Formulas & facts

These are the formulas and facts the AMC 10 uses again and again. Each one comes with what it means in plain words and a small example, because a formula only helps in a contest if you can use it on real numbers.

★ means must know: you should be able to write it down without stopping to think. The rest you should recognize when you see it.

Warning marks a mistake that a lot of students make on that idea.

Algebra

Factoring

• ★ Squaring a sum or difference. $(a+b)^2 = a^2+2ab+b^2$ and $(a-b)^2 = a^2-2ab+b^2$. Example: $51^2 = 50^2 + 2\cdot50\cdot1 + 1^2 = 2601$.
• ★ Difference of squares. $a^2-b^2 = (a-b)(a+b)$. Example: $47\cdot53 = (50-3)(50+3) = 2500-9 = 2491$.
• Sum and difference of cubes. $a^3+b^3 = (a+b)(a^2-ab+b^2)$ and $a^3-b^3 = (a-b)(a^2+ab+b^2)$.
• Squaring three terms. $(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)$.
• Simon's Favorite Factoring Trick. When you see $xy + ax + by$, add $ab$ to both sides so the left side factors: $xy+ax+by+ab = (x+b)(y+a)$. Example: $xy+2x+3y = 18$ becomes $(x+3)(y+2) = 24$, and now you only need the factor pairs of $24$.
• Finding $a^2+b^2$ without finding $a$ and $b$. $a^2+b^2 = (a+b)^2-2ab$. Example: if $a+b=5$ and $ab=3$, then $a^2+b^2 = 25-6 = 19$. In the same way, $\frac1a+\frac1b = \frac{a+b}{ab}$.

Quadratics

• ★ Quadratic formula. The solutions of $ax^2+bx+c=0$ are $x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$.
• ★ Vieta's Formulas. For $ax^2+bx+c=0$, the two roots add up to $-\frac{b}{a}$ and multiply to $\frac{c}{a}$. Example: the roots of $x^2-7x+10=0$ are $2$ and $5$; they add to $7$ and multiply to $10$. Often you never need the roots themselves, only their sum and product.
• The discriminant $b^2-4ac$ tells you about the roots before you find them: positive means two real roots, zero means one repeated root, negative means no real roots. If $a$, $b$, $c$ are integers and $b^2-4ac$ is a perfect square, the roots are rational.

Sequences and series

• ★ Counting the terms. From $a$ up to $b$ in steps of $d$, there are $\frac{b-a}{d}+1$ terms. Example: $3, 7, 11, \dots, 43$ has $\frac{43-3}{4}+1 = 11$ terms. Warning: forgetting the $+1$ is the most common slip here.
• ★ Arithmetic series (each term goes up by the same amount): sum $=$ (number of terms) $\times$ (average of the first and last terms). Example: $3+7+11+\dots+43 = 11\cdot\frac{3+43}{2} = 253$.
• ★ Adding the numbers from $1$ to $n$. $1+2+\dots+n = \frac{n(n+1)}{2}$. Example: $1+2+\dots+100 = 5050$.
• Adding squares. $1^2+2^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$. Example: $1+4+9+16 = \frac{4\cdot5\cdot9}{6} = 30$.
• ★ Geometric series (each term is the one before times the same common ratio $r$): $a+ar+ar^2+\dots+ar^{n-1} = a\cdot\frac{r^n-1}{r-1}$. If $r$ is between $-1$ and $1$, the series that goes on forever adds up to $\frac{a}{1-r}$. Example: $1+\frac12+\frac14+\frac18+\dots = \frac{1}{1-\frac12} = 2$.
• Telescoping. $\frac{1}{k(k+1)} = \frac1k-\frac1{k+1}$, so in a long sum almost everything cancels. Example: $\frac1{1\cdot2}+\frac1{2\cdot3}+\dots+\frac1{9\cdot10} = \left(1-\frac12\right)+\left(\frac12-\frac13\right)+\dots+\left(\frac19-\frac1{10}\right) = 1-\frac1{10} = \frac9{10}$.

Rates, percents, and work

• ★ Distance $=$ rate $\times$ time, and in the same way work $=$ rate $\times$ time. When people work together, add their rates, not their times. Example: Al paints a fence in $3$ hours and Bo in $6$. Together they paint $\frac13+\frac16 = \frac12$ of the fence each hour, so the job takes $2$ hours.
• ★ Average speed $=$ total distance $\div$ total time. Warning: it is not the average of the speeds. Example: driving $60$ mph there and $40$ mph back on the same road averages $48$ mph, not $50$.
• Percent changes multiply. Up $20\%$ means $\times1.2$, and down $20\%$ means $\times0.8$. So up $20\%$ and then down $20\%$ is $\times0.96$: you end $4\%$ lower, not back where you started.

