Formula sheet

The facts that come up again and again on the AMC 10. If you know these cold, the early problems get fast and the late ones get possible.

Notation: \binom{n}{k}=\frac{n!}{k!\,(n-k)!}, \lfloor x\rfloor is the floor (greatest integer \le x), and a\equiv b \pmod m means m divides a-b.

1. Algebra

Exponents and radicals

a^m a^n=a^{m+n}, \;\frac{a^m}{a^n}=a^{m-n}, \;(a^m)^n=a^{mn}, \;(ab)^n=a^nb^n
a^0=1\ (a\ne 0), \;a^{-n}=\frac{1}{a^n}, \;a^{m/n}=\left(\sqrt[n]{a}\right)^m
\sqrt{ab}=\sqrt a\,\sqrt b for a,b\ge 0; \;\sqrt{x^2}=|x|
Rationalize: \frac{1}{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}
Compare powers by matching exponents or bases, e.g. 2^{300}=8^{100} and 3^{200}=9^{100}, so 3^{200} is larger

Special products and factorizations

(a\pm b)^2=a^2\pm 2ab+b^2, and a^2+b^2=(a+b)^2-2ab
a^2-b^2=(a-b)(a+b)
a^3+b^3=(a+b)(a^2-ab+b^2), \;a^3-b^3=(a-b)(a^2+ab+b^2)
(a+b)^3=a^3+3a^2b+3ab^2+b^3, and a^3+b^3=(a+b)^3-3ab(a+b)
(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)
x^2+\frac{1}{x^2}=\left(x+\frac1x\right)^2-2
Simon's Favorite Factoring Trick: xy+ax+by+ab=(x+b)(y+a). Example: xy-3x-3y=k becomes (x-3)(y-3)=k+9
Sophie Germain identity (rare): a^4+4b^4=(a^2+2b^2+2ab)(a^2+2b^2-2ab)

Quadratics

The roots of ax^2+bx+c=0 are x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
Discriminant \Delta=b^2-4ac: \Delta>0 gives two real roots, \Delta=0 one double root, \Delta<0 no real roots. Integer coefficients with rational roots means \Delta is a perfect square
Vertex at x=-\frac{b}{2a}. When a>0 the minimum value is c-\frac{b^2}{4a}
Completing the square: ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}

Vieta's formulas

For ax^2+bx+c with roots r,s: \;r+s=-\frac ba and rs=\frac ca
For ax^3+bx^2+cx+d with roots r,s,t: \;r+s+t=-\frac ba, \;rs+st+tr=\frac ca, \;rst=-\frac da
r^2+s^2=(r+s)^2-2rs, \;\frac1r+\frac1s=\frac{r+s}{rs}, \;(r-s)^2=(r+s)^2-4rs

Polynomials

Remainder theorem: dividing P(x) by x-a leaves remainder P(a)
Factor theorem: x-a divides P(x) exactly when P(a)=0
Sum of the coefficients is P(1). Constant term is P(0)
Rational root theorem: a rational root \frac pq in lowest terms has p dividing the constant term and q dividing the leading coefficient
A degree-n polynomial has at most n roots

Sequences and series

Arithmetic: a_n=a_1+(n-1)d and S_n=\frac{n(a_1+a_n)}{2} (number of terms times the average of first and last)
Geometric: a_n=a_1r^{n-1}, \;S_n=a_1\frac{1-r^n}{1-r}\ (r\ne 1), infinite sum \frac{a_1}{1-r} when |r|<1
1+2+\cdots+n=\frac{n(n+1)}2, \;1+3+5+\cdots+(2n-1)=n^2
1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}6, \;1^3+2^3+\cdots+n^3=\left(\frac{n(n+1)}2\right)^2
Telescoping: \frac{1}{k(k+1)}=\frac1k-\frac1{k+1} and \frac{1}{k(k+d)}=\frac1d\left(\frac1k-\frac1{k+d}\right)
Counting terms: the list a, a+d, \dots, b has \frac{b-a}{d}+1 terms

Means, statistics, inequalities

Sum = mean \times count. This is usually the key step in average problems
Median: the middle value of the sorted list, or the average of the two middle values when the count is even. Mode: the most frequent value. Range: max minus min
For positive reals, \text{QM}\ge\text{AM}\ge\text{GM}\ge\text{HM}, e.g. \frac{a+b}{2}\ge\sqrt{ab}, with equality exactly when a=b
Harmonic mean of a and b: \frac{2ab}{a+b}. This is the average speed for two equal-length legs driven at speeds a and b
Trivial inequality: x^2\ge 0. For a fixed sum, a product is largest when the parts are equal

