed is total distance over total time, not the mean of speeds.
Factorials [H]	Multiply every whole number down to 1.

MOEMS - Deep (Divisions E & M) · 86 topics

LCM of Three Numbers [H]	Build up the least common multiple two numbers at a time.
GCF with Equal Remainders [H]	Subtract the remainder, then take a common factor.
Sum of Divisors [H]	Add up every factor, from 1 up to the number itself.
Counting Even Divisors [H]	An even divisor must keep at least one factor of 2.
Counting Primes [H]	A prime has exactly two divisors: 1 and itself.
Digit Products [H]	Multiply the digits together instead of adding them.
Reversing Three-Digit Numbers [H]	The gap is always a multiple of 99.
Counting Palindromes [H]	A palindrome reads the same in both directions.
Converting to Another Base [H]	Repeatedly divide, reading remainders from bottom to top.
Two Remainders at Once [H]	Step through one list of remainders until the other matches.
Summing Multiples [H]	Factor out the common multiple, then use a Gauss sum.
Counting Non-Multiples [H]	Count the total, then remove the multiples.
Divisible by One or the Other [H]	Add the two counts, then subtract the overlap once.
Counting Perfect Cubes [H]	Perfect cubes come from cubing whole numbers.
Highest Power that Divides [H]	Keep dividing by the prime until it no longer goes evenly.
Recovering a Number from GCF and LCM [H]	The product of two numbers equals GCF times LCM.
The Second Largest Divisor [H]	Divide by the smallest prime factor.
Digital Roots [H]	Adding digits repeatedly lands on the remainder mod 9.
Counting Subsets [H]	Each element is either in or out - two choices each.
Seating Around a Circle [H]	Fix one person to remove the identical rotations.
Arranging Letters with Repeats [H]	Divide out the reorderings of identical letters.
Choosing with a Required Member [H]	Seat the required person first, then fill the rest.
Sharing Identical Objects [H]	Place dividers in the gaps between the objects.
Diagonals of a Polygon [H]	Each vertex connects to all but itself and its two neighbors.
Triangles from Points [H]	Every choice of 3 points makes one triangle.
Coloring with Adjacency Rules [H]	The first region is free; each next avoids its neighbor.
Two-Digit Numbers by Digit Sum [H]	List the tens digit and read off the units digit.
How Many n-Digit Numbers [H]	The first digit can't be zero; the rest are free.
Choosing from Two Groups [H]	Multiply the independent choices from each group.
Spinner Probability by Angle [H]	A region's chance is its angle over the full 360°.
Probability with a Deck of Cards [H]	Count the favorable cards out of 52.
Probability of Both Events [H]	Multiply the probabilities of independent events.
Probability an Event Does Not Happen [H]	Subtract the event's probability from 1.
Expected Value [H]	Average the outcomes, weighted by how likely each is.
Geometric Probability [H]	Compare the target area to the total area.
Numbers from Their Sum and Difference [H]	Half the sum plus half the difference gives the larger number.
Finding Consecutive Even Numbers [H]	Write each even number in terms of the first.
Pattern to Formula (Toothpicks) [H]	Find how much each new figure adds, then build a rule.
Balance Puzzles [H]	Replace one object with its equal in the other.
The Median [H]	Sort the numbers and take the middle one.
The Mode [H]	The mode is the value that appears most often.
