Search & glossary

Every rule and concept in AMC 8 / MathCounts, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (60)

Add, then subtract the overlap: For two sets, $|A \cup B| = |A| + |B| - |A \cap B|$ .; Clever Counting & Logic › Venn Diagrams and Inclusion–Exclusion
Angle facts: A triangle's angles sum to $180^\circ$ ; a quadrilateral's to $360^\circ$ .; Geometry Essentials › Angles and Triangles
Angles, triangles, circles: Triangle angles sum to $180^\circ$ ; polygon interior sum $(n-2)180^\circ$ ; exterior angles sum to $360^\circ$ .; Formulas & Methods › Geometry
Area and circle formulas: Triangle area $= \tfrac12 bh = \tfrac12 ab\sin C$ .; Geometry Essentials › Similarity, Area, and Circles
Both hands move: Measure positions in degrees clockwise from $12$ .; Complete Syllabus & Gap Fillers › Clocks and Calendars
Casework that stays organized: Split the count into cases that do not overlap and together cover everything, then add the case counts.; Clever Counting & Logic › Counting Without Listing
Checking primality quickly: To test whether $n$ is prime you only need trial divisors up to $\sqrt n$ : if $n = ab$ with $a\le b$ then $a \le \sqrt n$ .; Number Theory Toolkit › Primes and Factorization
Choosing the method: Independent stages: multiply.; Counting & Probability › Putting It Together
Clock and calendar angles: The minute hand moves $6^\circ$ per minute, the hour hand $0.5^\circ$ per minute (and $30^\circ$ per hour).; Complete Syllabus & Gap Fillers › Gap Fillers
Clocks, calendars, patterns: Minute hand $6^\circ$ /min, hour hand $0.5^\circ$ /min.; Formulas & Methods › Geometry
Complementary counting: Count the opposite and subtract from the total: $\#(\text{at least one}) = \text{total} - \#(\text{none})$ .; Counting & Probability › Casework Without Double Counting
Count the opposite: For "at least one," compute (total) $-$ (ways it fails).; Counting & Probability › Complementary Counting and Probability
Counting and probability: Multiplication principle for independent choices; permutations $n!$ , with repeats $\tfrac{n!}{a!b!}$ ; combinations $\binom nk$ .; Formulas & Methods › Number Theory & Counting
Divisibility, primes, LCM/GCD: Divisible by $3$ / $9$ : digit sum; by $4$ / $8$ : last two/three digits; by $11$ : alternating digit sum.; Formulas & Methods › Number Theory & Counting
Enclose it in a rectangle: To find the area of a polygon drawn on a grid, surround it with the smallest rectangle whose sides run along grid lines, then subtract the pieces between the polygon and the rectangle.; Geometry Essentials › Grids and Coordinates
Estimate before you compute: Before calculating, decide roughly how big the answer is: "about $30$ ", "a bit less than half", "between $100$ and $200$ ".; Formulas & Methods › Putting It Together
Four questions before counting: Does order matter?; Clever Counting & Logic › Putting It Together
Handshakes and pairs: $n$ people each shaking every other hand once make $\dbinom{n}{2} = \dfrac{n(n-1)}{2}$ handshakes; the same count gives diagonals of a polygon, $\dfrac{n(n-3)}{2}$ , after removing the $n$ sides.; Clever Counting & Logic › Paths, Pairs and Handshakes
How many divisors a number has: Factor $n = p_1^{a_1}p_2^{a_2}\cdots$ ; then $n$ has $(a_1+1)(a_2+1)\cdots$ divisors, their sum is $\prod \dfrac{p_i^{a_i+1}-1}{p_i-1}$ , and their product is $n^{d(n)/2}$ .; Number Theory Toolkit › Counting Divisors, and the Last Digit
Lattice paths: Paths from $(0,0)$ to $(m,n)$ moving only right and up: $\dbinom{m+n}{n}$ , because a path is a word in $m$ Rs and $n$ Us.; Clever Counting & Logic › Paths, Pairs and Handshakes
