Math Kangaroo: Junior (Grades 9-10)

292 core topics + 86 prerequisite topics taught as needed · approximately 120 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 Kangaroo: Junior & Student · 6 topics

Clever Arithmetic [M]	Spot the structure before you compute.
Quadratic Tricks [H]	Complete the square; work with sums and products of roots.
Competition Sequences [H]	Equal spacing and equal ratios, exploited.
Angle Chasing [H]	Push known angles through triangles and polygons.
Grid Paths & Selections [H]	Shortest paths are just choices in disguise.
Divisibility Duels [H]	Factor counts, trailing zeros, digit sums.

Math Kangaroo: Junior - Deep (Grades 9-10) · 101 topics

Solving Linear Equations [H]	Undo the operations to isolate x.
Number Riddles [H]	Translate the words, then reverse the steps.
Systems by Elimination [H]	Add or subtract to cancel a variable.
Sharing in a Ratio [H]	Split into a + b equal parts.
Proportions [H]	Cross-multiply equal ratios.
Percent of a Quantity [H]	Percent means hundredths.
Percent Increase [H]	Multiply by (1 + p/100).
Percent Decrease [H]	Multiply by (1 − p/100).
Simple Interest [H]	I = P·r·t/100.
Finding a Missing Value [H]	Total = average × count.
Consecutive Integers [H]	The middle term is the average.
nth Term of an AP [H]	aₙ = a₁ + (n − 1)d.
Sum of an AP [H]	Sum = n(first + last)/2.
nth Term of a GP [H]	aₙ = a₁·rⁿ⁻¹.
Evaluating Expressions [H]	Substitute, then follow order of operations.
Multiplying Powers [H]	Same base: add the exponents.
Power of a Power [H]	(aᵐ)ⁿ = aᵐⁿ.
Negative Exponents [H]	a⁻ⁿ = 1/aⁿ.
Squaring a Sum [H]	(a + b)² = a² + 2ab + b².
Integer Roots by Factoring [H]	Find two numbers multiplying to c, summing to −b.
Axis of Symmetry [H]	The parabola is symmetric about x = −b/(2a).
Absolute Value Equations [H]	|x − a| = b gives x = a ± b.
Linear Inequalities [H]	Isolate x, then take the smallest integer that fits.
Direct Variation [H]	y = kx with constant k.
Inverse Variation [H]	y = k/x with constant product.
Add-the-Previous-Two [H]	Fibonacci rule: aₙ = aₙ₋₁ + aₙ₋₂.
Computing the Mean [H]	Mean = sum ÷ count.
Finding the Median [H]	Sort, then take the middle.
Weighted Average [H]	Weight each mean by its group size.
Distance = Rate × Time [H]	d = rt.
Combined Work Rates [H]	Rates add: 1/a + 1/b = 1/T.
Greatest Common Divisor [H]	Largest number dividing both.
Least Common Multiple [H]	lcm(a,b) = ab / gcd(a,b).
Counting Prime Factors [H]	Factor fully and count with repeats.
Counting Divisors [H]	Add one to each exponent and multiply.
Sum of Divisors [H]	Add every factor, 1 and n included.
Divisibility by 11 [H]	Alternating digit sum decides remainder mod 11.
Remainders [H]	What's left after the largest full multiple.
Last Digit of a Product [H]	Only the units digits matter.
Units Digit of a Factorial [H]	From 5! on, factorials end in 0.
Sum of Odd Numbers [H]	The first n odds sum to n².
Sum of Even Numbers [H]	The first n evens sum to n(n+1).
Counting Multiples [H]	There are ⌊N/k⌋ multiples of k up to N.
Counting Non-Multiples [H]	Subtract the multiples from the total.
