AMC 12 — Core Concepts
20 essential topic areas. Master these and you master the exam.
TOPIC 01
Quadratics & Vieta's Formulas
For \(ax^2+bx+c=0\) with roots \(r,s\):
\(r+s = -\dfrac{b}{a}\), \quad \(rs = \dfrac{c}{a}\)
Discriminant: \(\Delta = b^2 - 4ac\)
Discriminant: \(\Delta = b^2 - 4ac\)
🔑 Must Memorize
Sum of roots = −b/a
Product of roots = c/a
Real roots ⟺ Δ ≥ 0
Example
If \(x^2-5x+6=0\), then \(r+s=5\), \(rs=6\). Roots are 2, 3. ✓
TOPIC 02
Logarithms & Exponents
Key identities to know cold:
\(\log_b(xy)=\log_b x+\log_b y\)
\(\log_b\!\left(\tfrac{x}{y}\right)=\log_b x - \log_b y\)
\(\log_b(x^n)=n\log_b x\)
\(\log_b x = \dfrac{\ln x}{\ln b}\) (Change of base)
\(\log_b\!\left(\tfrac{x}{y}\right)=\log_b x - \log_b y\)
\(\log_b(x^n)=n\log_b x\)
\(\log_b x = \dfrac{\ln x}{\ln b}\) (Change of base)
log(product) = sum of logs
Change-of-base
Example
\(\log_2 8 + \log_2 4 = 3+2=5=\log_2 32\) ✓
TOPIC 03
Sequences & Series
Arithmetic (common difference \(d\)) and Geometric (common ratio \(r\)):
AP Sum: \(S_n = \dfrac{n}{2}(a_1+a_n)\)
GP Sum: \(S_n = \dfrac{a_1(1-r^n)}{1-r}\)
Infinite GP (\(|r|<1\)): \(S = \dfrac{a_1}{1-r}\)
GP Sum: \(S_n = \dfrac{a_1(1-r^n)}{1-r}\)
Infinite GP (\(|r|<1\)): \(S = \dfrac{a_1}{1-r}\)
AP: average × count
Infinite GP: a/(1−r)
Example
Sum of \(1+\tfrac{1}{2}+\tfrac{1}{4}+\cdots = \dfrac{1}{1-1/2}=2\) ✓
TOPIC 04
Counting & Combinatorics
Permutations and combinations:
\(P(n,k)=\dfrac{n!}{(n-k)!}\), \quad \binom{n}{k}=\dfrac{n!}{k!(n-k)!}\)
Inclusion-Exclusion: \(|A\cup B|=|A|+|B|-|A\cap B|\)
Inclusion-Exclusion: \(|A\cup B|=|A|+|B|-|A\cap B|\)
Order matters → P(n,k)
Order doesn't → C(n,k)
Example
\(\binom{5}{2}=\dfrac{5!}{2!\cdot 3!}=10\) ways to choose 2 from 5. ✓
TOPIC 05
Probability
Conditional probability and independence:
\(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\)
Independent: \(P(A\cap B)=P(A)\cdot P(B)\)
Complement: \(P(A^c)=1-P(A)\)
Independent: \(P(A\cap B)=P(A)\cdot P(B)\)
Complement: \(P(A^c)=1-P(A)\)
Complement trick: 1 − P(none)
Geometric probability
Example
P(at least one head in 3 flips) = 1 − P(all tails) = 1 − 1/8 = 7/8 ✓
TOPIC 06
Triangle Geometry
Area formulas and key theorems:
Area \(= \tfrac{1}{2}ab\sin C\)
Law of Cosines: \(c^2=a^2+b^2-2ab\cos C\)
Heron's: \(K=\sqrt{s(s-a)(s-b)(s-c)}\)
Law of Cosines: \(c^2=a^2+b^2-2ab\cos C\)
Heron's: \(K=\sqrt{s(s-a)(s-b)(s-c)}\)
30-60-90: 1:√3:2
45-45-90: 1:1:√2
Pythagorean triples
Example
3-4-5 right triangle: area = ½·3·4 = 6. ✓
TOPIC 07
Circles & Arcs
Circle theorems for AMC:
Inscribed angle = ½ × central angle
Power of a point: \(PA\cdot PB = PC\cdot PD\)
Arc length \(= r\theta\) (radians)
Power of a point: \(PA\cdot PB = PC\cdot PD\)
Arc length \(= r\theta\) (radians)
Inscribed angle theorem
Power of a point
Tangent ⊥ radius
Example
Central angle 60°, radius 6: arc length = 6·(π/3) = 2π ✓
TOPIC 08
Complex Numbers
Polar form and De Moivre's theorem:
\(z = a+bi = r(\cos\theta+i\sin\theta)\)
\(|z|=\sqrt{a^2+b^2}\)
De Moivre: \(z^n = r^n(\cos n\theta+i\sin n\theta)\)
\(|z|=\sqrt{a^2+b^2}\)
De Moivre: \(z^n = r^n(\cos n\theta+i\sin n\theta)\)
|z₁z₂| = |z₁||z₂|
arg(z₁z₂) = arg z₁ + arg z₂
Example
\((1+i)^2 = 2i\). Note \(|1+i|=\sqrt{2}\), arg = π/4. ✓
TOPIC 09
Functions & Polynomials
Polynomial facts and transformations:
Factor Theorem: \((x-a)\mid p(x)\) iff \(p(a)=0\)
Rational Root: \(\pm\dfrac{p}{q}\), \(p\mid a_0\), \(q\mid a_n\)
Sum of roots (degree \(n\)): \(-a_{n-1}/a_n\)
Rational Root: \(\pm\dfrac{p}{q}\), \(p\mid a_0\), \(q\mid a_n\)
Sum of roots (degree \(n\)): \(-a_{n-1}/a_n\)
Remainder theorem
Even/odd functions
Example
\(p(x)=x^3-6x^2+11x-6\). \(p(1)=0\), so \((x-1)\) is a factor. ✓
TOPIC 10
Number Theory
Divisibility, modular arithmetic, primes:
Euler: \(a^{\phi(n)}\equiv 1\pmod{n}\) (gcd=1)
\(\phi(p^k)=p^{k-1}(p-1)\)
# of divisors of \(p_1^{a_1}\cdots = (a_1+1)\cdots\)
\(\phi(p^k)=p^{k-1}(p-1)\)
# of divisors of \(p_1^{a_1}\cdots = (a_1+1)\cdots\)
Divisibility rules: 3,9,11
φ(n) Euler totient
Number of divisors
Example
12 = 2²·3¹ has (2+1)(1+1) = 6 divisors: 1,2,3,4,6,12. ✓
Answer Key & Solutions
Complete solutions with step-by-step explanations.
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