Multiply along a branch, add across branches: For two-stage experiments, make the sample space visible.; Counting & Probability › Probability With Tables and Trees
Multiply independent choices: If one choice has $m$ options and a second, independent choice has $n$ options, the pair has $m\cdot n$ options.; Counting & Probability › The Fundamental Principles
Name the unknown that the question asks about: Write one sentence of the problem as one equation.; Algebra & Manipulation › Word Problems Into Equations
Parity (odd/even) arguments: The parity of a sum depends only on how many ODD terms it has — an even number of odd terms gives an even sum.; Clever Counting & Logic › Counting Without Listing
Percents and ratios: Percent change $= \tfrac{\text{new} - \text{old}}{\text{old}}$ ; a $p\%$ increase then $p\%$ decrease leaves $(1 - \tfrac{p^2}{100^2})$ of the start.; Formulas & Methods › Arithmetic, Ratios & Percents
Permutations vs. combinations: Ordered picks of $k$ from $n$ : ${}_nP_k = \dfrac{n!}{(n-k)!}$ .; Counting & Probability › The Fundamental Principles
Prime factorization is the master key: Write $n = p_1^{a_1}\cdots p_k^{a_k}$ .; Number Theory Toolkit › Primes and Factorization
Probability is a ratio of counts: For equally likely outcomes, $P(\text{event}) = \dfrac{\text{favorable}}{\text{total}}$ .; Counting & Probability › Complementary Counting and Probability
Proof (Gauss pairing): $1 + 2 + \cdots + n = \dfrac{n(n+1)}{2}$: Write the sum forwards and backwards and add columns: every column totals $n + 1$ , and there are $n$ columns, so TWICE the sum is $n(n+1)$ .; Why the Formulas Work › The Sum Formulas, Proved
Proof by picture: the first $n$ odd numbers sum to $n^2$: Each new odd number $2k+1$ wraps an L-shape around a $k \times k$ square, growing it to $(k+1) \times (k+1)$ .; Why the Formulas Work › The Sum Formulas, Proved
Proof by rearrangement: $a^2 + b^2 = c^2$: Take a big square of side $a + b$ and tuck four copies of the right triangle into its corners.; Why the Formulas Work › The Pythagorean Theorem, Proved
Proof: a triangle’s angles sum to $180^\circ$: Draw the line through one vertex PARALLEL to the opposite side.; Why the Formulas Work › The Angle Formulas, Proved
Proof: an $n$ -gon’s angles sum to $(n-2)\cdot 180^\circ$: Pick one vertex and draw all diagonals from it.; Why the Formulas Work › The Angle Formulas, Proved
Proof: exterior angles always total $360^\circ$: Walk once around the polygon.; Why the Formulas Work › The Angle Formulas, Proved
Proof: the area of a triangle is $\tfrac12bh$: Every triangle is half a parallelogram.; Why the Formulas Work › Why the Area Formulas Work
Proof: the circle area $\pi r^2$ by slicing: Cut the disc into thin sectors and interleave them.; Why the Formulas Work › Why the Area Formulas Work
Rate problems: fill in a table: distance $=$ rate $\times$ time, work $=$ rate $\times$ time.; Algebra & Manipulation › Word Problems Into Equations
Rates and work: Distance $=$ rate $\times$ time; average speed is TOTAL distance over TOTAL time, never the average of speeds ( $60$ km at $30$ km/h and $60$ at $60$ km/h averages $40$ ).; Complete Syllabus & Gap Fillers › Gap Fillers
Rates, work, and averages: Distance $= rt$ .; Formulas & Methods › Arithmetic, Ratios & Percents
Ratios and proportions: A ratio $a:b$ means the amounts are $ak$ and $bk$ .; Algebra & Manipulation › Ratios and Clever Factoring
Same base, same height, same area: Triangles sharing a base and with apexes on a line parallel to it have equal areas; a median splits a triangle into two equal areas.; Geometry Essentials › Area by Subtraction and Rearrangement