The Range [H]	Subtract the smallest value from the largest.
Geometric Series Sums [H]	Use the doubling shortcut instead of adding term by term.
Inserting Arithmetic Means [H]	Count the equal gaps the inserted numbers create.
Iterating a Rule [H]	Apply the same operation repeatedly, tracking the result.
Average of Consecutive Integers [H]	The average is the midpoint of the run.
Area of a Parallelogram [H]	Base times height, using the straight-across height.
Area of a Trapezoid [H]	Average the parallel sides, then multiply by the height.
Area of a Square from its Diagonal [H]	A square's area is half the square of its diagonal.
Perimeter of an L-Shape [H]	Cutting a corner notch leaves the perimeter unchanged.
Rectangle from Area and Perimeter [H]	Half the perimeter is the sum of length and width.
Angles with Parallel Lines [H]	Same-side interior angles add to 180°.
Exterior Angles of a Polygon [H]	The exterior angles always add to 360°.
Complements and Supplements [H]	Complements make 90°; supplements make 180°.
Angles Around a Point [H]	Angles around a single point add to 360°.
Base Angles of an Isosceles Triangle [H]	The two equal sides sit opposite two equal angles.
Distance Between Points [H]	Make a right triangle from the horizontal and vertical gaps.
Midpoint of a Segment [H]	The midpoint averages the endpoints' coordinates.
Area from Coordinates [H]	The side lengths are the coordinate differences.
Cube Edge from Surface Area [H]	A cube has six equal square faces.
Cube Edge from Volume [H]	The volume of a cube is the edge cubed.
Euler's Formula for Solids [H]	Vertices minus edges plus faces always equals 2.
Scale Drawings [H]	Multiply the map distance by the scale factor.
Area of a Ring (Annulus) [H]	Subtract the inner circle's area from the outer circle's.
The Triangle Inequality [H]	The third side lies strictly between the sum and the difference.
Classifying Angles [H]	Compare the angle to 90° and 180°.
Approaching Objects [H]	Add the speeds to get the closing rate.
Catching Up [H]	Subtract the speeds to get the closing rate.
Boats and Currents [H]	The current helps one way and hinders the other.
Simple Interest [H]	Interest is principal times rate times time.
Mixture Prices [H]	The blended price is the total cost over the total weight.
Unit Conversion [H]	Multiply by how many small units fill one big unit.
Average Speed over Timed Legs [H]	Total distance divided by total time.
Filling Against a Drain [H]	Subtract the draining rate from the filling rate.
Elapsed Time [H]	Convert both clock times to minutes, then subtract.
Counting Days Inclusively [H]	Subtract the dates, then add one for both endpoints.
A Clock that Runs Fast [H]	The error grows by the same amount each hour.
Percent More Than [H]	Compare the increase to the original amount.
Fractions to Percents [H]	A percent is the fraction scaled to a denominator of 100.
From a Part to the Whole [H]	Find the value of one share, then count all the shares.
Shadows and Similar Triangles [H]	Height and shadow keep the same ratio for everything.
Meshing Gears [H]	The gear with fewer teeth spins faster.
Replacing One Member of an Average [H]	One swap shifts the total by n times the change in average.
A Percent of a Percent [H]	Take the percents one after the other.
Remainders Add [H]	The remainder of a sum is the sum of the remainders.
Sum of the First n Odd Numbers [H]	The running total of odd numbers is always a perfect square.

