AMC 12 Formulas To Know

Chinese Remainder Theorem

Link: https://www.youtube.com/watch?v=Y5RcMWiUyyE

Combinations

Number of Combintations: n!/[(k!)(n-k)!] Where n is the total numbers of objects from which you are choosing k objects (order doesn't matter)

Permutations

Number of Permutations: n!/(n-k)!. Where n is the total number of objects from which you are choosing k objects (order does matter).

Euler's Totient/Phi Function

Phi(n) = n (1 - 1/p₁)(1- 1/p₂)...(1 - 1/pₙ). Where n is any positive integer and pₙ are prime divisors of n. This gives the number of relatively prime positive integers less than or equal to some number n. This is often used in modular arithmetic problems since Euler's Theorem states a^(phi(n)) ≡ 1 mod n.

Sum of the first n terms of a Geometric Series

S = a₁ (1-rⁿ)/(1-r) Where r is the common ration, S is the sum, and a₁ is the first term.

Sum of an Infinite Geometric Series

S = a₁/(1-r) Where r is the common ration, S is the sum, a₁ is the first term

Sum of First Odd Numbers Sum of First N Numbers Sum of First Even Numbers

S = n²; S = n(n+1)/2; S = n(n+1)

Sum of an Arithmetic Series

S_n = n/2 (a₁ + aₙ) Where n is the number of terms, Sₙ is the sum, and a₁ is the first term.

Fundamental Properties of Cyclic Quadrilaterals

1. Opposite Angles Add to 180° 2. A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter. 3. In cyclic quadrilateral ABCD, <ABD = <ACD, <BCA = <BDA, <BAC = <BDC, <CAD = <CBD

Pythagorean Triples (x4)

3 - 4 - 5 5 - 12 - 13 7 - 24 - 25 8 - 15 - 17

Arcs and Angles in A Circle

An arc is a portion of the circle's circumference measured in degrees. The measure of an angle formed by the center of a circle and two point on the circumference is equal to the measure of the intercepted (subtended) arc. An angle formed by three points on the circumference of the circle is equal to 1/2 the measure of the subtended arc.

Area of a Regular Polygon

Area = (apothem)*(perimeter)/2; Area = ns^2/(4tan(180/n))

Area of a Triangle (4 formulas)

Area = (base)*(height)/2; Area = (inradius)*(semiperimeter); Area = (1/2)(a)(b)(sinθ) where θ is between a and b; Area = abc/4(Circumradius)

Area of a Trapezoid

Area = (base_1 + base_2)(height)/2

Area of Kite/Rhombus

Area = (diagonal_1)*(diagonal_2)/2

Area of a Regular Hexagon

Area = 3√3 (side)^2/2

Pick's Theorem

Area = I + B/2 - 1 Where I is the numbwer of lattic points in the interior, and B is the number of lattice points on the boundary of a figure in the coordinate plane.

Heron's Formula

Area = √(s)(s-a)(s-b)(s-c)

Brahmagupta's Formula

Area = √(s-a)(s-b)(s-c)(s-d) Where K is the area and s is the semiperimeter of the quadrilateral with sides a,b,c,d. For this formula to work, the quadrilateral much be cyclic

Ceva's Theorem

Ceva's Theorem states that in with points D, E, F on sides BC, AC, AB respectively, are concurrent if, and only if, BD/DC * CE/EA * AF/FB = 1

Fibonacci Numbers

Define a sequence Fₙ such that F₀ =1, F₁ = 1, Fₙ = F_(n-1) + F_(n-2). The first few terms look like 0,1,1,2,3,5,8 .... The quotient of two consecutive terms approaches the golden ration: (1+√5)/2

Distance Formula

Distance = √(x_2 - x_1)^2 +(y_2 - y_1)^2

Law of Cosines

For a triangle with sides a,b,c and opposite angles A,B,C respectively, the Law of Cosines states c² = a² + b² -2ab(cos(C))

Pigeonhole Principle

If we distribute n balls into k boxes such that n > k then at least one box must have multiple balls.

Number/Sum of Divisors

Let (p₁^x)*(p₂^y)...(p_m)ⁿ be the prime factorization of sum number a. The number of divisors a has is given by (x+1)(y+1)...(n+1). Similarly, the sum of the divisors of a is given by (p₁⁰ +p₁¹...p₁^x)*(p₂⁰ +p₂¹...p₂^y)...(p_m⁰+p_m¹...p_mⁿ).

Location of Roots (DeMoivre's Theorem)

Let cos(θ) + isin(θ) = cis(θ). DeMoivre's Theorem allows complex numbers in polar form - that is r * cis(θ) - to be raised to a power. It states that for a rational x and integral n, cis(x)ⁿ = cis(nx). This can be used to find the nth root of a number/polynomial.

Shoelace Theorem

Suppose the polygon has vertices (a_1, b_1), (a_2, b_2), ..., (a_n, b_n) listed in clockwise order. Then the area of P is (1/2) |(a_1b_2 + a_2b_3 + ... + a_nb_1) - (b_1a_2 + b_2a_3 + ... b_na_1|. The Shoelace Theorem gets its name because if one lists the coordinates in a column and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes. (a_1, b_1) (a_2, b_2) ... (a_n , b_n) (a_1 , b_1)

Stewart's Theorem

Take ∆ABC with sides of length a,b,c opposite vertices A,B,C respectively. If cevian AD is drawn so that BD = m, DC = n, AD = d we have that man + dad = bmb + cnc, which can be rememberd using the mnemonic device, "A man and his dad put a bomb in the sink." This theorem isn't used very often in competition math, but it can trivialize otherwise difficult problems when it does come up.

