MATHCOUNTS (Grades 6-8)

294 core topics · approximately 99 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

MATHCOUNTS - Number Theory & Algebra · 52 topics

Sum of All Divisors [H]	Add up every divisor of a number using its prime factorization.
The GCF·LCM Identity [H]	Recover a hidden number from gcd(a,b)·lcm(a,b) = a·b.
Counting Square Divisors [H]	Count the divisors of n that are themselves perfect squares.
Trailing Zeros of a Factorial [H]	Trailing zeros come from factors of 10 = 2·5, and 5s are the bottleneck.
Prime Powers in a Factorial [H]	Legendre's formula counts how many times a prime divides n!.
Divisibility by 9: Missing Digit [H]	A number is divisible by 9 exactly when its digit sum is.
Divisibility by 11: Missing Digit [H]	Use the alternating digit sum to test and fill divisibility by 11.
Simultaneous Remainders [H]	Find a number matching two remainder conditions at once.
Modular Inverses [H]	Find x whose product with a is 1 more than a multiple of m.
Last Two Digits of a Power [H]	Track a power modulo 100 by watching its two-digit endings cycle.
Counting Relatively Prime Numbers [H]	Euler's totient counts integers up to n that are coprime to n.
Counting Primes [H]	Sieve or test-divide to count primes up to a bound.
Perfect Squares in a Range [H]	Count squares in [a,b] with floor square roots at the endpoints.
Multiples by Inclusion-Exclusion [H]	Count 'divisible by a or b' by adding and removing the overlap.
Summing Multiples [H]	Sum the multiples of k up to N as k times a triangular number.
GCF of Three Numbers [H]	Extend the greatest common factor to three numbers by pairing.
LCM of Three Numbers [H]	Build the least common multiple of three numbers step by step.
Converting to Another Base [H]	Repeatedly divide to rewrite a base-ten number in a new base.
Reversing a Two-Digit Number [H]	The gap between a two-digit number and its reverse is 9 times a digit gap.
Consecutive Integer Products [H]	Recover consecutive integers from their product using a square estimate.
Systems: Value of x + y [H]	Solve a 2×2 linear system, then combine the results.
Age Problems [H]	Set both ages t years later and translate the 'times as old' condition.
Triangular Numbers [H]	Invert the formula 1 + 2 + ⋯ + n = n(n+1)/2 to find n.
Finding a Missing Data Value [H]	Use total = mean × count to solve for an unknown entry.
Weighted Averages [H]	Average with weights: multiply each value by its weight before dividing.
Geometric Sequence Terms [H]	Each term multiplies the previous by the common ratio.
Finite Geometric Series [H]	Sum a geometric series with a(rⁿ − 1)/(r − 1).
Infinite Geometric Series [H]	A ratio with |r| < 1 gives a finite total a/(1 − r).
Vieta: Sum of Squared Roots [H]	Combine root sum and product without ever finding the roots.
Sum and Difference to Product [H]	Recover two numbers from their sum and difference, then multiply.
Solving Exponential Equations [H]	Match bases to read off the exponent.
Laws of Exponents [H]	Add exponents when multiplying, subtract when dividing (same base).
Defined Operations [H]	Follow an invented rule literally, substituting the given values.
Function Composition [H]	Evaluate the inner function first, then feed the result into the outer.
Undoing a Linear Function [H]	Solve f(x) = y for x by reversing the operations.
Absolute-Value Equations [H]	An absolute-value equation splits into two cases with a symmetric sum.
The Floor Function [H]	The floor is the quotient of integer (long) division.
Telescoping Sums [H]	Split each term so consecutive pieces cancel in a chain.
Sum of Consecutive Squares [H]	Use 1² + ⋯ + n² = n(n+1)(2n+1)/6.
Summing Even Numbers [H]	The first n even numbers add to n(n+1).
Net Rate: Fill and Drain [H]	Combine an inflow and an outflow by subtracting their rates.
