Vieta's FormulasFor ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with roots r1,r2r_1, r_2r1,r2:r1+r2=−ba,r1r2=car_1 + r_2 = -\frac{b}{a},\qquad r_1r_2 = \frac{c}{a}r1+r2=−ab,r1r2=ac
AM-GM InequalityFor non-negative reals a1,a2,…,ana_1, a_2, \ldots, a_na1,a2,…,an:a1+a2+⋯+ann≥a1a2⋯ann\frac{a_1+a_2+\cdots+a_n}{n} \geq \sqrt[n]{a_1a_2\cdots a_n}na1+a2+⋯+an≥na1a2⋯an
Fermat's Little TheoremIf ppp is prime and gcd(a,p)=1\gcd(a,p)=1gcd(a,p)=1:ap−1≡1(modp)a^{p-1} \equiv 1 \pmod pap−1≡1(modp)
Euler's TotientFor n=p1e1p2e2⋯pkekn = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}n=p1e1p2e2⋯pkek:φ(n)=n∏i=1k(1−1pi)\varphi(n)=n\prod_{i=1}^{k}\left(1-\frac{1}{p_i}\right)φ(n)=n∏i=1k(1−pi1)
Heron's FormulaFor sides a,b,ca,b,ca,b,c and s=a+b+c2s=\frac{a+b+c}{2}s=2a+b+c:Area=s(s−a)(s−b)(s−c)\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}Area=s(s−a)(s−b)(s−c)
Power of a PointFor PPP at distance ddd from the center of a circle of radius RRR, with a secant through A,BA,BA,B:PA⋅PB=∣d2−R2∣PA\cdot PB = \left|d^2-R^2\right|PA⋅PB=d2−R2
Stars and BarsWays to place nnn identical items into kkk bins:(n+k−1k−1)\binom{n+k-1}{k-1}(k−1n+k−1)
Pigeonhole PrincipleIf nnn items go into mmm containers with n>mn>mn>m:∃ a container with at least ⌈nm⌉ items\exists\text{ a container with at least }\left\lceil\frac{n}{m}\right\rceil\text{ items}∃ a container with at least ⌈mn⌉ items