Binary Digit Count [H]	Split into powers of two.
Digit Sums [H]	Add the digits.
Digital Root [H]	Repeated digit sums land on n mod 9.
Next Perfect Square [H]	Find the next whole-number root.
Triangular Numbers [H]	Tₙ = n(n+1)/2.
Fibonacci Numbers [H]	Each term sums the two before it.
Clock Arithmetic [H]	Hours wrap around modulo 12.
Days of the Week (mod 7) [H]	Weekdays cycle with period 7.
Leftover Problems [H]	Group into full sets; count the remainder.
Factor Pairs [H]	Pair each small divisor with its cofactor.
Counting Primes [H]	Sift out the composites.
Permutations [H]	nPk = n·(n−1)···(n−k+1).
Combinations [H]	nCk = n! / (k!(n−k)!).
Factorials [H]	n! multiplies 1 through n.
Arrangements with Repeats [H]	Divide n! by the factorial of each repeat count.
Counting k-Digit Numbers [H]	9 choices lead, 10 for each of the rest.
Handshakes [H]	Each pair shakes once: C(n,2).
Polygon Diagonals [H]	Diagonals = n(n−3)/2.
The Multiplication Principle [H]	Multiply the choices at each stage.
Counting Rectangles [H]	Choose 2 of the vertical and 2 of the horizontal lines.
Basic Probability [H]	P = favorable / total.
Complementary Probability [H]	P(not A) = 1 − P(A).
Independent Events [H]	Multiply probabilities of independent events.
Drawing Without Replacement [H]	Multiply, shrinking the pool each draw.
Dice Sum Probability [H]	Count the 36 equally likely pairs.
Expected Value [H]	Average the equally likely outcomes.
Odds to Probability [H]	Odds a:b means probability a/(a+b).
Pigeonhole (Socks) [H]	One more than the number of colors.
Counting Subsets [H]	Each element is in or out: 2ⁿ.
Committees with a Fixed Member [H]	Seat the required person, then fill the rest.
Round-Table Seating [H]	Fix one person: (n−1)! ways.
Codes Without Repeats [H]	Fewer choices remain after each digit.
Sharing Identical Items (Two) [H]	Choose how many the first child gets: n + 1 ways.
Pythagoras: Hypotenuse [H]	c = √(a² + b²).
Pythagoras: Missing Leg [H]	leg = √(c² − a²).
Area of a Triangle [H]	Area = ½ · base · height.
Area of a Parallelogram [H]	Area = base · height.
Area of a Trapezoid [H]	Area = ½(a + b)·h.
Area of a Circle [H]	Area = πr².
Circumference of a Circle [H]	C = 2πr.
Perimeter of a Rectangle [H]	P = 2(l + w).
Triangle Angle Sum [H]	Angles of a triangle add to 180°.
Exterior Angle Theorem [H]	Exterior angle = sum of remote interiors.
Polygon Angle Sum [H]	Total interior angle = (n − 2)·180°.
Complements & Supplements [H]	Complements sum to 90°, supplements to 180°.
Parallel Lines & a Transversal [H]	Co-interior angles are supplementary.
Similar Triangles [H]	Corresponding sides share one ratio.
Scaling Perimeter [H]	Perimeter scales by the linear factor k.
Volume of a Box [H]	V = l · w · h.
Volume of a Cylinder [H]	V = πr²h.
Surface Area of a Box [H]	S = 2(lw + lh + wh).
Distance Between Points [H]	d = √((Δx)² + (Δy)²).
Midpoint [H]	Midpoint coordinate = average of the endpoints.
Slope of a Line [H]	slope = rise / run.
Clock Hand Angles [H]	Each hour mark is 30° apart.
30-60-90 Triangles [H]	Sides are in ratio 1 : √3 : 2.
45-45-90 Triangles [H]	Equal legs; area = leg²/2.