Approaching Travelers [H]	Two objects closing in shrink the gap at their combined speed.
Diluting a Solution [H]	Adding water keeps the solute fixed while the total volume grows.
Inverse Proportion [H]	More workers means fewer days: the product worker·days stays fixed.
What Percent Is It? [H]	The percent one number is of another is their ratio times 100.
Percent of a Percent [H]	Chain percentages by multiplying their decimal forms.
Three-Part Ratios [H]	Split a total into ratio parts by finding the value of one share.
Quadratic Patterns [H]	Constant second differences reveal a quadratic rule to extend.
Counting Whole-Number Solutions [H]	Count non-negative solutions of ax + by = n by stepping through one variable.
The Remainder Theorem [H]	Dividing a polynomial by (x − a) leaves the remainder f(a).
Identifying Primes [H]	Spot the prime by ruling out numbers with small factors.
Divisibility Rules [H]	Apply quick tests for 4, 6, 8, and 9 to pick the multiple.

MATHCOUNTS - Geometry & Counting · 51 topics

Area from Coordinates (Shoelace) [H]	Find a triangle's area directly from its vertex coordinates.
Area of a Trapezoid [H]	Average the parallel sides, then multiply by the height.
Area from Diagonals [H]	A rhombus's (or kite's) area is half the product of its diagonals.
Composite (L-Shaped) Areas [H]	Break a compound figure into rectangles and add or subtract.
Fraction of a Region Shaded [H]	Compare the shaded area to the whole as a reduced fraction.
Area of an Equilateral Triangle [H]	An equilateral triangle of side s has area (s²√3)/4.
Heron's Formula [H]	Find a triangle's area from its three sides.
Area of a Ring (Annulus) [H]	Subtract the inner circle's area from the outer one.
Volume of a Cylinder [H]	A cylinder's volume is the base circle's area times the height.
Volume of a Cone [H]	A cone holds one-third of the cylinder with the same base and height.
Volume of a Sphere [H]	A sphere of radius r has volume (4/3)πr³.
Surface Area of a Cylinder [H]	Add the two circular caps to the wrapped-around rectangle.
Space Diagonal of a Box [H]	The 3-D diagonal squared is the sum of the squared edge lengths.
Space Diagonal of a Cube [H]	A cube of edge s has space diagonal s√3, so its square is 3s².
Interior Angle of a Regular Polygon [H]	Split the total interior angle sum equally among the vertices.
Sum of Interior Angles [H]	Any convex n-gon splits into n−2 triangles.
Exterior Angle of a Regular Polygon [H]	The exterior angles of any polygon always total 360°.
Counting Diagonals [H]	Each vertex connects to all but itself and two neighbours.
Inscribed Angle Theorem [H]	An inscribed angle is half the arc it opens onto.
Clock Hand Angles [H]	Track each hand's position in degrees, then take the difference.
The Triangle Inequality [H]	The third side lies strictly between the difference and sum of the others.
Midpoints [H]	A midpoint's coordinates are the averages of the endpoints'.
Slope Between Two Points [H]	Slope is the rise over the run between two points.
Finding the y-Intercept [H]	Get the slope, then back up to where x = 0.
Squared Distance [H]	The squared distance is the sum of the squared coordinate gaps.
Reflections in the Plane [H]	Reflecting flips a sign or swaps the coordinates.
The Centroid of a Triangle [H]	A triangle's centroid averages its three vertices.
30-60-90 Triangles [H]	The sides of a 30-60-90 triangle are in the ratio 1 : √3 : 2.
Similar Triangles: Missing Side [H]	Corresponding sides of similar figures share one scale factor.
Shadow (Similar-Triangle) Problems [H]	Height and shadow length stay in a fixed ratio for everything at once.
Tangent Length from an External Point [H]	A tangent meets the radius at a right angle, forming a right triangle.