Rates, ratios, percents

d=rt. Average speed =\frac{\text{total distance}}{\text{total time}}, which is not the average of the speeds
Closing speed: rates add when objects move toward each other and subtract when one chases the other
Work: rates add, so \frac1T=\frac1{t_1}+\frac1{t_2} for two workers finishing alone in t_1 and t_2
Increasing by p\% means multiplying by 1+\frac p{100}. Successive changes multiply. A 20% rise followed by a 20% drop leaves 0.96 of the start
Percent change =\frac{\text{new}-\text{old}}{\text{old}}\times 100\%

Floor, absolute value, functions

\lfloor x\rfloor\le x<\lfloor x\rfloor+1, and x=\lfloor x\rfloor+\{x\} with 0\le\{x\}<1
|x-a| is the distance from x to a on the number line. |x|<c \iff -c<x<c
Composition: (f\circ g)(x)=f(g(x)). Inverse: f(f^{-1}(x))=x. Swap x and y and solve to get the inverse
Logarithms are rare on the AMC 10 (they belong to the AMC 12): \log_b(xy)=\log_b x+\log_b y, \;\log_b x^k=k\log_b x, \;\log_b a=\frac{\log a}{\log b}

2. Counting and Probability

Core counting

Multiplication principle: independent stages multiply. Addition principle: disjoint cases add
Arranging n distinct objects: n! ways. Ordered choice of k from n: \frac{n!}{(n-k)!}
Unordered choice (committees, subsets): \binom nk, and \binom nk=\binom n{n-k}
An n-element set has 2^n subsets
Words with repeated letters: \frac{n!}{n_1!\,n_2!\cdots n_k!} (for example, MISSISSIPPI gives \frac{11!}{4!\,4!\,2!})
Circular seatings of n distinct people: (n-1)!. If flipping over also counts as the same (bracelets), use \frac{(n-1)!}{2} for n\ge 3
Handshakes among n people: \binom n2. Diagonals of an n-gon: \frac{n(n-3)}2

Standard techniques

Complementary counting: (wanted) = (total) - (unwanted). Use it for "at least one" and "not all"
Adjacency: glue objects that must sit together into one block. For objects that must not touch, place the others first and choose gaps for them
Stars and bars: x_1+\cdots+x_k=n has \binom{n+k-1}{k-1} solutions in nonnegative integers and \binom{n-1}{k-1} in positive integers
Inclusion-exclusion: |A\cup B|=|A|+|B|-|A\cap B| and |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|
Grid paths using a steps right and b steps up: \binom{a+b}{a}
Derangements (nobody gets their own item): D_1=0, D_2=1, D_3=2, D_4=9, D_5=44, with D_n=(n-1)(D_{n-1}+D_{n-2})
Pigeonhole: placing kn+1 objects into n boxes forces some box to hold at least k+1

Pascal's triangle and the binomial theorem

(x+y)^n=\sum_{k=0}^{n}\binom nk x^{n-k}y^k
Pascal's rule: \binom nk=\binom{n-1}{k-1}+\binom{n-1}{k}. Row sum: \sum_k\binom nk=2^n
Hockey stick: \binom rr+\binom{r+1}r+\cdots+\binom nr=\binom{n+1}{r+1}

Recursion

Build the count for size n from smaller sizes. Example: binary strings of length n with no two adjacent 1s satisfy a_n=a_{n-1}+a_{n-2} (Fibonacci)
For probability or state problems, give each state an unknown and write one equation per state

Probability

For equally likely outcomes, P=\frac{\#\text{favorable}}{\#\text{total}}
Independent events: P(A\text{ and }B)=P(A)\,P(B). \;P(\text{at least one})=1-P(\text{none})
Conditional: P(A\mid B)=\frac{P(A\cap B)}{P(B)}. Restrict the sample space to the outcomes where B happened
Geometric probability: favorable length, area, or volume divided by the total
Expected value: E[X]=\sum x\,P(X=x). Linearity: E[X+Y]=E[X]+E[Y] always holds, even for dependent variables, so count expected hits one position at a time
The expected number of trials until the first success, when each trial succeeds with probability p, is \frac1p

3. Number Theory

Divisors and factorization

If n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}, then n has (a_1+1)(a_2+1)\cdots(a_k+1) positive divisors
Sum of divisors: \prod_i\frac{p_i^{a_i+1}-1}{p_i-1}. Product of all divisors: n^{d(n)/2}
n is a perfect square exactly when every exponent is even, which is also exactly when n has an odd number of divisors. For a cube, every exponent is a multiple of 3
\gcd(a,b)\cdot\operatorname{lcm}(a,b)=ab. Take the minimum exponents for the gcd and the maximum exponents for the lcm
Euclidean algorithm: \gcd(a,b)=\gcd(b,\ a \bmod b)=\gcd(a-b,\ b)
Bezout: ax+by=c has integer solutions exactly when \gcd(a,b) divides c
Chicken McNugget: for coprime a,b, the largest number that is not a nonnegative combination ax+by is ab-a-b, and \frac{(a-1)(b-1)}{2} numbers are unreachable
Primes below 100 (25 of them): 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