Math Kangaroo (Grades 9-12) - Algebra and Number Theory · 47 topics

Adding Every Equation at Once [H]	Add the equations of a symmetric system so the total appears in one step, then peel off one unknown.
Balance Puzzles With a Chain of Animals [H]	Convert along a chain of balances by keeping one common unit of weight instead of solving for each animal.
Asking for a Combination, Not the Parts [H]	Add or subtract the given equations so the requested combination of unknowns appears without solving for each one.
Powers of a Number Plus Its Reciprocal [H]	Square or cube the relation x + 1/x = k to reach the higher power sums without ever finding x.
Two Ratios Joined at the Middle [H]	Rescale two ratios so their shared quantity matches, then treat the result as one three-part ratio.
Reading the Sum and Product of the Roots [H]	Take the sum and product of the roots straight from the coefficients instead of solving the quadratic.
Symmetric Powers of the Two Roots [H]	Rewrite a symmetric expression in the roots using their sum and product, then substitute the coefficients.
Reciprocals of the Roots [H]	Put the reciprocals of the roots over their common product so the answer is a ratio of the coefficients.
Shifting and Squaring the Roots [H]	Express the sum and product of transformed roots through the sum and product of the original ones.
The Root Two Equations Share [H]	Subtract two quadratics so their square terms cancel and the shared root satisfies the linear equation left behind.
The Extreme Value of a Quadratic [H]	Complete the square so the extreme value of a quadratic is read directly from the constant left outside.
The Least Perimeter for a Given Area [H]	Minimise a sum whose product is fixed by making the two parts equal, unless a whole-number condition forces a search.
Splitting a Total to Make the Product Largest [H]	Make the parts of a fixed sum as equal as the conditions allow, and move to the nearest allowed split when a bound blocks the balanced one.
Counting the Whole Numbers in a Window [H]	Turn a condition into an interval for the unknown and count the whole numbers strictly inside it.
Pushing One Value to Its Limit [H]	Fix the total from the average, then make the other values as small or as large as the conditions allow.
A List Built From the Number Before [H]	Apply a one-term recursion forwards step by step, or undo it to travel back to the start.
Recursions That Come Back Round [H]	Detect the period of a recursion and use the remainder of the position to jump straight to the required term.
Each Number From the Two Before It [H]	Use the add-the-previous-two rule forwards, or express a later term through the first two and solve for the missing one.
Arithmetic Growth Inside a Story [H]	Recover the first term and the common difference from two stated terms, then reach any term or any total.
Multiplying by the Same Factor Each Step [H]	Find the common ratio from two known terms by taking the root of their quotient, then step on to any other term.
Sums That Collapse in Pairs [H]	Split each term into a difference of two simple fractions so all the middle parts cancel.
Products That Cancel Step by Step [H]	Write each factor as a ratio of consecutive whole numbers so every numerator cancels the next denominator.
Chains of Differences of Squares [H]	Factor each difference of squares into a sum of consecutive whole numbers so the chain becomes a short arithmetic sum.
Alternating Sums Taken in Blocks [H]	Group an alternating sum into blocks of equal value so the total is the number of blocks times that value.
The Last Digit of a Power [H]	Use the repeating cycle of last digits of the powers of a number and the remainder of the exponent.
The Last Two Digits [H]	Work modulo 100, using the cycle of the last two digits of the powers or the point where every later term ends in 00.
Solving a Remainder Equation [H]	Find the multiplier that leaves a required remainder by walking through one full cycle of remainders.
Two Remainder Conditions at Once [H]	Combine two remainder conditions by shifting the number so both divisions come out exactly, or by stepping through one condition and testing the other.
What Always Divides an Expression [H]	Factor an expression into consecutive whole numbers and count the factors of two and three that are then guaranteed.
The Quantity a Move Cannot Change [H]	Track the quantity a move leaves unchanged, such as the parity of the total, and rule out every value that disagrees with it.
Counting Divisors From the Exponents [H]	Multiply one more than each exponent in the prime factorisation to count the divisors, and drop a prime to count a restricted family.
The Smallest Number With So Many Divisors [H]	Split the required divisor count into factors, read them as exponents, and give the largest exponent to the smallest prime.
Multiplying Up to a Perfect Power [H]	Repair the prime exponents so every one of them becomes a multiple of the required power.
Zeros at the End of a Product [H]	Count the pairs of factors two and five in a product, since each pair contributes one final zero.
Spotting What Cannot Be a Square [H]	Rule out a perfect square by its final digit, by its position between two consecutive squares, or by an odd exponent in its factorisation.
Numbers Tied to Their Own Digits [H]	Write a number as 100a + 10b + c so a condition on its digits becomes a short equation with few candidates.
Counting in Another Base [H]	Read a numeral as a sum of powers of its base, or divide repeatedly by the base to write a number in it.
How Long a Numeral Is [H]	Compare a number with the powers of the base to find its length, and count the numerals of a given length as a product of digit choices.
Counting Numbers That Read Both Ways [H]	Count palindromes by choosing only the free digits, then impose the extra divisibility condition on those choices.
The Unknown in the Exponent [H]	Rewrite both sides as powers of one base so the exponents can be compared directly.
Which Power Is the Largest [H]	Compare powers by matching either their bases or their exponents before looking at the numbers themselves.
Logarithms That Add and Subtract [H]	Combine logarithms of the same base into a single logarithm of a product or quotient before evaluating.
Chains of Logarithms [H]	Multiply logarithms in a chain so each base cancels the argument before it and only the two ends survive.
Average Speed Over a Journey [H]	Divide the total distance by the total time, since equal distances and equal times give different averages.
Two Travellers Coming Together [H]	Add the speeds of two travellers approaching each other so the gap closes at one combined rate.
Two Percentage Changes in a Row [H]	Multiply the two change factors, since a rise followed by a fall of the same percentage does not return to the start.
Ages Now, Before and Later [H]	Name one age as the unknown and write every other age as that unknown shifted by the stated number of years.