Intersecting Chords [H]	When two chords cross, the products of their pieces are equal.
Recognizing Right Triangles [H]	A triangle is right exactly when its sides fit the Pythagorean theorem.
Arrangements with Repeated Letters [H]	Divide the total arrangements by the factorials of the repeat counts.
Arrangements with People Together [H]	Glue the pair into one block, arrange, then swap the pair.
Circular Permutations [H]	Fix one person to remove rotational duplicates.
Lattice Paths [H]	Every shortest path is a rearrangement of the required right and up moves.
Connecting Points in Pairs [H]	Each segment is an unordered pair of points: C(n,2) of them.
Counting Subsets [H]	Each element is independently in or out, giving 2ⁿ subsets.
Committees with a Required Member [H]	Seat the required person first, then fill the rest.
Choosing from Two Groups [H]	Independent choices multiply.
Counting Rectangles in a Grid [H]	A rectangle is set by choosing two vertical and two horizontal lines.
The Multiplication (Product) Rule [H]	Multiply the number of choices at each independent slot.
Forming Numbers Without Repetition [H]	Fill each place in turn with one fewer choice remaining.
The Pigeonhole Principle [H]	Prepare for the worst case, then one more forces a match.
Complementary Counting [H]	Count the easy opposite event and subtract from 1.
Probability Without Replacement [H]	Multiply stage probabilities, updating the counts as you draw.
'At Least One' by Complement [H]	'At least one' is 1 minus the probability of none.
Geometric Probability on a Segment [H]	On a line, probability is favorable length over total length.
Geometric Probability by Area [H]	In a region, probability is favorable area over total area.
Expected Value [H]	Expected value is the probability-weighted average of the payoffs.

MATHCOUNTS - Advanced Problem Solving · 51 topics

Vieta on a Cubic [H]	Sum of squares of the roots from the cubic's coefficients.
Sum of Reciprocal Roots [H]	1/p + 1/q rewrites as (p+q)/(pq), both known from Vieta.
The x + 1/x Trick [H]	Square the given expression to reach x² + 1/x².
Cubing x + 1/x [H]	x³ + 1/x³ comes from the cube of x + 1/x.
Infinite Nested Radicals [H]	Name the whole expression and solve the equation it satisfies.
Telescoping Products [H]	Factor each term as a difference of squares so factors cancel.
Telescoping with Partial Fractions [H]	Split 1/(k(k+2)) into halves that cancel two steps apart.
Alternating Sum of Squares [H]	Pair consecutive terms to collapse the alternating sum.
Sum of Consecutive Cubes [H]	The sum of the first n cubes is the square of the nth triangular number.
Telescoping Factorials [H]	Use k·k! = (k+1)! − k! so the sum collapses.
The Hockey-Stick Identity [H]	A diagonal sum in Pascal's triangle folds into one binomial coefficient.
Maximizing a Product (AM-GM) [H]	A product with a fixed sum is largest when the factors are as equal as possible.
Minimizing x + N/x (AM-GM) [H]	AM-GM makes x + N/x smallest when the two terms are equal.
Maximum Area for a Fixed Perimeter [H]	Among rectangles of fixed perimeter, the most square-like has the largest area.
Extreme Value of a Quadratic [H]	Complete the square (or find the vertex) to read off the extreme value.
Fibonacci-Style Recurrences [H]	Add the two previous terms repeatedly to reach the target index.
Linear Recurrences [H]	Apply the rule aₙ₊₁ = p·aₙ + q step by step.
Inclusion-Exclusion (Three Sets) [H]	Subtract the multiples of each prime, add back the double-counted overlaps.
The Chinese Remainder Theorem [H]	Three coprime remainder conditions pin down one class mod their product.
Fermat's Little Theorem [H]	Reduce a huge exponent modulo p−1 before powering.
Remainders of Sums of Powers [H]	Reduce each power modulo m separately, then add.