Divisibility rules

2: last digit even. 4: last two digits. 8: last three digits
3 and 9: digit sum. 5: ends in 0 or 5. 10: ends in 0
11: the alternating digit sum is a multiple of 11
6 means 2 and 3. 12 means 3 and 4. 7, 11, 13: note 1001=7\cdot 11\cdot 13

Modular arithmetic

Congruences mod m can be added, subtracted, multiplied, and raised to powers
n\equiv(\text{digit sum})\pmod 9, and the same holds mod 3
Perfect squares are \equiv 0,1 \pmod 3 and \equiv 0,1\pmod 4. Odd squares are \equiv 1\pmod 8. Squares end in 0, 1, 4, 5, 6, or 9
Units digits of powers repeat with period dividing 4. For 2 the cycle is 2,4,8,6. For 3 it is 3,9,7,1. For 7 it is 7,9,3,1. For 8 it is 8,4,2,6. For 4 it is 4,6 and for 9 it is 9,1. Powers of 0, 1, 5, 6 keep the same units digit
Fermat: for prime p not dividing a, a^{p-1}\equiv 1\pmod p
Euler: when \gcd(a,n)=1, a^{\varphi(n)}\equiv 1 \pmod n, where \varphi(n)=n\prod_{p\mid n}\left(1-\frac1p\right)
Chinese remainder theorem: when \gcd(m,n)=1, a pair of conditions x\equiv a\pmod m and x\equiv b\pmod n has exactly one solution mod mn
Days of the week work mod 7, clock hours mod 12, and minutes mod 60

Factorials, digits, bases

Legendre's formula: the exponent of prime p in n! is \left\lfloor\frac np\right\rfloor+\left\lfloor\frac n{p^2}\right\rfloor+\left\lfloor\frac n{p^3}\right\rfloor+\cdots. The number of trailing zeros of n! is the exponent of 5
A two-digit number \overline{ab} equals 10a+b. Reversing the digits changes the number by a multiple of 9: (10a+b)-(10b+a)=9(a-b)
Base b: \overline{d_k\cdots d_1d_0}_{\,b}=d_kb^k+\cdots+d_1b+d_0, with each digit between 0 and b-1
Repeating decimals: 0.\overline{ab}=\frac{ab}{99} and 0.\overline{abc}=\frac{abc}{999}
Parity: a sum is odd exactly when it contains an odd number of odd terms. A product is odd only when every factor is odd

Year numbers worth knowing

2023=7\cdot 17^2, \;2024=2^3\cdot 11\cdot 23, \;2025=3^4\cdot 5^2=45^2, \;2026=2\cdot 1013 (1013 is prime), \;2027 is prime

4. Geometry

Angles and polygons

The angles of a triangle add to 180^\circ. An exterior angle equals the sum of the two remote interior angles
Interior angles of an n-gon add to (n-2)\cdot 180^\circ. Each angle of a regular n-gon is \frac{(n-2)180^\circ}{n}, and exterior angles always total 360^\circ
Parallel lines: corresponding and alternate interior angles are equal. Base angles of an isosceles triangle are equal
A regular hexagon with side s splits into 6 equilateral triangles, giving area \frac{3\sqrt3}{2}s^2. A regular octagon with side s has area 2(1+\sqrt2)s^2

Right triangles

Pythagorean theorem: a^2+b^2=c^2. Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29, 9-40-41, and their multiples
45-45-90 sides are in ratio 1:1:\sqrt2. 30-60-90 sides are in ratio 1:\sqrt3:2
The altitude to the hypotenuse is h=\frac{ab}{c} and splits the triangle into two triangles similar to it, with h^2=pq where p,q are the hypotenuse pieces
The hypotenuse is a diameter of the circumcircle, so R=\frac c2. The inradius is r=\frac{a+b-c}{2}

Triangle area and centers

[ABC]=\frac12bh=\frac12ab\sin C=rs=\frac{abc}{4R}, where s=\frac{a+b+c}2
Heron's formula: [ABC]=\sqrt{s(s-a)(s-b)(s-c)}. For example, sides 13, 14, 15 give area 84
Equilateral triangle with side s: height \frac{\sqrt3}2s, area \frac{\sqrt3}4s^2, inradius \frac{s\sqrt3}6, circumradius \frac{s\sqrt3}3
The centroid cuts each median in ratio 2:1 from the vertex. The three medians split the triangle into 6 regions of equal area
Median length: m_a=\frac12\sqrt{2b^2+2c^2-a^2}
Triangle inequality: each side is less than the sum of the other two and greater than their difference

