Formulas & Methods

Study Sheet

Formulas & Methods

The AMC 10 formula sheet with the methods behind each entry

Algebra

Tip
Quadratics and Vieta

x=−b±b2−4ac2ax = \tfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}; roots sum to −ba-\tfrac ba, multiply to ca\tfrac ca; vertex at x=−b2ax = -\tfrac{b}{2a}. For a cubic x3+px2+qx+rx^3 + px^2 + qx + r: roots sum to −p-p, pairwise products qq, product −r-r. Difference of squares, sum/difference of cubes, (a+b+c)2=∑a2+2∑ab(a + b + c)^2 = \sum a^2 + 2\sum ab.

Reminder — Vieta's formulas:  r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1 r_2=\frac{c}{a}

Use Vieta whenever a problem asks about roots without asking for them. Complete the square for vertex/extremum questions. Factor by grouping or by testing rational roots.

Tip
Exponents, radicals, logs (light)

aman=am+na^ma^n = a^{m+n}, (am)n=amn(a^m)^n = a^{mn}, a−n=1ana^{-n} = \tfrac{1}{a^n}, a1/n=ana^{1/n} = \sqrt[n]a. Rationalize with conjugates. log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_bx + \log_by, log⁡bxn=nlog⁡bx\log_bx^n = n\log_bx, change of base log⁡bx=ln⁡xln⁡b\log_bx = \tfrac{\ln x}{\ln b}.

Rewrite everything in one base before comparing. Substitute u=2xu = 2^x (or similar) to turn exponential equations into polynomials.

Tip
Sequences and telescoping

Arithmetic: Sn=n(a1+an)2S_n = \tfrac{n(a_1 + a_n)}{2}. Geometric: Sn=a1−rn1−rS_n = a\tfrac{1 - r^n}{1 - r}, infinite a1−r\tfrac{a}{1 - r} for ∣r∣<1|r| < 1. ∑k2=n(n+1)(2n+1)6\sum k^2 = \tfrac{n(n+1)(2n+1)}{6}, ∑k3=(n(n+1)2)2\sum k^3 = \left(\tfrac{n(n+1)}{2}\right)^2. Telescope: 1k(k+1)=1k−1k+1\tfrac{1}{k(k+1)} = \tfrac1k - \tfrac{1}{k+1}.

Write out the first three and last term of any sum. Products of fractions often telescope after factoring numerators and denominators.

Counting, Probability & Number Theory

Tip
Counting

Permutations, combinations, (nk)=(nn−k)\binom nk = \binom{n}{n-k}, Pascal (nk)=(n−1k−1)+(n−1k)\binom nk = \binom{n-1}{k-1} + \binom{n-1}{k}. Stars and bars: (n+k−1k−1)\binom{n+k-1}{k-1} nonnegative solutions of x1+⋯+xk=nx_1 + \cdots + x_k = n. Inclusion–exclusion for two or three sets. Circular arrangements (n−1)!(n-1)!.

Decide ordered vs unordered first. Complement for "at least." Casework on a stated criterion. Bijection to a simpler set when the direct count is messy.

Tip
Probability and expected value

Conditional P(A∣B)=P(A∩B)P(B)P(A\mid B) = \tfrac{P(A\cap B)}{P(B)}; independent events multiply. Expected value ∑x P(x)\sum x\,P(x); linearity: E[X+Y]=E[X]+E[Y]E[X + Y] = E[X] + E[Y] always. Geometric probability: area ratios.

Draw the sample space; for continuous problems draw the region. Use indicators for expected counts.

Tip
Number theory

Modular arithmetic; Fermat ap−1≡1a^{p-1} \equiv 1; Euler aφ(n)≡1a^{\varphi(n)} \equiv 1; φ(n)=n∏(1−1p)\varphi(n) = n\prod(1 - \tfrac1p). Divisor count and sum. Base conversion. Legendre: vp(n!)=∑⌊n/pk⌋v_p(n!) = \sum\lfloor n/p^k\rfloor. CRT for simultaneous congruences.

Reduce mod small numbers to kill cases. Factor NN before counting divisors. For trailing zeros count factors of 55.

Geometry

Tip
Triangles

Heron: s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}. Area=rs=abc4R=12absin⁡C\text{Area} = rs = \tfrac{abc}{4R} = \tfrac12ab\sin C. Law of cosines c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C. Angle bisector theorem BDDC=ABAC\tfrac{BD}{DC} = \tfrac{AB}{AC}. Similar triangles: ratios of corresponding sides; areas scale by the square.

Reminder — Law of Cosines:  c2=a2+b2−2abcos⁡Cc^{2}=a^{2}+b^{2}-2ab\cos C

Look for similar triangles created by parallel lines. Compute area two ways to find an unknown height or inradius. Special triangles 3030-6060-9090 and 4545-4545-9090 by sight.

Tip
Circles and polygons

Inscribed angle =12= \tfrac12 arc; tangent ⊥\perp radius; power of a point PA⋅PB=PC⋅PD=PT2PA\cdot PB = PC\cdot PD = PT^2; chord length 2r2−d22\sqrt{r^2 - d^2}. Regular nn-gon: interior angle (n−2)180∘n\tfrac{(n-2)180^\circ}{n}, area 12\tfrac12 perimeter ×\times apothem. Sector area θ360πr2\tfrac{\theta}{360}\pi r^2.

Reminder — Inscribed angle theorem:  inscribed angle=12 central angle on the same arc\text{inscribed angle}=\tfrac12\,\text{central angle on the same arc}

Connect centers to tangent points and to chord midpoints (right angles appear). For two circles, the line of centers, the radii, and the tangent form right triangles.

Tip
Coordinates and 3D

Distance, midpoint, slope; perpendicular slopes multiply to −1-1; point-to-line ∣ax0+by0+c∣a2+b2\tfrac{|ax_0 + by_0 + c|}{\sqrt{a^2+b^2}}; shoelace for polygon area. Solids: prism/cylinder BhBh, pyramid/cone 13Bh\tfrac13Bh, sphere 43πr3\tfrac43\pi r^3 and 4πr24\pi r^2; cube diagonal s3s\sqrt3.

Place the figure so that a vertex is the origin and a side lies on an axis. For 3D shortest paths, unfold the surface into a plane.

Concept
The universal problem-solving loop

Read twice. Restate the goal in your own words. List the givens. Pick a representation. Try small cases. Look for symmetry, an invariant, or an extremal object. Compute, then CHECK against a second method or a sanity bound.

At AMC 10/12 and AIME level, the second method is what separates a 44-second guess from a 1212-second certainty: compute a probability two ways (complement and direct), a length via two theorems, a count via a recurrence and a formula. Casework must be organized by a stated criterion so nothing is double-counted; algebra must be checked by substituting back.