Shaded regions: subtract, do not chase: Area of a shaded region $=$ area of the enclosing shape $-$ areas of the pieces removed.; Geometry Essentials › Area by Subtraction and Rearrangement
Similar triangles: Equal-angle triangles have proportional sides; if the ratio is $k$ , areas scale by $k^2$ .; Geometry Essentials › Similarity, Area, and Circles
Simon's Favorite Factoring Trick: The expression $xy + ax + by$ factors as $(x+b)(y+a) - ab$ .; Algebra & Manipulation › Ratios and Clever Factoring
Special right triangles and triples: $45\text{-}45\text{-}90$ has sides $1:1:\sqrt2$ ; $30\text{-}60\text{-}90$ has $1:\sqrt3:2$ .; Geometry Essentials › Angles and Triangles
Split into cases that cannot overlap: Choose a feature that every arrangement has exactly one value of (the largest element, the position of a special item, the number of red balls).; Counting & Probability › Casework Without Double Counting
Statistics in one box: Mean = total $\div$ count (so "the mean of $7$ numbers is $10$ , remove one, mean becomes $9$ " means the removed number is $70 - 54 = 16$ ).; Complete Syllabus & Gap Fillers › Gap Fillers
The checklist: Arithmetic and fractions/decimals/percents; ratios, rates, and unit conversion; basic number theory (divisibility, primes, LCM/GCD, remainders, digit problems); counting (lists, complements, simple casework, permutations of small sets); probability of simpl…; Complete Syllabus & Gap Fillers › What the AMC 8 Tests
The complement trick: Sometimes the easiest way to count what you want is to count what you do NOT want: $|A| = |\text{total}| - |\text{not } A|$ .; Clever Counting & Logic › Counting Without Listing
The GCD–LCM identity: For positive integers, $\gcd(a,b)\cdot\operatorname{lcm}(a,b) = ab$ .; Number Theory Toolkit › GCD, LCM, and Units Digits
The geometry checklist: Draw the figure, larger than you think, and mark every given.; Geometry Essentials › Putting It Together
The rules worth memorizing: Divisible by 3 or 9 when the digit sum is; by 4 when the last two digits form a multiple of $4$ ; by 8 when the last three do; by 11 when the alternating digit sum (units $-$ tens $+$ hundreds $\dots$ ) is a multiple of $11$ .; Number Theory Toolkit › Divisibility Rules
The translation checklist: Name the unknown the question asks for.; Algebra & Manipulation › Putting It Together
The universal problem-solving loop: Read twice.; Formulas & Methods › Geometry
Three kinds of middle: The mean is the total divided by the count.; Complete Syllabus & Gap Fillers › Mean, Median and Mode
Units digits cycle: The last digit of $a^n$ repeats with a period dividing $4$ : powers of $2$ cycle $2,4,8,6$ ; of $3$ cycle $3,9,7,1$ .; Number Theory Toolkit › GCD, LCM, and Units Digits
Vieta for quadratics: For $ax^2+bx+c=0$ with roots $r,s$ : sum $r+s = -\dfrac ba$ and product $rs = \dfrac ca$ .; Algebra & Manipulation › Setting Up and Solving
Which tool?: Divisors, squares, cubes, gcd and lcm: factor into primes first and read the exponents.; Number Theory Toolkit › Putting It Together
Why the $3$ - $4$ - $5$ family never ends: Scaling every side of a right triangle by $k$ scales both sides of $a^2 + b^2 = c^2$ by $k^2$ , so $(3k, 4k, 5k)$ is a right triangle for every $k$ — one proof, infinitely many triangles.; Why the Formulas Work › The Pythagorean Theorem, Proved
Why the digit-sum rule works: Because $10 \equiv 1 \pmod 9$ , every power of ten leaves remainder $1$ , so a number and its digit sum leave the same remainder mod $9$ (and mod $3$ ).; Number Theory Toolkit › Divisibility Rules