← Topic 1 · Algebra
AMC 10/12 · Topic Cheatsheet

Topic 1 · Algebra

46 key results accumulated across 7 chapters.

Vieta (quadratic)
Ch 1
ax2+bx+c:  r+s=−ba,  rs=caax^2+bx+c:\; r+s=-\tfrac{b}{a},\; rs=\tfrac{c}{a}
Vieta (degree n)
Ch 1
∑r=−an−1an,    ∏r=(−1)na0an\sum r = -\tfrac{a_{n-1}}{a_n},\;\; \prod r = (-1)^n\tfrac{a_0}{a_n}
Symmetric identity
Ch 1
r2+s2=(r+s)2−2rsr^2+s^2 = (r+s)^2 - 2rs
SFFT
Ch 1
xy+ax+by=(x+b)(y+a)−abxy+ax+by = (x+b)(y+a) - ab
Add abab to factor; then divisor casework.
Telescoping
Ch 1
1n(n+1)=1n−1n+1\tfrac{1}{n(n+1)} = \tfrac1n - \tfrac1{n+1}
Middle terms cancel; only boundary survives.
Average speed (equal $d$)
Ch 1
vˉ=2v1v2v1+v2\bar v = \tfrac{2v_1v_2}{v_1+v_2}
Harmonic mean — not arithmetic.
Combined work
Ch 1
1a+1b=1T,    T=aba+b\tfrac1a + \tfrac1b = \tfrac1T,\;\; T = \tfrac{ab}{a+b}
Mixture
Ch 1
c=V1c1+V2c2V1+V2c = \tfrac{V_1 c_1 + V_2 c_2}{V_1 + V_2}
Weighted average by volume.
Floor / fractional part
Ch 1
x=⌊x⌋+{x},    {x}∈[0,1)x = \lfloor x\rfloor + \{x\},\;\;\{x\}\in[0,1)
Floor shift
Ch 1
⌊x+n⌋=⌊x⌋+n\lfloor x+n\rfloor = \lfloor x\rfloor + n
For integer nn.
Hermite ($n=2$)
Ch 1
⌊x⌋+⌊x+12⌋=⌊2x⌋\lfloor x\rfloor + \lfloor x+\tfrac12\rfloor = \lfloor 2x\rfloor
Multiples in $\{1,..,n\}$
Ch 1
∣{k≤n:d∣k}∣=⌊n/d⌋|\{k\le n : d\mid k\}| = \lfloor n/d\rfloor
Legendre
Ch 1
vp(n!)=∑i≥1⌊n/pi⌋v_p(n!) = \sum_{i\ge1}\lfloor n/p^i\rfloor
Arithmetic sum
Ch 2
Sn=n2(a1+an)S_n = \tfrac{n}{2}(a_1 + a_n)
Geometric sum
Ch 2
Sn=a1rn−1r−1  (r≠1)S_n = a_1\tfrac{r^n-1}{r-1}\;(r\neq1)
Infinite geometric
Ch 2
S∞=a11−r  (∣r∣<1)S_\infty = \tfrac{a_1}{1-r}\;(|r|<1)
AM–GM
Ch 3
a+b2≥ab,  =  ⟺  a=b\tfrac{a+b}{2} \ge \sqrt{ab},\; = \iff a=b
Log laws
Ch 3
log⁡(mn)=log⁡m+log⁡n,  log⁡mk=klog⁡m\log(mn)=\log m+\log n,\; \log m^k = k\log m
Change of base
Ch 3
log⁡bx=log⁡cxlog⁡cb\log_b x = \tfrac{\log_c x}{\log_c b}
Abs as distance
Ch 3
∣x−a∣=dist(x,a)|x-a| = \text{dist}(x,a)
Two-point min
Ch 3
min⁡(∣x−a∣+∣x−b∣)=∣a−b∣\min(|x-a|+|x-b|) = |a-b|
Median rule
Ch 3
min⁡∑i=1n∣x−ai∣\min\sum_{i=1}^n |x-a_i| at x=median⁡(ai)x = \operatorname{median}(a_i).
Triangle inequality
Ch 3
∣a+b∣≤∣a∣+∣b∣|a+b|\le |a|+|b|
Powers of i
Ch 4
i, −1, −i, 1  (cycle 4)i,\,-1,\,-i,\,1\;(\text{cycle }4)
Modulus
Ch 4
∣a+bi∣=a2+b2|a+bi| = \sqrt{a^2+b^2}
Product
Ch 4
(a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di) = (ac-bd)+(ad+bc)i
Polar form
Ch 4
z=reiθ=r(cos⁡θ+isin⁡θ)z = re^{i\theta} = r(\cos\theta + i\sin\theta)
De Moivre
Ch 4
(cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)
$n^{\text{th}}$ roots of unity
Ch 4
ωk=e2πik/n,    k=0,1,…,n−1\omega_k = e^{2\pi i k/n},\;\; k=0,1,\dots,n-1
Sum of roots of unity
Ch 4
∑k=0n−1ωk=0    (n≥2)\sum_{k=0}^{n-1}\omega_k = 0\;\;(n\ge 2)
Remainder theorem
Ch 5
P(x)=(x−a)Q(x)+P(a)P(x)=(x-a)Q(x)+P(a)
Factor theorem
Ch 5
(x−a)∣P(x)  ⟺  P(a)=0(x-a)\mid P(x) \iff P(a)=0
Rational roots
Ch 5
Candidate p/qp/q: p∣p\mid constant, q∣q\mid leading coeff.
Forward difference
Ch 5
ΔP(n)=P(n+1)−P(n)\Delta P(n) = P(n+1) - P(n)
Degree-dd poly ⇒ ΔdP\Delta^d P is constant.
Lagrange formula
Ch 5
P(x)=∑iyi∏j≠ix−xjxi−xjP(x) = \sum_i y_i \prod_{j\ne i}\tfrac{x-x_j}{x_i-x_j}
Newton forward
Ch 5
P(n+k)=∑i=0d(ki)ΔiP(n)P(n{+}k)=\sum_{i=0}^d\binom{k}{i}\Delta^i P(n)
Uniqueness
Ch 5
Degree ≤n\le n ⇒ uniquely determined by n+1n+1 values.
Composition
Ch 6
f(g(x))  — inner machine firstf(g(x))\;\text{— inner machine first}
Inverse
Ch 6
Swap x,yx,y then solve; f(f−1(x))=xf(f^{-1}(x))=x.
Domain
Ch 6
Exclude division by 0 and negatives under even roots.
Fixed point
Ch 6
f(x∗)=x∗f(x^*)=x^*
Iteration freezes there.
Cycle of period $k$
Ch 6
fk(x0)=x0f_k(x_0)=x_0, no smaller jj. Value depends on n mod kn \bmod k.
Counting pre-images
Ch 6
ff is mm-to-1 ⇒ fnf_n typically mnm^n-to-1.
Newton's identity
Ch 7
pk=e1pk−1−e2pk−2+⋯+(−1)k−1k ekp_k = e_1 p_{k-1} - e_2 p_{k-2} + \cdots + (-1)^{k-1} k\,e_k
Power sum from Vieta
Ch 7
p2=e12−2e2,    p3=e1p2−e2p1+3e3p_2 = e_1^2 - 2 e_2,\;\; p_3 = e_1 p_2 - e_2 p_1 + 3 e_3
Cubic shortcut
Ch 7
r3+s3+t3−3rst=(r+s+t)(r2+s2+t2−rs−rt−st)r^3+s^3+t^3 - 3 rst = (r+s+t)(r^2+s^2+t^2 - rs - rt - st)