Exponents, roots, and absolute value

• ★ Exponent rules. $a^m\cdot a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$, $a^{-n} = \frac{1}{a^n}$, and $a^{1/n} = \sqrt[n]{a}$. Example: $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$.
• Simplifying square roots. Pull out the biggest perfect square: $\sqrt{72} = \sqrt{36\cdot2} = 6\sqrt2$. The rule $\sqrt a\cdot\sqrt b = \sqrt{ab}$ works when $a$ and $b$ are zero or positive.
• Absolute value is distance. $|x-a|$ is how far $x$ is from $a$ on the number line. Example: $|x-3| < 2$ means $x$ is less than $2$ away from $3$, so $1 < x < 5$. With several absolute values, split into cases at the numbers that make each one zero.
• Logarithms (rare on the AMC 10). $\log_b x = y$ means the same thing as $b^y = x$. Example: $\log_2 32 = 5$ because $2^5 = 32$.

AM-GM

• AM-GM Inequality. For two numbers that are zero or positive, the average is at least the geometric mean: $\frac{a+b}{2} \ge \sqrt{ab}$, with equality only when $a = b$. In practice: if the product is fixed, the sum is smallest when the numbers are equal. Example: if $x$ and $y$ are positive and $xy = 36$, then $x+y$ is at least $12$, reached when $x = y = 6$.

Number theory

Primes, divisors, gcd and lcm

• ★ Start with the prime factorization. Most number theory problems get easier once every number is written as a product of primes. Example: $360 = 2^3\cdot3^2\cdot5$.
• ★ Counting divisors. If $n = p^a\cdot q^b\cdot r^c$ (with $p$, $q$, $r$ different primes), then $n$ has $(a+1)(b+1)(c+1)$ positive divisors. Example: $360 = 2^3\cdot3^2\cdot5^1$ has $4\cdot3\cdot2 = 24$ divisors. Why: a divisor chooses how many $2$s to use (from $0$ to $3$, which is $4$ choices), how many $3$s ($3$ choices), and how many $5$s ($2$ choices).
• Adding the divisors. Multiply one sum for each prime. Example: the divisors of $360$ add up to $(1+2+4+8)(1+3+9)(1+5) = 15\cdot13\cdot6 = 1170$.
• ★ GCD and LCM. $\gcd(a,b)\cdot\text{lcm}(a,b) = a\cdot b$. Using prime factorizations, the gcd takes the smaller power of each prime and the lcm takes the larger. Example: $12 = 2^2\cdot3$ and $18 = 2\cdot3^2$, so the gcd is $2\cdot3 = 6$, the lcm is $2^2\cdot3^2 = 36$, and $6\cdot36 = 12\cdot18 = 216$.
• Euclidean Algorithm. $\gcd(a,b)$ is the same as $\gcd(b, r)$, where $r$ is the remainder when $a$ is divided by $b$. Repeat until the remainder is $0$. Example: $\gcd(84,36) = \gcd(36,12) = \gcd(12,0) = 12$.
• Primes under $100$ (there are $25$): $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$.
• Is it prime? Only try dividing by primes up to its square root. Example: any number below $121$ is prime if none of $2, 3, 5, 7$ divides it, since $11^2 = 121$.

Divisibility rules

★ Know all of these.

Remainders and units digits

• Remainders can be added and multiplied. To find a remainder, you can work with the remainders of the pieces. AoPS writes $a \equiv b \pmod m$ for "$a$ and $b$ leave the same remainder when divided by $m$." Example: $17\cdot23$ divided by $5$: the remainders are $2$ and $3$, and $2\cdot3 = 6$ leaves $1$. Check: $17\cdot23 = 391 = 5\cdot78+1$.
• ★ Units digits repeat in cycles. To find the units digit of a big power, find where you are in the cycle (use the table below). Example: the units digit of $7^{2026}$: $2026 = 4\cdot506+2$, so it is the 2nd in the cycle $7, 9, 3, 1$, which is $9$.

A fun one: $n^5$ always ends in the same digit as $n$.

Squares and consecutive numbers

• Facts about perfect squares. A perfect square has an odd number of divisors (divisors come in pairs, except the square root, which pairs with itself). A square divided by $4$ leaves remainder $0$ or $1$, never $2$ or $3$. A square never ends in $2$, $3$, $7$, or $8$.
• Consecutive integers. The product of any $k$ consecutive integers is divisible by $k!$. Example: the product of any $3$ consecutive integers is divisible by $3! = 6$.

