← Topic 1 · Algebra
AMC 10/12 · Cheatsheet

Topic 1 · Algebra

Chapter 1 · Core algebra tools

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Vieta (quadratic)
ax2+bx+c:  r+s=−ba,  rs=caax^2+bx+c:\; r+s=-\tfrac{b}{a},\; rs=\tfrac{c}{a}
Vieta (degree n)
∑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
r2+s2=(r+s)2−2rsr^2+s^2 = (r+s)^2 - 2rs
SFFT
xy+ax+by=(x+b)(y+a)−abxy+ax+by = (x+b)(y+a) - ab
Add abab to factor; then divisor casework.
Telescoping
1n(n+1)=1n−1n+1\tfrac{1}{n(n+1)} = \tfrac1n - \tfrac1{n+1}
Middle terms cancel; only boundary survives.
Average speed (equal $d$)
vˉ=2v1v2v1+v2\bar v = \tfrac{2v_1v_2}{v_1+v_2}
Harmonic mean — not arithmetic.
Combined work
1a+1b=1T,    T=aba+b\tfrac1a + \tfrac1b = \tfrac1T,\;\; T = \tfrac{ab}{a+b}
Mixture
c=V1c1+V2c2V1+V2c = \tfrac{V_1 c_1 + V_2 c_2}{V_1 + V_2}
Weighted average by volume.
Floor / fractional part
x=⌊x⌋+{x},    {x}∈[0,1)x = \lfloor x\rfloor + \{x\},\;\;\{x\}\in[0,1)
Floor shift
⌊x+n⌋=⌊x⌋+n\lfloor x+n\rfloor = \lfloor x\rfloor + n
For integer nn.
Hermite ($n=2$)
⌊x⌋+⌊x+12⌋=⌊2x⌋\lfloor x\rfloor + \lfloor x+\tfrac12\rfloor = \lfloor 2x\rfloor
Multiples in $\{1,..,n\}$
∣{k≤n:d∣k}∣=⌊n/d⌋|\{k\le n : d\mid k\}| = \lfloor n/d\rfloor
Legendre
vp(n!)=∑i≥1⌊n/pi⌋v_p(n!) = \sum_{i\ge1}\lfloor n/p^i\rfloor