Ratios inside triangles

Similar figures with length ratio k have area ratio k^2 and volume ratio k^3
Two triangles with the same height have areas in the ratio of their bases
Angle bisector theorem: if the bisector from A meets BC at D, then \frac{BD}{DC}=\frac{AB}{AC}
Stewart's theorem (rare): for a cevian AD=d with BD=m, DC=n, BC=a, we have b^2m+c^2n=a(d^2+mn)
Mass points: if D on BC has BD:DC=m:n, put mass n at B and mass m at C. Masses add at D

Circles

Area \pi r^2, circumference 2\pi r. A sector with central angle \theta^\circ has area \frac{\theta}{360}\pi r^2 and arc length \frac{\theta}{360}2\pi r
An inscribed angle is half its intercepted arc, so an angle inscribed in a semicircle is 90^\circ
In a cyclic quadrilateral, opposite angles add to 180^\circ
Two chords crossing inside a circle make an angle equal to half the sum of the intercepted arcs. Two secants from an outside point make an angle equal to half the difference
A tangent is perpendicular to the radius at the point of tangency. The two tangent segments from an outside point have equal length
For tangent circles, join the centers. The distance between centers is r_1+r_2 for external tangency and |r_1-r_2| for internal
Power of a point: PA\cdot PB=PC\cdot PD for two chords through P or two secants from P. For a tangent PT, PT^2=PA\cdot PB
Ptolemy (rare), for cyclic ABCD: AC\cdot BD=AB\cdot CD+AD\cdot BC

Coordinates

Distance \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. Midpoint \left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right). Slope \frac{y_2-y_1}{x_2-x_1}
Perpendicular lines have slopes that multiply to -1. Parallel lines have equal slopes
Distance from (x_0,y_0) to the line Ax+By+C=0 is \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}
Circle: (x-h)^2+(y-k)^2=r^2. Complete the square to find the center and radius
Shoelace: the area of the polygon with vertices (x_1,y_1),\dots,(x_n,y_n) taken in order is \frac12\left|\sum_{i}(x_iy_{i+1}-x_{i+1}y_i)\right|
Pick's theorem for lattice polygons: A=I+\frac B2-1, where I counts interior lattice points and B counts boundary lattice points
|x|+|y|\le k is a square (a diamond) with diagonals 2k, so its area is 2k^2
Reflection trick: the shortest path from A to a line and then to B has the length of the straight segment from A to the mirror image of B

Solids

Box: volume lwh, surface area 2(lw+wh+hl), space diagonal \sqrt{l^2+w^2+h^2}. A cube with edge s has space diagonal s\sqrt3
Prism or cylinder: V=(\text{base area})\cdot h. Cylinder: V=\pi r^2h and lateral area 2\pi rh
Pyramid or cone: V=\frac13(\text{base area})\cdot h. Cone lateral area: \pi r\ell with slant height \ell=\sqrt{r^2+h^2}
Sphere: V=\frac43\pi r^3, surface area 4\pi r^2
Regular tetrahedron with edge s: V=\frac{s^3\sqrt2}{12}, height \frac{s\sqrt6}3
Euler's formula for convex polyhedra: V-E+F=2

Trigonometry (officially outside the AMC 10, but sometimes a shortcut)

Law of cosines: c^2=a^2+b^2-2ab\cos C. Law of sines: \frac a{\sin A}=2R
\sin 30^\circ=\frac12, \sin 45^\circ=\frac{\sqrt2}2, \sin 60^\circ=\frac{\sqrt3}2

Memorize

Squares	Powers of 2	Factorials	Approximations
11^2=121 through 25^2=625	2^{10}=1024	5!=120	\pi\approx 3.1416
13^2=169,\ 17^2=289,\ 19^2=361	2^{8}=256, 2^{12}=4096	6!=720, 7!=5040	\sqrt2\approx1.414
23^2=529,\ 24^2=576	2^{16}=65536	8!=40320	\sqrt3\approx1.732
45^2=2025	3^5=243, 3^6=729	10!=3628800	\sqrt5\approx 2.236

Scoring reminder: a correct answer earns 6 points, a blank earns 1.5, and a wrong answer earns 0. For the 2026–27 cycle, the AIME qualifying score on the AMC 10 is fixed at 100. Fourteen correct answers plus eleven blanks gives 100.5.