Math Kangaroo (Grades 9-12) - Geometry and Trigonometry · 46 topics

The Points of a Star [H]	Find the angle at a point of a star from the arc that the two chords through that point cut off.
Inscribed Angles and the Arcs They Stand On [H]	Measure an angle at, inside, or outside a circle as half the sum or half the difference of the arcs it cuts off.
Quadrilaterals With All Four Corners on a Circle [H]	Use the fact that opposite angles of a quadrilateral inscribed in a circle add to 180 degrees.
The Angle Between a Tangent and a Chord [H]	Equate the angle between a tangent and a chord with the inscribed angle standing in the alternate segment.
Where the Bisectors and the Altitude Meet [H]	Compute the angle at the incentre as 90 degrees plus half the opposite angle, and the bisector to altitude angle as half the difference of the other two angles.
Angles Made by the Diagonals of a Regular Polygon [H]	Treat the vertices of a regular n-gon as points on its circumcircle, where one step of the polygon is an arc of 360/n degrees.
A Chain of Isosceles Triangles [H]	Apply the exterior angle theorem along a chain of isosceles triangles so the tilt grows by the same amount at every step.
An Equilateral Triangle Meets a Regular Polygon [H]	Combine the interior angle of a regular polygon with the 60 degree angle of an equilateral triangle at a shared vertex.
The Median to the Hypotenuse [H]	Use the fact that the midpoint of the hypotenuse is the same distance from all three vertices of a right triangle.
A Line Drawn Parallel to a Side [H]	Use the equal ratios of a triangle cut by a line parallel to one side, where lengths scale by the ratio and areas by its square.
The Four Triangles Made by the Diagonals of a Trapezium [H]	Exploit the similar triangles at the crossing of the diagonals of a trapezium, where the four pieces have areas in the pattern a squared, ab, ab, b squared.
Two Crossed Wires Between Poles [H]	Find the height where two wires cross by adding the reciprocals of the two pole heights.
The Largest Square Inside a Triangle [H]	Set the shrunken top triangle similar to the whole one to get a square of side bh divided by b plus h.
A Lamp, a Walker and a Shadow [H]	Match the triangle of the lamp with the triangle of the walker to get a shadow of length pd divided by H minus p.
A Line From a Vertex Cuts the Area [H]	Split areas with the rule that two triangles of the same height have areas in the ratio of their bases.
A Point Inside a Parallelogram [H]	Use the fact that the two triangles on opposite sides of a point inside a parallelogram together cover half of it.
Distances to the Four Corners of a Rectangle [H]	Apply the rule that for any point and any rectangle the squared distances to one pair of opposite corners equal those to the other pair.
Two Squares That Overlap at a Center [H]	Use the quarter turn symmetry of a square to show that a congruent square with a corner at its center always overlaps exactly a quarter of it.
Rings Between Two Circles [H]	Compute a ring as the difference of two circle areas, which a tangent chord fixes without either radius being known.
Regions Made From Semicircles [H]	Add and subtract semicircle areas, which are proportional to the squares of their diameters, so that Pythagoras makes the pi cancel.
Areas and Points on Grid Paper [H]	Find the area of a lattice polygon by the shoelace sum and relate it to the counts of boundary and interior grid points.
Chords and Their Distance From the Center [H]	Drop the perpendicular from the center to a chord, which bisects it and makes a right triangle with the radius.
Equal Tangents From a Point [H]	Use the equal lengths of the two tangents from a point, together with the right angle at each point of contact.
Circles That Touch Each Other [H]	Set the distance between the centers of touching circles equal to the sum or the difference of their radii.
The Circle Inside a Right Triangle [H]	Use the tangent lengths of a right triangle, which make the inradius equal to half of the sum of the legs minus the hypotenuse.
Circles Drawn Inside and Around a Polygon [H]	Compare a circle with a polygon by writing both areas in terms of the same length so that the shared length cancels.
The Longest Straight Line Inside a Box [H]	Apply Pythagoras twice so that the space diagonal of a box is the square root of the sum of the squares of its three edges.
The Shortest Crawl Over a Box [H]	Unfold the surface of a box flat so that a shortest crawling route becomes a straight line, then compare every unfolding.
The Altitude to the Hypotenuse [H]	Use the three similar triangles made by the altitude to the hypotenuse, where each length is a geometric mean of two others.
Distances Inside a Cube [H]	Place a cube on coordinate axes so that any distance inside it is a square root of a sum of three squares.
The Missing Corner on the Grid [H]	Locate a missing vertex with vectors, adding a side vector for a parallelogram and turning one a quarter turn for a square.
The Shortest Path That Touches a Wall [H]	Reflect one endpoint in the wall so that a bent shortest path becomes a straight line.
A Straight Cut That Halves a Shape [H]	Use the fact that a straight line halves a rectangle exactly when it passes through the center of that rectangle.
Where the Three Medians Meet [H]	Average the three vertices to find the centroid, which cuts each median in the ratio two to one and the triangle into six equal pieces.
Reading a Ratio Off a Right Triangle [H]	Name the sides as opposite, adjacent and hypotenuse for the marked angle, then read the required ratio.
Looking Up at an Angle [H]	Turn an angle of elevation into the exact side ratios of a 45 45 90 or a 30 60 90 triangle.
The Cosine Rule at Friendly Angles [H]	Extend Pythagoras with the cosine rule, which adds ab at 120 degrees and subtracts ab at 60 degrees.
The Sine Rule and the Circle Through the Corners [H]	Apply the sine rule in the form a over sin A equals the diameter of the circle through the three corners.
Area From Two Sides and the Angle Between Them [H]	Compute a triangle area as half the product of two sides times the sine of the angle between them, and a parallelogram as twice that.
Slicing a Cube [H]	Identify the cross section of a cube by finding the vertices the plane passes through and measuring its edges with Pythagoras.
A Painted Block Cut Into Unit Cubes [H]	Sort the unit cubes of a painted block by position, since corner, edge, face and inside cubes carry three, two, one and no painted faces.
Surface Area After Cutting and Gluing [H]	Track only the faces that appear and disappear when solids are joined or pieces are removed.
Cones, Balls and Cans [H]	Use the fixed ratio one to two to three for a cone, a ball and a cylinder that share a base and a height.
Water That Keeps Its Volume [H]	Hold the volume of water fixed while the shape it fills changes, so the new depth follows from the new cross section.
Symmetry That Halves the Work [H]	Turn a figure onto itself so that congruent pieces can be counted instead of measured one by one.
A Bamboo That Bends Over [H]	Split a bent upright into a standing part and a leaning part, which form the legs and hypotenuse of a right triangle.