Olympiad - Number Sense and Algebraic Reasoning · 48 topics

Digits Forced by a Divisor [H]	Split the divisor into coprime parts and let each part pin down a digit.
Divisors That Are Always There [H]	Among k consecutive integers, every residue class modulo k appears exactly once.
GCD of Two Linear Forms [H]	A common divisor of two linear forms also divides any integer combination of them.
The 1001 Trick [H]	Because 1001 = 7 x 11 x 13, a six-digit number with a repeated block is divisible by all three.
Prime Powers Inside a Factorial [H]	Count multiples of p, then of p squared, then of p cubed, and add.
Which Claim Holds for Every n [H]	A statement about all integers needs a residue argument, while one counterexample kills it.
Forcing a Divisor to Be Constant [H]	Reduce the dividend modulo the divisor until only a constant remains, then list its divisors.
A Parity That Survives Every Move [H]	Replacing two numbers by their difference leaves the parity of the total unchanged.
Coloring a Board [H]	Color the board like a chessboard: every domino covers one square of each color.
Flipping a Fixed Number of Switches [H]	Each move changes the number of ON switches by an amount with the same parity as the move size.
Choosing the Argument That Settles It [H]	An impossibility proof needs a quantity that no move can change, not an arithmetic coincidence.
The Handshake Sum Must Be Even [H]	Adding everyone's handshake count double counts each handshake, so that total is even.
Reachable Positions Form One Progression [H]	Swapping one jump for the other always changes the finish by the same fixed amount.
A Total That Drifts by a Fixed Amount [H]	Track how much the total changes at each move, not which numbers were chosen.
Adding One Turns a Sum Rule into a Product Rule [H]	Since ab + a + b + 1 = (a+1)(b+1), the product of all numbers increased by one is preserved.
Color Counts Modulo Three [H]	Each meeting changes two color counts by one and the third by two, so differences modulo three are preserved.
Every Amount Is a Combination of the Two Jugs [H]	Any amount reachable with two jugs is a multiple of the greatest common divisor of their sizes.
A Score That Does Not Depend on Your Choices [H]	Each split scores exactly the pairs of stones it separates, so the total counts all pairs once.
Undoing a Repeated Rule [H]	Reverse each step in turn: undoing halve-then-subtract means add back, then double.
Three Rounds Run in Reverse [H]	Undo a doubling round by halving the receivers and returning what was given.
Shortest Route to a Target Number [H]	Work backwards from the target: undo a multiplication when possible, otherwise undo an addition.
Losing Positions Found Backwards [H]	A position is losing exactly when every move from it leads to a winning position.
Make the Others as Small as Possible [H]	To maximise one member of a set with a fixed total, minimise everything else.
Pushing the Middle Value Up [H]	To maximise the median, make the values below it minimal and those above it as tight as allowed.
Pair Up the Forbidden Partners [H]	Split the numbers into pairs that add to the forbidden total and take one from each pair.
How Many Can Clear the Bar [H]	Count how many members can meet a threshold by giving everyone the least the rules allow.
Squeezing the Largest Member Down [H]	If the biggest number is M, the whole set fits inside M, M-1, ..., M-k+1, which caps the total.
Remainders as Pigeonholes [H]	Two numbers differ by a multiple of m exactly when they leave the same remainder on division by m.
Pigeonholes of Different Sizes [H]	A color with fewer than k socks caps the worst case at its own supply.
None Divides Another [H]	Writing each number as an odd number times a power of two sorts them into chains.
Something Beats the Average [H]	Some box holds at least the average, rounded up, and an even spread shows no more is guaranteed.
Factoring an Equation into a Product [H]	Adding the right constant turns xy + ax + by into a product of two brackets.
Counting Differences of Squares [H]	Every way of writing N as a difference of squares comes from a factor pair of N whose two factors have the same parity.
Building Symmetric Expressions [H]	Any symmetric expression in x and y can be rebuilt from their sum and product alone.
Powers of x Plus Its Reciprocal [H]	Multiplying x^n + 1/x^n by x + 1/x produces the next power and the previous one.
A Sum of Fourth Powers That Factors [H]	The identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2) factors a sum of fourth powers.
Alternating Squares Collapse [H]	Pair the terms so each pair is a difference of squares equal to the sum of its two bases.
Splitting a Power Minus One [H]	For every divisor d of e, the number m^d - 1 divides m^e - 1.
Spotting the Hidden Factorisation [H]	A number close to a square or to a round power usually hides an algebraic factorisation.
Telescoping with a Gap [H]	Split 1/(k(k+d)) as (1/d)(1/k - 1/(k+d)) so terms cancel d places apart.
A Product That Telescopes [H]	Factor each term as (k-1)(k+1)/k^2 so neighbouring numerators and denominators cancel.
Rationalising Makes It Telescope [H]	Multiplying by the conjugate turns 1/(root k + root (k+1)) into a difference of square roots.
Three Factors in the Denominator [H]	Half the difference of two neighbouring products of pairs gives 1/(k(k+1)(k+2)).
Adding Rates, Not Times [H]	Let the unknown be the tank per hour rate of each pipe; the three pairwise sums then add to twice the total rate.
One Worker Leaves Partway [H]	Measure the fraction of the job finished before the change, then divide what remains by the surviving rate.
The Escalator Steps You Never Tread [H]	The escalator supplies the steps you do not walk, so steps walked plus steps carried is a constant.
Meeting, Then Finishing [H]	Let the meeting time be the unknown; each rider covers the other's first leg in the time stated.
Grass That Grows While the Cows Eat [H]	Take one cow's daily ration as the unit and treat the starting grass and the daily growth as two unknowns.