Decimals and bases

• Repeating decimals. When two digits repeat, put them over $99$; three digits, over $999$. Example: $0.\overline{27} = \frac{27}{99} = \frac{3}{11}$. Worth knowing: $\frac17 = 0.\overline{142857}$.
• Decimals that end. A fraction in lowest terms has a decimal that ends exactly when its denominator has no prime factors except $2$ and $5$. Example: $\frac{3}{40}$ ends ($40 = 2^3\cdot5$), but $\frac{1}{12}$ repeats forever ($12$ has a factor of $3$).
• Other bases. In base $b$, the places stand for $1, b, b^2, \dots$ Example: $213_5 = 2\cdot25+1\cdot5+3 = 58$. To go the other way, divide by $b$ again and again; the remainders, read from last to first, are the digits: $58 = 11\cdot5+3$, $11 = 2\cdot5+1$, $2 = 0\cdot5+2$, so $58 = 213_5$.
• The digit-sum trick. A number and the sum of its digits leave the same remainder when divided by $9$. (In base $b$, the same is true for $b-1$.)

Numbers to memorize

• ★ Squares: $11^2=121$, $12^2=144$, $13^2=169$, $14^2=196$, $15^2=225$, $16^2=256$, $17^2=289$, $18^2=324$, $19^2=361$, $21^2=441$, $22^2=484$, $23^2=529$, $24^2=576$, $25^2=625$.
• ★ Powers of $2$: $2^1$ to $2^{12}$ are $2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096$.

Counting & probability

Basic counting

• ★ Multiply when you make one choice and then another. If the first choice can be made $m$ ways and the second $n$ ways, together there are $m\cdot n$ ways. Example: $3$ shirts and $4$ pairs of pants make $12$ outfits.
• ★ Permutations: order matters. Lining up $k$ of $n$ different things: $n(n-1)(n-2)\cdots$ ($k$ numbers multiplied), which is $\frac{n!}{(n-k)!}$. Example: gold, silver and bronze among $8$ runners: $8\cdot7\cdot6 = 336$.
• ★ Combinations: order doesn't matter. Choosing $k$ of $n$ things: $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$, read "$n$ choose $k$". Example: a $3$-person committee from $8$ people: $\binom83 = \frac{8\cdot7\cdot6}{3\cdot2\cdot1} = 56$.
• Choosing who is in is the same as choosing who is out: $\binom{n}{k} = \binom{n}{n-k}$. Example: $\binom{10}{8} = \binom{10}{2} = 45$.
• Each row of Pascal's Triangle adds up to a power of $2$: $\binom n0+\binom n1+\dots+\binom nn = 2^n$, because each of the $n$ things is either in or out. Example: a set of $5$ things has $2^5 = 32$ subsets.

Arrangements

• ★ Repeated letters. Divide by the factorial of each repeat count. Example: BANANA has $6$ letters with three A's and two N's, so it has $\frac{6!}{3!\,2!} = 60$ arrangements.
• Around a circle. $n$ people around a round table: $(n-1)!$ ways, since turning the table doesn't make a new seating. If flipping over also doesn't count as new (beads on a bracelet), divide by $2$ more: $\frac{(n-1)!}{2}$ for $n\ge3$.
• Glue things that must be together. Treat them as one block, arrange, then multiply by the ways to order the inside of the block. Example: $5$ people in a row with Ann and Ben side by side: $4!\cdot2! = 48$.
• Paths on a grid. Walking from $(0,0)$ to $(m,n)$ by steps right and up: $\binom{m+n}{m}$ paths, because you only choose which of the $m+n$ steps go right. Example: from $(0,0)$ to $(3,4)$ there are $\binom73 = 35$ paths.
• Handshakes and diagonals. If $n$ people each shake hands once with everyone else, there are $\binom{n}{2}$ handshakes. A polygon with $n$ sides has $\frac{n(n-3)}{2}$ diagonals. Example: a hexagon has $\frac{6\cdot3}{2} = 9$ diagonals.