Math Kangaroo (Grades 9-12) - Combinatorics and Probability · 46 topics

Standing Together as One Block [H]	Glue the people who must stay together into a single block, arrange the blocks, then arrange the people inside the block.
Two Who Refuse to Stand Together [H]	Count every arrangement and subtract those in which the forbidden pair is adjacent, adding back whatever a double subtraction removed twice.
Seats That Are Already Reserved [H]	Place the people who carry a condition first, count the choices they have, then arrange everybody else freely in the seats that remain.
Boys and Girls in Alternate Places [H]	Fix the pattern of places first and count how many patterns are possible, then arrange each kind of person inside its own places.
Choosing a Team With a Condition [H]	Split a restricted selection into the groups the condition names, count each group separately and multiply, or count the complement when the condition says at least.
Choosing Seats With a Gap Between Them [H]	Count selections that must keep a gap by removing the compulsory empty places first and then choosing freely from what is left.
At Least One, Counted Backwards [H]	Count the arrangements that fail the condition and subtract them from the total, subtracting each forbidden family once and repairing the overlap.
Two Clubs and One Overlap [H]	Use that the size of a union equals the sum of the two groups minus their overlap, and read whichever of the four regions the question leaves unknown.
Three Activities at the Camp [H]	Add the three groups, subtract the three pairwise overlaps and add the triple overlap back, then read whichever region the question asks for.
Numbers Divisible by This or That [H]	Count multiples of each divisor separately and correct for the numbers counted twice, which are the multiples of the least common multiple.
House Numbers That Avoid a Digit [H]	Count the numbers that avoid the digit position by position, then subtract that count from the total to reach the numbers that contain it.
Beads of the Same Color Look Alike [H]	Divide by the factorial of each repeat count to avoid counting identical objects twice, then apply the restriction by fixing a place or by gluing a forbidden pair.
Neighbouring Stripes Must Differ [H]	Color a row one place at a time, giving the first place every color and each later place every color except the one just used.
When the Dice Cannot Be Told Apart [H]	Count unordered results as multisets, choosing how many dice show each face rather than which die shows which face.
Bracelets That Look the Same When Turned [H]	Count circular arrangements by grouping together the ones that a turn or a flip carries into each other, averaging the arrangements each symmetry leaves fixed.
Sharing Identical Sweets Between Children [H]	Hand out the compulsory items first and count the ways to place the dividing bars among what is left, subtracting the sharings that break an upper limit.
Different Toys Into Different Boxes [H]	Send each distinct item to a box independently, then repair the count by subtracting the distributions that leave a box empty.
Paying the Exact Price [H]	Count payments by fixing the number of the largest coins first and counting the ways the rest of the price can be made from the smaller coins.
Hopping Up a Staircase [H]	Count the ways to reach a step by adding the counts for the steps a single hop could have come from, or choose the places of the long hops directly.
Shortest Routes Through the Streets [H]	Count shortest grid routes as choices of which hops go east, splitting the route at a compulsory corner and subtracting the routes that use a forbidden corner or block.
Sliding Down a Triangle of Cells [H]	Count the routes into each cell by adding the routes into the two cells above it, which makes the row of counts the binomial coefficients.
Handshakes That Did Not Happen [H]	Count every possible pair and subtract the pairs inside each group that never shake hands, and read the group size backwards when the total is given instead.
Corners, Diagonals and Triangles [H]	Turn a question about diagonals or triangles into a choice of corners, since each crossing point comes from four corners and each triangle from three.
Dots, Lines and Triangles [H]	Count pairs and triples of dots, then correct for a group of dots on one line, which contributes a single line and no triangles at all.
Two Dice and One Clever Event [H]	Work in the sample space of ordered rolls, count the outcomes that satisfy the event directly or by its complement, and divide by the size of that space.
Two Spinners at the Fair [H]	List the equally likely pairs of results from the two spinners and count the pairs in which the stated event happens.
Two Socks Out of the Drawer [H]	Take the unordered selections as the sample space, count the selections of each color pattern with combinations and divide.
Sitting Next to a Friend by Chance [H]	Compare the number of arrangements in which the friends are together with the number of all arrangements, or count the places one friend can take beside the other.
Heads, Tails and Runs [H]	Treat the tosses as equally likely strings of heads and tails, count the strings with the stated pattern and divide by two to the power of the number of tosses.
What the Extra Information Changes [H]	Throw away the outcomes ruled out by the information given, then count the favourable outcomes among the survivors only.
Which Bag Did the Sweet Come From? [H]	Split the experiment by which bag was used, weigh each branch by its own chance, and compare one branch with the total to answer the backwards question.
Given at Least One Is a Girl [H]	List the equally likely families as strings, keep only those matching the information given, and count the favourable ones among them.
Tokens With Two Colored Faces [H]	Make the equally likely outcomes the faces rather than the tokens, since a token with two red faces can show red in two ways.
Is the Game Worth the Ticket? [H]	Average the prizes weighted by their chances to get the expected prize, compare it with the price of a go, and set the two equal to make the game fair.
The Average of the Better Roll [H]	Add the score of every equally likely outcome and divide by the number of outcomes, since each outcome carries the same weight.
Counting the Average One Item at a Time [H]	Add the separate chances of the individual items, because an average total is the sum of the averages even when the items are not independent.
How Many Draws Until the Prize? [H]	Use the symmetry of a shuffled row to place the winners evenly, or sum the chances of still searching when the drawing puts every ticket back.
The Number That Survives on the Board [H]	Find the quantity the move never changes, evaluate it at the start and read the final number off that unchanged quantity.
Chameleons That Change Color [H]	Track the differences of the three counts modulo three, since a meeting changes each count by one or two and leaves those differences unchanged.
Dominoes on a Board With Holes [H]	Color the board like a chessboard: every domino covers one square of each color, so the smaller color count bounds how many dominoes can fit.
Where the Jumping Kangaroo Can Land [H]	Every landing point is a whole number of right jumps minus a whole number of left jumps, so only multiples of the greatest common divisor are reachable.
How Many to Be Certain [H]	Build the unluckiest draw that still fails the requirement, count how many socks it contains and take one more.
The Biggest Set With No Forbidden Pair [H]	Split the numbers into the chains or pairs the forbidden relation creates, then take the largest allowed share of each chain independently.
The Fewest Questions That Always Work [H]	Count how many different answer patterns a plan can produce and demand at least as many patterns as possibilities, which fixes the smallest number of steps.
Splitting a Number for the Biggest Product [H]	Keep the parts as equal as possible, and when the number of parts is free use as many threes as possible with at most two twos.
The Most and the Fewest Friendships [H]	Count each friendship from both ends so that the sum of the numbers of friends is twice the number of friendships, then push that sum to its limit.