Olympiad - Geometry and Combinatorial Reasoning · 47 topics

Areas Along a Divided Side [H]	Triangles with the same apex and bases on one line have areas in the ratio of those bases.
Shrinking a Triangle at One Corner [H]	Cutting both sides at a vertex multiplies the area by the product of the two ratios.
A Chain of Midpoints [H]	Each median drawn in a triangle halves the area of the triangle it is drawn in.
A Point Inside a Rectangle [H]	Opposite triangles from an interior point of a rectangle have areas summing to half the rectangle.
The Midpoint Quadrilateral [H]	Joining the midpoints of a convex quadrilateral produces a parallelogram of exactly half the area.
Pick's Theorem on a Lattice [H]	A lattice polygon has area I + B/2 - 1, where I and B count interior and boundary lattice points.
Which Fact Forces Equal Areas [H]	Two triangles have equal areas exactly when equal bases are paired with equal heights.
Rectangles Inside a Grid [H]	A rectangle in a grid is fixed by choosing two horizontal lines and two vertical lines.
Tilted Squares in a Point Array [H]	Every square in a lattice sits inside a unique upright square that circumscribes it.
Counting Inside a Fan [H]	In a fan of cevians every triangle is fixed by choosing two of the rays from the apex.
Parallelograms From Two Line Families [H]	Two lines from each of two parallel families bound exactly one parallelogram.
Triangles With Collinear Points Removed [H]	Count all vertex triples, then subtract the triples that are collinear and so degenerate.
Intersections of a Family of Lines [H]	Each pair of lines meets once unless the pair is parallel, so subtract the parallel pairs.
Rectangles in an L-Shape [H]	Count rectangles in the full grid and subtract exactly those that reach into the removed block.
A Board With Two Squares Removed [H]	A domino always covers one square of each color, so a color imbalance blocks any covering.
Counting Tilings of a Strip [H]	Tilings of a 2 by n strip satisfy a recurrence found by looking at the last column only.
Deficient Boards and L-Trominoes [H]	A tromino covering of a deficient board uses exactly one third of the remaining unit squares.
Covering a Board With Long Pieces [H]	An m by n board takes 1 by k pieces exactly when k divides m or k divides n.
Fewest Tiles on a Floor [H]	Fewest tiles means most 2 by 2 tiles, and an even-coordinate invariant caps how many fit.
Which Covering Argument Is Sound [H]	An impossibility needs an invariant that every piece respects; a possibility needs an actual covering.
Degrees Add Up to Twice the Edges [H]	The friendship counts of all members add to twice the number of friendly pairs.
When Everyone Has the Same Degree [H]	Equal degrees d for n people are achievable exactly when d is at most n - 1 and n times d is even.
Walking Every Path Exactly Once [H]	A connected network can be walked in one route exactly when it has no odd corner or exactly two.
How Many Strokes to Draw a Figure [H]	A connected figure with 2k odd points needs exactly k strokes, and k is never smaller.
Counting Routes Through a Network [H]	Routes to a town total the routes to each town with a road into it.
Roads You Can Close [H]	A connected network on n towns keeps only n - 1 roads at minimum, so the rest are removable.
Can These Handshake Counts Happen [H]	A list of handshake counts needs an even total and no entry above the number of other people.
Regions Cut by Straight Lines [H]	Each new line adds one region for every region it crosses, that is one more than its intersections.
Regions Inside a Circle of Chords [H]	Each interior crossing comes from a unique set of four marked points, which drives the region count.
Pentagons Forced by Euler's Formula [H]	Counting each edge and vertex through the faces turns Euler's formula into a fixed pentagon count.
Regions Cut by Circles [H]	A new circle meeting the earlier ones in 2(k - 1) points gains exactly that many regions.
Integer Triangles of a Given Perimeter [H]	List sides in increasing order and let the triangle inequality bound the largest side.
Counting Coin Combinations by Cases [H]	Fix the number of the largest coin first; the rest of each case is then a short count.
Splitting a Number into Three Parts [H]	Order the three parts to count each split once, then sweep the smallest part.
Numbers With Ordered Digits [H]	A strictly increasing digit string is just a choice of digits, since the order is then forced.
Choosing an Exhaustive Case Split [H]	A case split must cover every possibility, overlap nowhere, and be fine enough to settle the claim.
Largest Product With a Fixed Sum [H]	Parts that differ by more than one can be evened out to increase the product, so the best split is level.
Kings That Never Touch [H]	Cutting the board into 2 by 2 blocks caps the kings, and an odd-row odd-column placement attains the cap.
Bishops on Their Diagonals [H]	No two bishops may share a diagonal, and two of the corner diagonals can never both be used.
A Set With No Fixed Difference [H]	Splitting the numbers into chains that step by d turns the problem into alternating along each chain.
Shading Without a Full 2 by 2 Block [H]	Disjoint 2 by 2 blocks each need one blank square, and blanking the even-even squares attains that.
Does the Construction Attain the Bound [H]	An extremal answer needs both an upper bound and one arrangement that reaches it and obeys every rule.
Bracelets Up to Rotation [H]	Averaging the colorings left unchanged by each rotation corrects the overcount from turning the ring.
Painting the Faces of a Cube [H]	The 24 rotations of a cube each fix their own colorings, and the answer is the average of those counts.
Shadings Up to Rotation and Reflection [H]	Group shadings into families joined by the eight symmetries of the square and count one per family.
Rook Placements That Mirror Themselves [H]	A self-mirroring rook placement pairs up its columns, so it is built from fixed points and swapped pairs.
Using Symmetry Without Overcounting [H]	Dividing by the number of symmetries is valid only when every object is repeated that many times.