Dividing in Modular Arithmetic [H]	Multiply by the modular inverse to 'divide' modulo a prime.
Solving for an Unknown Base [H]	Set up the place-value equation and solve for the base.
Smallest Multiplier for a Square [H]	Supply the primes whose exponents are odd to even out the factorization.
Counting Cube Divisors [H]	A cube divisor uses each prime to a multiple-of-three power.
Counting Pairs with a Given LCM [H]	Handle each prime independently: at least one number must hit the top power.
Sums of Consecutive Integers [H]	Representations correspond to the odd divisors of the number.
Reversing a Handshake Count [H]	Solve n(n−1)/2 = games for n with a square-root estimate.
Stars and Bars [H]	Distribute n identical units into k slots with dividers.
Lattice Paths Avoiding a Point [H]	Subtract the paths through the forbidden point from all paths.
The Binomial Theorem [H]	Each term of (x + c)ⁿ is C(n,k) xᵏ cⁿ⁻ᵏ.
Subsets of Even Size [H]	Exactly half of all subsets have an even number of elements.
Counting Coin Combinations [H]	Fix the larger coins, then the rest must fill in - count systematically.
Lattice Points in a Disk [H]	Count integer points column by column within the circle.
Figurate Numbers [H]	Polygonal numbers follow fixed quadratic formulas.
Comparing Powers [H]	Bring powers to a common base or exponent to compare their sizes.
Expected Value by Linearity [H]	The expected sum is the sum of the expected values.
Expected Value of a Maximum [H]	Average the maximum over every equally likely pair of rolls.
Probability All Are Different [H]	Multiply the shrinking chance each new roll avoids the earlier ones.
Probability via Geometric Series [H]	Sum the chances of success on the 1st, 3rd, 5th, … trials.
The Law of Total Probability [H]	Weight each scenario's probability by the chance of that scenario.
The Inradius Formula [H]	A triangle's area equals its inradius times its semiperimeter.
The Circumradius Formula [H]	The circumradius is the product of the sides over four times the area.
Shortest Surface Path (Unfolding) [H]	Flatten the box so the path becomes a straight line, then use Pythagoras.
When Clock Hands Overlap [H]	The minute hand laps the hour hand at a steady relative rate.
Three-Variable Systems [H]	Combine equations cleverly to isolate one variable at once.
Repeated Replacement Mixtures [H]	Each replacement scales the remaining original by the same fraction.
Average Speed Over Legs [H]	Average speed is total distance over total time - never average the speeds.
Combined Work Rates (Three) [H]	Rates add; the combined time is the reciprocal of the total rate.
Reversing Percent Changes [H]	Divide out the multipliers to undo a chain of percent changes.
Repeating Decimals to Fractions [H]	A repeating block over as many 9s converts a repeating decimal to a fraction.

MathCounts Sprint Round - Deep · 46 topics

Decimals to Common Fractions [H]	Read a terminating decimal as hundredths or thousandths and reduce it in one step.
Percents That Are Really Eighths [H]	Replace an awkward percent by the simple fraction it equals before multiplying.
Ordering Fractions, Decimals and Percents [H]	Convert every number to one common form before comparing sizes.
What Fraction Is Left [H]	Track the fraction that survives each stage and multiply those survivors together.
Net Effect of Two Percent Changes [H]	Multiply the change factors instead of adding the percents.
Keeping the Total Spending Fixed [H]	Total spending is price times quantity, so the two change factors must multiply to the target factor.
Percent More as a Fraction [H]	Turn every comparison into a multiplier so the two directions of a comparison stay distinct.
From a Ratio to the Whole [H]	Name the common part size in a ratio and read every other quantity as a multiple of it.
Better Buy by Unit Rate [H]	Divide price by size to compare packages of different sizes on equal terms.
Scaling Workers, Time and Output [H]	Reduce the given information to output per machine per minute, then rebuild any other combination.