Math Kangaroo (Grades 9-12) - Functions, Logic and Advanced Reasoning · 46 topics

Running a Number Machine Twice [H]	Composing a linear machine with itself, forwards to find the output and backwards to find the input.
Two Machines, Two Orders [H]	Composition is not commutative: f(g(x)) and g(f(x)) are compared and their difference is used to recover an unknown input.
Following an Arrow Diagram [H]	A function on a finite set read from a diagram of arrows, iterated to find cycle lengths and points fixed by a composition.
Rules That Undo Themselves [H]	A reflection rule x to k minus x is its own inverse, so an even number of applications returns the start and two different reflections compose to a shift.
The Number a Machine Leaves Alone [H]	A fixed point satisfies f(x) = x, and one fixed point together with one other value determines a linear rule completely.
Choosing the Rule That Undoes It [H]	The inverse of a chain of operations reverses the order of the steps and replaces each step by its opposite.
Substituting the Reciprocal [H]	Replacing x by 1/x turns one functional equation into a pair of linear equations in f(x) and f(1/x).
Substituting the Opposite [H]	Replacing x by minus x pairs the equation with a second one, and adding or subtracting the pair isolates f(x).
A Rule That Steps by One [H]	A recurrence f(n+1) = f(n) + g(n) is summed by telescoping, which turns a step rule into a closed formula.
Adding the Inputs [H]	From f(x+y) = f(x) + f(y) + c one value of f generates every other by repeated substitution of y = 1.
A Rule Linking Two Shifts [H]	A relation between f(x) and a shifted value forces a period, after which only the remainder of the argument matters.
How Many Times the Line Meets the Graph [H]	The number of common points of a piecewise linear graph and a horizontal line is the number of solutions of an absolute-value equation.
Where the Marked Point Lands [H]	A transformation of a graph moves each of its points in a predictable way, with the change inside the bracket acting on x and the change outside acting on y.
Reading a Journey Graph [H]	A distance-time graph is read stage by stage: heights give positions, differences give distances hopped, and steepness gives speed.
Two Equal Heights Fix the Axis [H]	Two inputs with the same value on a parabola are mirror images, so their midpoint is the axis of symmetry and every other value comes in a mirrored pair.
Counting the Whole Numbers That Fit [H]	An absolute-value inequality describes an interval, so counting its whole-number solutions means counting integers between two endpoints.
Solutions That Come in Mirror Pairs [H]	The solutions of an absolute-value equation are symmetric about a center, so their sum follows from the center alone without solving.
A Tariff in Two Pieces [H]	A charge with a fixed part and a rate for the rest is a piecewise linear function, read forwards for a cost and backwards for a distance.
A Function Given in Two Pieces [H]	A piecewise rule must be evaluated on the branch that owns the input, and each branch contributes its own solutions to an equation or inequality.
Lattice Points Inside a Diamond [H]	The region described by an absolute-value inequality is counted row by row, each row contributing an arithmetic number of lattice points.
Adding the Digits Again and Again [H]	Repeatedly replacing a number by its digit sum reaches the digital root, which is the number reduced modulo nine with nine in place of zero.
Where a Repeating Pattern Lands [H]	A repeating pattern is read by the remainder of the position, and two patterns of different lengths agree with the period of their least common multiple.
A Frog Hopping Round a Ring [H]	Repeated equal jumps round a ring of m stones visit exactly the multiples of the greatest common divisor of m and the jump length.
Iterating a Halving and Adding Rule [H]	Iterating a rule on whole numbers eventually repeats, and the sequence consists of a tail followed by a cycle that decides every later term.