Olympiad - Advanced Arguments & Constructions · 18 topics

Inversions Fall One at a Time [H]	Count the pairs standing in the wrong order: one adjacent swap changes that count by exactly one, so it measures the work exactly.
A Quantity That Only Falls [H]	A positive whole number that strictly decreases at every step cannot decrease forever, so the process must halt and its length can be counted.
What Rows and Columns Cannot Change [H]	Adding 1 along a whole row or column leaves every corner-to-corner cross difference of a rectangle untouched.
Pressing a Lamp and Its Neighbours [H]	Only the parity of the number of presses at each lamp matters, so a solution is a SET of lamps and the whole search is finite.
Most Friendships Without a Triangle [H]	Split the people into equal groups and join every pair from different groups: that construction meets the counting bound exactly.
Pushing the Closest Pair Apart [H]	The gaps between neighbouring markers add up to the whole length, so the smallest gap can never beat the average gap - and even spacing attains it.
Where the Claim First Breaks [H]	A claim about every n is settled by the smallest n that breaks it, and the largest modulus that never breaks is the gcd of the first few values.
Making the Heaviest Group as Light as Possible [H]	The largest group can never be lighter than the average group, and the question is whether some split actually reaches that floor.
Every Long Sequence Turns Monotone [H]	Label each term by the longest increasing run ending there; two terms with the same label are forced to decrease.
More Subsets Than Possible Totals [H]	Count the subsets and count the totals they could land on: when the first count wins, two subsets must share a total.
How Many Columns Force a Repeat [H]	Count the painting patterns a single column can have: that count is the number of pigeonholes.
The Two Colors a Knight Alternates [H]	A knight always lands on the opposite color, which fixes the parity of any journey and organises the search into layers.
A Coloring That Caps the Tiling [H]	Color cell (i, j) by the remainder of i + j: a straight piece of k cells then covers one cell of every color.
How Long Until Everything Returns [H]	Break the shuffle into cycles: each cycle returns after its own length, so everything returns after the least common multiple.
Painting a Row and Painting a Ring [H]	Build the coloring one cell at a time and count the choices left at each step; a ring needs a correction because its last cell sees the first.
Stepping Along a Recurrence [H]	A rule that builds each term from the one before is an induction already written down, and the closed form is found by shifting the constant.
Routes That Never Cross the Diagonal [H]	Count the routes by filling in the grid corner by corner, discarding every point that breaks the rule before it is ever used.
Counting the Outcomes to Get a Bound [H]	Each question has a fixed number of possible answers, so q questions can tell apart at most that many possibilities raised to the power q.

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 · 4 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.

Expressions & Equations · 4 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.

Quadratics & Polynomials · 8 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 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.

Math Kangaroo: Benjamin & Cadet · 2 topics

Systematic Counting	Counting without missing or double-counting.
Number Patterns	Finding the rule behind a sequence.

Olympiad (Elementary) - Reasoning Puzzles and Strategy · 8 topics

Which Clue Is Enough	A clue decides the answer only when exactly one candidate survives it.
An Invariant on the Board	Track a quantity the move changes in a fixed way, so the order of moves cannot matter.
Why a Cut Board Cannot Be Covered	Color the board and compare the color counts a covering would need.
Counting Squares of One Color	Colors alternate, so an odd total leaves the corner color one square ahead.
Patterns with Growing Gaps	When the gaps grow by a fixed amount, add up the gaps instead of guessing.
Counting Inside a Repeating Pattern	Count whole repeats first, then handle the part-repeat at the end.
Grid Routes Past a Closed Corner	Count every route, then subtract the routes that use the closed corner.
Fewest Weighings on a Balance	Each weighing has three outcomes, so it can cut the suspects to a third.

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