Angle Bisector Theorem

The Angle Bisector Theorem states that given ∆ABC and angle bisector AD, where D is on the side BC, then AB/BD = AC/CD. Likewise, the converse of thie theorm holds as well

Chicken McNugget Theorem

The Chicken McNugget Theorem states that for any two relatively prime postive integers, m, n, the greatest integer that cannot be written in the form a*m + b*n (the greatest number that can be expressed as a sum of the two numbers) where a,b are positive integers is m*n - m - n. It follow that there are (m-1)(n-1)/2 positive integers which cannot be expressed as the sum of some number of m and n.

Triangle Inequality

The Triangle Inequality says that in nondegenerate ∆ABC: AB + BC > AC, BC + AC > AB, AC + AB > BC

Binomial Theorem

The binomial theorem states that (a+b)ⁿ = Sum: (n k) * (a^(n-k)) * (b^k)

Stars and Bars

The number of ways to distribute n indistinguishable items into k distinguishable boxes, where each box must receive at least one item is (n-1 k-1). Alternatively, the number of ways to distribute indistinguishable balls in distinguishable boxes, where some box(es) may remain rempty, is (n+k-1 k+1)

Power of a Point

There are three cases for this theorem: 1. Two Intersecting Internal Chords (AC and BD that Intersect at E): AE * EC = BE * ED; 2. A Tangent (AB) and a Secant (BD where it intersects the circle at C) That Meet At A Point: AB^2 = BC * BD 3. Secants (AC and EC where AC intersects the circle at D and EC intersects the circle at D) That Intersect Outside The Circle: CB * CA = CD * CE

Menelaus' Theorem

This configuration shows up from time to time, though not often. The theorem states that given a configuration as shown below, BP * CQ * PC = QA * RB * AR

Vieta's Formulas (Quadratic and Cubic)

Vieta's formulas relate the coefficients of a polynomial to its roots. This set of formulas is one of the most useful in competition math. These state that the sum of the roots for q quadratic polynomial ax² + bx + c is -b/a and that the product of the roots is c/a. This extends to higher degree polynomials as well. For example, if the cubic polynomial ax³ + bx² + cx + d has roots r,s,t, r + s + t = -b/a, rst = -d/a, and rs + rt + st = c/a. Notice the signs alternate (with b being negative and alternating thereafter). This takes some practice to get used to and I recommend you visit the AoPS page on Vieta's Formulas for further details.

Volume/Surface Area of a Sphere

Volume = (4)(π)(r^3)/3; Surface Area = 4(π)(r^2)

Volume/Surface Area of a Pyramid

Volume = (Area of base)(height)/3; Surface Area = 2(slant height)(side length) + (Area of Base)

Volume/Surface Area of a Prism

Volume = (length)(width)(height); Surface Area = 2(length)(width) + 2(length)(height) + 2(width)(height)

Volume/Surface Area of a Cube

Volume = (side)^3; Surface Area = 6(side)^2

Volume/Surface Area of a Cylinder

Volume = (π)(radius)^2(height) Surface Area = (2)(π)(radius)^2 + (2)(π)(radius)(height)

Volume/Surface Area of a Cone

Volume = (π)(radius^2)(height)/3; Surface Area =(π)(radius^2) + (π)(radius)(length)

(Extended) Law of Sines

a/(sin A) = b/(sin B) = c/(sin C) = 2R. Where a, b, c are sides of a triangle, each opposite its respective angle A,B,C. R is the circumradius

Ptolemy's Theorem

ab + cd = ef Where ABCD is a cyclic quadrilateral with side lengths a,b,c,d and diagonals e, f (with a opposite b and c opposite d)

Pythagoream Theorem

a² + b² = c²

Common Factorizations and Simon's Favorite Factoring Trick a² - b² a³ - b³ a³ + b³ ab + a + b + 1 ab - a - b + 1

a² - b² = (a + b)(a - b); a³ - b³ = (a - b)(a² + ab + b²); a³ + b³ = (a + b)(a² - ab + b²); ab + a + b + 1 = (a + 1)(b + 1); ab - a - b + 1 = (a - 1)(b - 1) Simon's Favorite Factoring Trick refers to the the strategy of adding some number to both sides of a polynomial so that it will factor and then breaking the side that is an integer into its prime factors to find possible solutions.

Logarithm Rules: Logarithmic to Exponential: Addition: Subtraction: Exponent Reducing: Change of Base: Reciprocals:

log_a(b) = x → a^x = b; log_a(b) + log_a(c) = log_a(bc); log_a(b) - log_a(c) = log_a(b/c); log_a(bⁿ) = n*log_a(b); log_a(b) = log_c(b)/log_c(a); log_a(b) = 1/log_b(a)

Trigonometry Double Angle Formulas

sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos²(θ) - sin²(θ)

Addition/Subraction Identities

sin(a+B) = sin(a)*cos(B) + sin(B)*cos(a); cos(a+B) = cos(a)*cos(B) - sin(a)*sin(B)

Half Angle Identities

sin(θ/2) = ±√(1-cos(θ)/2; cos(θ/2) = ±√(1+cos(θ)/2

Trigonometry Pythagorean Identities

sin²(θ) + cos²(θ) = 1; cot²(θ) + 1 = csc²(θ); tan²(θ) + 1 = sec²(θ)

Quadratic Formula and Discriminant

x = [-b ± √(b² - 4ac)]/2a Discriminant = b² - 4ac Where a,b,c are the coefficients of the x², x¹, x⁰ terms respectively. If the discriminant is 0, the quadratic will have one real solution. If the discriminant is negative, the quadratic will have no real solutions. If the discriminant if positive, the quadratic will have two real solutions.