A Sequence That Comes Back Round [H]	The recurrence a(n+2) = a(n+1) minus a(n) repeats with period six and every six consecutive terms add to zero.
Counting the Liars [H]	Statements about how many of the speakers are lying are tested against every possible number of liars, and only one number survives.
Only Some of the Statements Are True [H]	When the number of true statements is known, each candidate is tested against all of them and only one candidate gives the right tally.
Exactly One of Them Is Telling the Truth [H]	Each suspect is assumed guilty in turn and the number of true statements is counted, which leaves exactly one possible culprit.
Putting Them in Order [H]	Comparative clues are combined by testing orders against every clue until only the possible finishing orders remain.
Filling the Gap in a Tournament Table [H]	A tournament table obeys global identities: every match awards three points unless it is drawn, and the wins and losses in the table must balance.
Plus and Minus Signs in a Row [H]	Changing a plus sign to a minus changes the total by an even amount, so the parity of the total is fixed by the sum of the numbers.
Dominoes on a Damaged Board [H]	Every domino covers one dark and one light square, so the number that fits is limited by the smaller of the two color counts.
Dark Squares and Where a Hopper Can Land [H]	Chessboard coloring counts the squares of one color and also fixes the parity of the squares a hopper can reach in a given number of steps.
Handshakes and Odd Numbers [H]	Every handshake is counted by both of its participants, so the total of all handshake counts is twice the number of handshakes and is always even.
Turning Cups Over in Groups [H]	Turning over a fixed number of cups changes the count of upright cups by an amount of fixed parity, which decides what is reachable and in how few moves.
Two Numbers Replaced by One [H]	Each move of the process changes the total on the board by the same fixed amount, so the final number depends only on the starting total and the number of moves.
Splitting Piles and Scoring Products [H]	The total score of a splitting or merging process counts every pair of stones exactly once, so it does not depend on the choices made.
Chameleons That Change Color [H]	A meeting changes each color count by minus one or plus two, so the differences of the counts keep their remainders on division by three.
Adding One to Two of Them [H]	Each move raises the total by a fixed amount, so the reachable common value is limited by the total and by the fact that one number can gain at most one per move.
What Two Jugs Can Measure [H]	Every amount obtainable with two jugs is a whole number of times their greatest common divisor, and every such amount up to the larger capacity is obtainable.
Socks in the Dark [H]	A guarantee must survive the worst possible order of draws, so the answer is one more than the largest number of socks that can still fail.
Turning Cards Until Two Match [H]	The guaranteed number of draws is one more than the size of the largest collection that avoids the required pattern.
How Many Boxes Must Be Opened [H]	A guarantee over unknown boxes is decided by the worst choice of boxes, so the count comes from the smallest possible haul rather than the typical one.
Questions That Halve the Search [H]	Each yes or no answer can at best halve the number of possibilities and each weighing can at best divide it by three, which fixes how many are needed in the worst case.
Terms That Pair Up to One [H]	When f(x) + f(1 - x) is constant, a long sum is finished by pairing the first term with the last rather than by evaluating any term.
Odd Powers Cancel in Pairs [H]	A polynomial in odd powers alone changes sign when x is replaced by minus x, so the unknown coefficients cancel when the two values are added.