Counting smarter

• ★ Complementary counting. When a problem says "at least one," count the opposite and subtract: (at least one) $=$ (all) $-$ (none). Example: three-digit numbers with at least one $7$: there are $900$ in all, and $8\cdot9\cdot9 = 648$ with no $7$, so $900-648 = 252$.
• ★ Principle of Inclusion-Exclusion (PIE). (number in $A$ or $B$) $=$ (number in $A$) $+$ (number in $B$) $-$ (number in both). With three groups: add all three, subtract the three overlaps of two, then add back the overlap of all three. Example: the numbers from $1$ to $100$ divisible by $2$ or $3$: $50+33-16 = 67$.
• ★ Stars and bars (AoPS also calls it balls and urns, or sticks and stones). The number of ways to hand out $n$ identical things to $k$ people is $\binom{n+k-1}{k-1}$. If everyone must get at least one: $\binom{n-1}{k-1}$. Example: $10$ identical candies among $3$ kids: $\binom{12}{2} = 66$ ways, or $\binom92 = 36$ ways if each kid gets at least one.

Probability

• ★ Probability $=$ (outcomes you want) $\div$ (all outcomes), as long as every outcome is equally likely.
• ★ Independent events multiply. If one event doesn't affect the other, multiply their probabilities. Example: two heads in two coin flips: $\frac12\cdot\frac12 = \frac14$.
• "Or" probabilities. $P(A\text{ or }B) = P(A)+P(B)-P(\text{both})$, the same idea as PIE.
• Two dice. There are $36$ equally likely rolls. A sum of $s$ can be made in $6-|s-7|$ ways: $7$ is the most common ($6$ ways), and $2$ and $12$ are the rarest ($1$ way each).
• Expected value is the long-run average: multiply each outcome by its probability and add. Expected values add up, even when the events affect each other. Example: the expected number of heads in $10$ coin flips is $10\cdot\frac12 = 5$.
• Geometric probability. When a point is picked at random from a region, probability $=$ (length or area you want) $\div$ (total length or area).

Geometry

Triangles

• ★ Area $= \frac12\cdot$ base $\cdot$ height. Two more ways: area $= rs$, where $r$ is the radius of the inscribed circle and $s$ is half the perimeter; and, if you know two sides $a$, $b$ and the angle $C$ between them, area $= \frac12ab\sin C$.
• Heron's Formula. Area $= \sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is half the perimeter. Example: the $13$-$14$-$15$ triangle has $s = 21$ and area $\sqrt{21\cdot8\cdot7\cdot6} = 84$.
• ★ Equilateral triangle with side $a$: height $\frac{\sqrt3}{2}a$, area $\frac{\sqrt3}{4}a^2$.
• ★ Pythagorean Theorem. In a right triangle, $a^2+b^2 = c^2$, where $c$ is the hypotenuse. Pythagorean triples to know: $3$-$4$-$5$, $5$-$12$-$13$, $8$-$15$-$17$, $7$-$24$-$25$, $20$-$21$-$29$, $9$-$40$-$41$, and their multiples such as $6$-$8$-$10$.
• ★ Special right triangles. $45$-$45$-$90$: sides $x$, $x$, $x\sqrt2$. $30$-$60$-$90$: sides $x$, $x\sqrt3$, $2x$. The shortest side is across from the $30^\circ$ angle, and the hypotenuse is twice the shortest side.
• ★ Similar triangles. Same shape, different size. If the sides are $k$ times as long, the area is $k^2$ times as big. Two pairs of equal angles are enough to make triangles similar (AA). A line parallel to one side of a triangle cuts off a smaller similar triangle.
• Triangle Inequality. Any two sides add up to more than the third side. Example: $3$, $4$, $8$ can't be a triangle, because $3+4 < 8$.
• Angle Bisector Theorem. The bisector of angle $A$ cuts the opposite side $BC$ in the ratio $AB : AC$. Example: if $AB = 6$, $AC = 9$ and $BC = 10$, the pieces are $4$ and $6$.
• Medians and the centroid. The three medians meet at the centroid, which is $\frac23$ of the way from each corner to the midpoint of the opposite side. The medians cut the triangle into $6$ small triangles with equal area.
• The altitude to the hypotenuse. In a right triangle, it splits the big triangle into two smaller ones similar to it. If it cuts the hypotenuse into pieces $p$ and $q$, then the altitude $h$ satisfies $h^2 = pq$. Example: in the $3$-$4$-$5$ triangle, the pieces are $\frac95$ and $\frac{16}5$, and $h = \frac{12}5$.
• Inscribed circle of a right triangle. Its radius is $r = \frac{a+b-c}{2}$ ($a$, $b$ the legs, $c$ the hypotenuse). Example: the $3$-$4$-$5$ triangle has $r = 1$.

