Complete Syllabus & Gap Fillers

Study Sheet

Complete Syllabus & Gap Fillers

Everything the AMC 12 tests, with the pieces that were missing

What the AMC 12 Tests

Concept
The checklist

Everything on the AMC 10, plus: trigonometry (identities, sum/product formulas, law of sines/cosines, trig equations); logarithms and exponential equations; complex numbers (de Moivre, roots of unity, modulus geometry); polynomials in depth (Vieta for cubics/quartics, symmetric sums, roots of unity filter); conic sections; sequences and series including telescoping products and recursions with characteristic equations; combinatorics with generating-function intuition; probability with geometric probability and expected value; functions (composition, inverses, functional-equation style). Twenty-five questions, seventy-five minutes.

Reminder — Vieta's formulas:  r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1 r_2=\frac{c}{a}
Reminder — Law of Sines:  asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

Gap Fillers

Tip
Conics on the AMC 12

Ellipse x2a2+y2b2=1\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1 with c2=a2−b2c^2 = a^2 - b^2 (foci), hyperbola with c2=a2+b2c^2 = a^2 + b^2, parabola y2=4pxy^2 = 4px with focus (p,0)(p, 0). Reflective properties and the focus–directrix definition appear as "sum of distances" problems in disguise.

Tip
Geometric probability

Probability == favorable area ÷\div total area. Two numbers in [0,1][0,1] with ∣x−y∣<14|x - y| < \tfrac14: the complement is two right triangles of legs 34\tfrac34, so 1−916=7161 - \tfrac{9}{16} = \tfrac7{16}. Draw the region; the answer is usually a polygon or a circle piece.

Tip
Complex numbers as geometry

Multiplication rotates and scales; ∣z−a∣|z - a| is a distance; roots of zn=rnz^n = r^n form a regular nn-gon of circumradius rr (the roots of z6=64z^6 = 64 enclose area 636\sqrt3). Sums of roots of unity vanish, which collapses many trig sums.