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 · 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.

Linear Functions · 2 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.

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.

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 · 7 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.
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 · 4 topics

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.
Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

Trigonometry · 4 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.
The Pythagorean Identity	sin²θ + cos²θ = 1 links sine and cosine of the same angle.
Basic Trig Identities	Quotient, reciprocal, and even-odd identities.

Math Olympiad: MOEMS & AMC 8 · 6 topics

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

Math Kangaroo: Student - Deep (Grades 11-12) · 27 topics

Sum of Squares of Roots	r² + s² = (r+s)² − 2rs from Vieta.
Recovering a Quadratic	Two points fix b and c.
Factorial Telescoping	k·k! = (k+1)! − k!.
Linear Recurrences	Iterate aₙ₊₁ = 2aₙ + q.
Periodic Recurrences	1/(1 − x) cycles with period 3.
Continued Fractions	Simplify from the bottom up.
Multiplicative Functional Equation	f(x+y) = f(x)f(y) ⇒ f(n) = f(1)ⁿ.
Finite Differences	Constant second differences ⇒ quadratic.
Minimizing a Sum of Distances	The median point minimizes total distance.
Rationalizing Telescope	1/(√k+√(k+1)) = √(k+1) − √k.
Sum of Coefficients	Substitute x = 1.
Linear Congruences	Multiply by the inverse of a.
Counting Perfect Squares	There are ⌊√N⌋ squares up to N.
Last Two Digits	Work modulo 100.
Sum of Cubes	Σk³ = (n(n+1)/2)².
Frobenius (Chicken McNugget)	Largest unmakeable = ab − a − b.
Permutations of a Multiset	n! divided by each repeat factorial.
Stars and Bars	Solutions = C(n + k − 1, k − 1).
Inclusion-Exclusion (Two Sets)	|A∪B| = |A| + |B| − |A∩B|.
Inclusion-Exclusion (Three Sets)	Add singles, subtract pairs, add the triple.
Derangements	Dₙ = (n−1)(Dₙ₋₁ + Dₙ₋₂).
Pigeonhole with Pairs	n complementary pairs, so n+1 forces a match.
Paths Through a Point	Multiply the two leg counts.
At Least One (Committees)	All minus the all-male committees.
Counting Functions	Each input independently picks an output: kⁿ.
Point-to-Line Distance	d = |ax₀ + by₀ + c| / √(a² + b²).
Rotation by 90°	(x, y) ↦ (−y, x).

🖨 Print / save as PDF ← All courses