Circles

• ★ Circumference and area. $C = 2\pi r$ and $A = \pi r^2$. An arc or sector of $\theta$ degrees is $\frac{\theta}{360}$ of the whole circle. Example: a $90^\circ$ sector of a circle with radius $4$ has area $\frac14\cdot16\pi = 4\pi$.
• ★ Inscribed Angle Theorem. An angle with its corner on the circle is half the central angle that cuts off the same arc. So an angle whose sides pass through the two ends of a diameter is a right angle.
• ★ Tangents. A tangent line is perpendicular to the radius at the point where it touches. Two Tangent Theorem: the two tangent segments from a point outside a circle are equal in length.
• Power of a Point. If two chords cross at $P$, then $PA\cdot PB = PC\cdot PD$: the two pieces of one chord multiply to the same number as the two pieces of the other. From a point $P$ outside the circle, with tangent $PT$ and a line through the circle at $A$ and $B$: $PT^2 = PA\cdot PB$.

Polygons and areas

• ★ Angles in a polygon. The inside angles of an $n$-sided polygon add up to $(n-2)\cdot180^\circ$. The outside angles always add up to $360^\circ$, so each outside angle of a regular $n$-gon is $\frac{360^\circ}{n}$. Example: a regular hexagon has outside angles of $60^\circ$ and inside angles of $120^\circ$.
• Regular hexagon. It is six equilateral triangles, so with side $a$ its area is $6\cdot\frac{\sqrt3}{4}a^2 = \frac{3\sqrt3}{2}a^2$.
• Trapezoid: area $=$ (average of the two parallel sides) $\times$ height. Rhombus or kite: area $=$ half the product of the diagonals.
• Shoelace Theorem (area from coordinates). List the corners in order around the shape and repeat the first one at the end. Add $x_1y_2+x_2y_3+\dots$, subtract $y_1x_2+y_2x_3+\dots$, and take half of the result (made positive). Example: the triangle $(0,0)$, $(4,0)$, $(1,3)$: $(0+12+0)-(0+0+0) = 12$, so the area is $6$.

Coordinates

• ★ Distance, midpoint, slope. Distance $= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, which is just the Pythagorean Theorem. The midpoint averages the coordinates. Slope $= \frac{y_2-y_1}{x_2-x_1}$.
• ★ Parallel and perpendicular lines. Parallel lines have the same slope. Perpendicular slopes multiply to $-1$. Example: a line perpendicular to slope $\frac23$ has slope $-\frac32$.
• Distance from a point to a line (rare). From $(x_0,y_0)$ to the line $ax+by+c = 0$, the distance is $\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$.

Solids

• ★ Volume. A prism or cylinder is (area of the base) $\times$ height. A pyramid or cone is $\frac13$ of that. A sphere is $\frac43\pi r^3$, and its surface area is $4\pi r^2$.
• Diagonals of a box. The long diagonal of an $a\times b\times c$ box is $\sqrt{a^2+b^2+c^2}$. For a cube with edge $a$ it is $a\sqrt3$.
• Cones. The slant height is $\ell = \sqrt{r^2+h^2}$, and the curved side has area $\pi r\ell$.

Shortest paths

• Reflect to straighten the path. To find the shortest path from one point to another that has to touch a line, reflect one point across the line and draw a straight segment.

Averages, clocks, and calendars

• ★ Work with the sum, not the average. Sum $=$ average $\times$ count. Example: if $4$ tests average $85$, the scores add up to $340$. So "the average is a whole number" really means "the sum is divisible by the count."
• Median and mode. The median is the middle value once the numbers are in order; with an even count, it is the average of the two middle values. The mode is the value that appears most often.
• Adding one more number. If $n$ numbers average $m$ and you add the number $v$, the average changes by $\frac{v-m}{n+1}$. Example: $4$ numbers average $10$; adding $20$ raises the average by $\frac{20-10}{5} = 2$, to $12$.
• Clocks. The minute hand moves $6^\circ$ per minute and the hour hand $\frac12^\circ$ per minute. At $h$:$m$ the angle between them is $|30h-5.5m|$ degrees; if that is more than $180$, subtract it from $360$. Example: at $3$:$30$ the angle is $|90-165| = 75^\circ$. The hands are on top of each other every $\frac{720}{11}$ minutes, a little more than an hour.
• Calendars. A year of $365$ days is $52$ weeks and $1$ day, so a date moves one weekday later each year, or two if a February 29 comes in between. Example: January 1, 2026 is a Thursday, so January 1, 2027 is a Friday. Leap years are the years divisible by $4$, except century years, which must be divisible by $400$ ($2000$ was a leap year, $1900$ was not).