Solving a Proportion Quickly [H]	Cross multiply once, and recognize the middle term of a continued proportion as a geometric mean.
The Gap Between Mean and Median [H]	The mean uses every value while the median uses only position, so the two separate whenever the data is lopsided.
Statistics from a Frequency Table [H]	A frequency table is a compressed list, so every total must be weighted by its frequency.
How One Value Moves the Mean [H]	Work with the total rather than the mean: adding or removing a value changes the total by exactly that value.
Evaluating a Nested Fraction [H]	Simplify a continued fraction from the innermost denominator outward, and undo it in the same order when solving.
Evaluating by Regrouping [H]	Reshape an arithmetic expression with the distributive property or the difference of squares before computing anything.
Scaling a Given Relation [H]	One linear relation fixes the value of every constant multiple of itself, so no individual variable is ever needed.
Values of Expressions from a Ratio [H]	A ratio fixes the value of any expression whose numerator and denominator are both sums of first-degree terms.
Clearing Fractions in an Equation [H]	Multiply every term by the least common denominator so the equation becomes whole-number linear.
Variables on Both Sides [H]	Collect the variable terms on one side and the constants on the other, and read a vanished variable as a statement about the coefficients.
Solving a Formula for a Letter [H]	Isolate the requested letter by undoing the operations applied to it, treating every other letter as a constant.
Equations with Decimal Coefficients [H]	Clear the decimals by multiplying through by a power of ten, or read each decimal as the simple fraction it equals.
Terms of an Arithmetic Sequence [H]	The nth term is the first term plus n minus one common differences, so count the gaps rather than the terms.
Position in a Repeating Pattern [H]	Divide the position by the length of the repeating block and let the remainder locate the item inside the block.
Each Term from the One Before [H]	Apply the stated rule one step at a time, and invert the rule step by step when the sequence must be run backwards.
Counting Divisors Fast [H]	Read the divisor count off the exponents of the prime factorization instead of listing divisors.
The Largest Prime Factor [H]	Factor the expression algebraically before multiplying it out, then take the largest prime among the pieces.
Rectangles with a Given Area [H]	Each rectangle with a given whole-number area corresponds to one factor pair of that area.
Between Perimeter and Area [H]	Recover the two side lengths first, because perimeter and area are two different functions of the same pair.
A Cube's Volume and Surface Area [H]	Travel between a cube's volume and its surface area through the edge length, and scale each measure by the matching power.
A Box from Its Face Areas [H]	Multiply the three face areas that meet at a corner to obtain the square of the volume.
Perimeter of a Curved Region [H]	Add the fraction of the circumference that the arc represents to the straight edges that close the region.
Angles Given by Expressions [H]	Set the sum of the labeled expressions equal to the angle total the configuration forces, then finish the question that was asked.
Angles from Parallel Lines [H]	Corresponding and alternate angles are equal while same-side interior angles are supplementary, and an auxiliary parallel line resolves a bend.
Angles Inside a Triangle [H]	The three interior angles total 180 degrees, so an exterior angle equals the sum of the two remote interior angles.
How Many Sides from an Angle [H]	The exterior angles of any polygon total 360 degrees, so one exterior angle divides 360 to give the number of sides.
Two-Digit Numbers with a Given Property [H]	Count by running through the allowed leading digit and testing what the condition forces the other digit to be.
Games in a Tournament [H]	Count a knockout by eliminations and a round robin by pairs, then add the stages of a mixed format separately.
Coloring a Strip so Neighbours Differ [H]	Color the stripes left to right: the first is free and each later stripe avoids only the one before it.
Probability of a Single Draw [H]	Divide the number of favorable items by the total number of items, and remember that adding items changes both counts.
Probability of a Sum on Two Dice [H]	Count ordered outcomes out of the full grid of possibilities rather than counting sums.
Probability from a Two-Way Table [H]	Read the favorable count off the table and divide by the grand total, subtracting the overlap when two categories are joined by the word or.
Converting Through Several Units [H]	Multiply by conversion factors written so the unwanted units cancel, including both units of a rate.
Square and Cubic Unit Conversions [H]	A length conversion factor must be squared for areas and cubed for volumes.
How Far Off Is the Estimate [H]	Compare a rounded estimate with the exact value to see how large the rounding error can grow.
How Many Digits, How Many Zeros [H]	Pair every 2 with a 5 to build powers of ten, then count the digits of the leftover factor and the zeros it sits on.

MathCounts Target Round - Deep · 47 topics

Two Purchases, Two Prices [H]	Two mixed purchases of the same two items determine both unit prices by elimination.
Coins by Count and Value [H]	A count equation and a value equation pin down how many coins of each denomination there are.
Numbers from Pairwise Sums [H]	Adding all the given group sums recovers the grand total, and each number is that total minus one group sum.
Balance-Scale Systems [H]	Each balance is an equation, so converting every object into one common unit answers the question.
Work With a Head Start [H]	Rates add, so the fraction of the job already done decides how much time the remaining rate must cover.
Work Rates Run Backwards [H]	Subtracting a known rate from a combined rate recovers the rate, and therefore the time, of the other agent.
Sharing Pay in Proportion to Work [H]	Pay splits in the ratio of the work done, which for equal working times is the ratio of the rates.
Catching Up After a Head Start [H]	A head start is a fixed gap that closes at the difference of the two speeds.
Blending Two Solutions [H]	The amount of pure acid is conserved, so acid before equals acid after in every mixing step.
Evaporation and Concentration [H]	Evaporating water leaves the amount of solute unchanged, so the concentration rises exactly as the volume falls.
Alloys and the Mixing Ratio [H]	The masses of two alloys mix in the inverse ratio of their distances from the target percent.
Pricing a Blend [H]	The value of a blend per pound is the weighted average of the two prices, and a selling price marked up by a known percent reveals that cost.
Recovering an Original Price [H]	A discount and a tax are successive multipliers, so dividing the final amount by both recovers the original price.
Percent More Against Percent Less [H]	A percent is always taken of a stated base, so reversing the comparison changes the base and changes the percent.
Undoing a Chain of Percent Changes [H]	Successive percent changes multiply, so an unknown starting value or an unknown middle percent is recovered by division.
Group Percentages Before and After [H]	A percent of a group is a count, so a change is tracked by writing the new count over the new total.
Areas of Similar Figures [H]	Areas of similar figures are in the ratio of the squares of corresponding lengths.
Areas Cut by a Point on a Side [H]	Triangles that share an altitude have areas in the ratio of their bases.
A Square Cut by Points on Its Sides [H]	Subtracting the three corner right triangles from the square gives the area of the inner triangle.
The Four Triangles of a Trapezoid [H]	The diagonals of a trapezoid cut it into two similar triangles and two triangles of equal area that are the geometric mean of them.
A Rectangle Inscribed in a Right Angle [H]	The far vertex lies on the hypotenuse, so its coordinates satisfy the intercept equation of that line.
A Missing Endpoint and a Distance [H]	The midpoint is the average of the endpoints, so doubling the midpoint and subtracting one endpoint gives the other.
Midsegments on the Coordinate Plane [H]	Each midsegment is parallel to a side and half its length, so the medial triangle has half the perimeter of the original.
Points Equidistant From Two Others [H]	Setting two squared distances equal cancels the squared unknowns and leaves a linear equation.
The Fourth Vertex of a Parallelogram [H]	In a parallelogram the diagonals bisect each other, so opposite vertices have equal coordinate sums.
The Smallest Number With a Given Divisor Count [H]	The divisor count is the product of one more than each exponent, so the smallest number puts the largest exponents on the smallest primes.
Counting Residues in a Range [H]	Numbers with a fixed remainder form an arithmetic sequence, and two independent remainder conditions merge into one modulus equal to the product.
Counting Factor Pairs [H]	Divisors pair off around the square root, so unordered factorisations number half the divisor count.
Repeating Signals That Coincide [H]	Events that repeat coincide on a cycle equal to the least common multiple of their periods, shifted by their starting offsets.
Arrangements With a Forbidden Adjacency [H]	Count all arrangements and subtract those in which the forbidden pair is glued together.
Three-Digit Numbers With Digit Conditions [H]	Counting the numbers that avoid a digit is direct, so the numbers that contain it come from the complement.
Committees With a Restriction [H]	Subtracting the committees that violate a condition from all committees is faster than building the valid ones.
Lattice Paths With Streets Closed [H]	Routes through a corner multiply the counts of the two legs, and a closed street is removed by subtracting the routes that use it.
Identical Objects Under a Cap [H]	Bars between stars count the distributions, and an upper limit is imposed by subtracting the cases that break it.
Draws Without Replacement [H]	Counting the favorable committees of marbles and dividing by all committees settles a draw without replacement.
Compound Events on Two Dice [H]	Every ordered pair of faces is equally likely, so a compound event is counted in that grid of outcomes.
Heads, Tails and Forbidden Runs [H]	Each sequence of flips is equally likely, so a probability is a count of sequences over two to the number of flips.
Independent Events, Forwards and Backwards [H]	For independent events the probability that neither occurs is the product of the two complements.
Arithmetic Sums With a Missing Piece [H]	The sum of an arithmetic series is the number of terms times the average of the first and last terms.
An Arithmetic Sequence From Two Terms [H]	The difference between two given terms divided by the gap in their positions is the common difference.
Geometric Terms and Sums [H]	The ratio of two terms of a geometric sequence is the common ratio raised to the gap in their positions.
The Bouncing Ball [H]	Rebound heights form a geometric sequence, and every rebound is traveled twice, once up and once down.
A Tapering Stack [H]	A stack whose rows decrease by a fixed amount is an arithmetic series, so its total is the number of rows times the average row.
Fencing Against a Wall [H]	Area against a wall is a quadratic in the width, so the maximum sits at the vertex, halfway between the roots.
Price Against Demand [H]	Revenue is price times quantity, a quadratic in the number of price increases, so its extremum sits at the vertex.
Shortest Reflected Paths [H]	Reflecting one endpoint across the line straightens the path, so the minimum equals the straight distance between the reflected points.
Least Cost From Two Pack Sizes [H]	Only whole packs may be bought, so the least cost is found by testing every admissible number of large packs.

MathCounts Team & Countdown Round - Deep · 47 topics

Locating a Term by Its Value [H]	Find the position of a value in an arithmetic sequence by dividing the gap by the common difference.
Reversing a Quadratic Pattern [H]	Work backwards in a constant-second-difference sequence to find a term number.
Position in a Repeating Block [H]	Divide by the block length: the quotient counts whole blocks and the remainder locates the term.
Units Digits of Powers and Sums [H]	Units digits of powers repeat in a short cycle, so reduce the exponent modulo the cycle length.
Grouped Alternating Sums [H]	Group the terms of a signed sum into repeating blocks and add the equal block totals.
Growing Figures Built from Toothpicks [H]	Read the constant increase from a table of figures and write the count as a linear rule.
Handshakes with Exceptions [H]	Count every pair, then subtract the pairs that do not shake hands.
Diagonal Lengths in a Regular Polygon [H]	Classify a diagonal of a regular polygon by how many vertices it skips.
Crossings of the Diagonals [H]	Every interior crossing of diagonals comes from one choice of four vertices.
Routes Forced Through a Corner [H]	A route through a required corner splits into independent legs whose counts multiply.
Paths That Stay Below the Diagonal [H]	Count monotone paths obeying a running majority condition step by step.
Round Tables with a Restriction [H]	Fix one person to kill the rotations, then impose the seating restriction.
Symmetry in Dice Sums [H]	Replacing every face value by its complement reflects a dice sum about its center.
Splitting a Group into Pairs or Teams [H]	Arrange everyone in a row, cut into groups, then divide out the orders that do not matter.
Lattice Points by Symmetry [H]	Count solutions in one quadrant and multiply by the symmetry, correcting for points on the axes.
Undoing a Chain of Operations [H]	Reverse a described process by applying the opposite operations in the opposite order.
Half of What Is Left, Plus a Few More [H]	Undo a give-away day by reversing the fixed amount first and then the fraction.
Transfers That Double What Others Hold [H]	Undo a doubling transfer by halving the receivers and returning the difference to the giver.
Reversing a Recursive Rule [H]	Run a term-to-term rule backwards by inverting it once for each step.
Counting Numbers with a Given Digit Sum [H]	Count digit triples with a fixed sum by fixing the leading digit and counting the rest.
Counting Palindromes [H]	A palindrome is built from its front half, so count the free digits only.
Numbers Against Their Reversals [H]	Subtracting a reversal cancels the middle digits and leaves a multiple of 99.
Counting a Digit's Appearances [H]	Count a digit place by place, and count numbers containing it by complement.
Digits Used in Page Numbers [H]	Group page numbers by how many digits they have and count each group separately.
Counting Weekdays in a Span [H]	Weekdays repeat every 7 days, so reduce the day count modulo 7.
When the Hands Form a Given Angle [H]	The minute hand gains on the hour hand at 11/2 degrees per minute.
A Clock That Runs Fast [H]	A clock gaining a fixed amount per hour runs on a constant ratio to true time.
Dates and the Weekdays They Fall On [H]	Shift a known date to the first of the month, then move whole months using their lengths modulo 7.
A Value That Shifts the Mean [H]	Work with totals: the mean times the count is the only quantity that matters.
Removing a Value from a Data Set [H]	Removing values subtracts from the total and from the count at the same time.
The Score Needed Next [H]	Compare the total you have with the total the target mean demands.
Making the Mean Meet the Median [H]	Locate the new value in the sorted order first, then set the mean equal to the median.
Averages of Evenly Spaced Numbers [H]	In an evenly spaced list the mean equals the middle value, so the sum is the middle value times the count.
Overlapping Regions [H]	Area covered equals the sum of the pieces minus the overlaps counted twice.
The Area of a Border [H]	A uniform border makes a larger rectangle, so subtract the inner area from the outer one.
Triangles Inside a Rectangle [H]	A triangle on a full side of a rectangle takes half the rectangle, wherever its apex sits.
Perimeter of a Staircase Figure [H]	Slide the segments of a right-angled figure outward: the perimeter matches its bounding rectangle.
Areas from Midpoints [H]	Joining the midpoints of a quadrilateral halves the area; joining a triangle's midpoints quarters it.
Regions Between Circles and Squares [H]	Decompose a circle-and-square figure into whole shapes and subtract, keeping the answer in terms of pi.
The Painted Cube [H]	Sort the unit cubes by position: corners, edges, faces, and hidden interior.
Differences of Squares [H]	x squared minus y squared factors as (x − y)(x + y), turning a subtraction into a factor pair.
Roots by Size and Units Digit [H]	Pin down a root's tens digit by size and its units digit by the last digit of the power.
Sums of Two Squares [H]	Search for representations as a sum of two squares by testing squares up to half the number.
Products Near a Round Number [H]	Write each factor as a round number plus or minus a small amount before multiplying.
Two Purchases, One Combination [H]	Combine the given equations directly to build the bundle you are asked about.
Posts, Cuts, and the Gaps Between [H]	Count the gaps first: along a line the posts number one more than the gaps, and around a loop they match.
Sums of All the Pairs [H]	Every number appears in exactly n − 1 of the pair sums, so the grand total is (n − 1) times the total.

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