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Olympiad Toolkit

A growing collection of essential formulas, identities, and techniques for mathematical olympiad problem solving.

  1. 1.
    Sum of the first n integers
    1+2+⋯+n=n(n+1)21+2+\cdots+n=\frac{n(n+1)}{2}
  2. 2.
    Sum of squares
    12+22+⋯+n2=n(n+1)(2n+1)61^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}
  3. 3.
    Sum of cubes
    13+23+⋯+n3=(n(n+1)2)21^3+2^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2
  4. 4.
    Geometric Sequence and Series
    an=a1rn−1a_n=a_1r^{n-1}
    an2=an−1an+1a_n^2=a_{n-1}a_{n+1}
    Sn=a1+a1r+a1r2+⋯+a1rn−1=a1(rn−1)r−1S_n=a_1+a_1r+a_1r^2+\cdots+a_1r^{n-1}=\frac{a_1(r^n-1)}{r-1}
    ∣r∣<1:S∞=a1+a1r+a1r2+⋯=a11−r|r|<1:\quad S_\infty=a_1+a_1r+a_1r^2+\cdots=\frac{a_1}{1-r}
  5. 5.
    Telescoping series
    11⋅2+12⋅3+⋯+1n(n+1)=(1−12)+⋯+(1n−1n+1)=1−1n+1\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{n(n+1)}=\left(1-\frac{1}{2}\right)+\cdots+\left(\frac{1}{n}-\frac{1}{n+1}\right)=1-\frac{1}{n+1}
  6. 6.
    Square of a sum
    (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2
  7. 7.
    Square of a difference
    (a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2
  8. 8.
    Cube of a sum
    (a+b)3=a3+3a2b+3ab2+b3(a+b)^3=a^3+3a^2b+3ab^2+b^3
  9. 9.
    Cube of a difference
    (a−b)3=a3−3a2b+3ab2−b3(a-b)^3=a^3-3a^2b+3ab^2-b^3
  10. 10.
    Difference of squares
    a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)
  11. 11.
    Difference of cubes
    a3−b3=(a−b)(a2+ab+b2)a^3-b^3=(a-b)(a^2+ab+b^2)
  12. 12.
    Sum of cubes (factored)
    a3+b3=(a+b)(a2−ab+b2)a^3+b^3=(a+b)(a^2-ab+b^2)
  13. 13.
    General difference of powers
    an−bn=(a−b)(an−1+an−2b+⋯+bn−1)a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+\cdots+b^{n-1}\right)
  14. 14.
    Sum of odd power
    n odd:an+bn=(a+b)(an−1−an−2b+⋯+bn−1)n\text{ odd}:\quad a^n+b^n=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots+b^{n-1}\right)
  15. 15.
    Difference of even power
    n even:an−bn=(a+b)(an−1−an−2b+⋯−bn−1)n\text{ even}:\quad a^n-b^n=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots-b^{n-1}\right)
  16. 16.
    Hockey Stick Identity
    (kk)+(k+1k)+(k+2k)+⋯+(nk)=(n+1k+1)\binom{k}{k}+\binom{k+1}{k}+\binom{k+2}{k}+\cdots+\binom{n}{k}=\binom{n+1}{k+1}
  17. 17.
    Vandermonde's Identity
    (m0)(nk)+(m1)(nk−1)+(m2)(nk−2)+⋯+(mk)(n0)=(m+nk)\binom{m}{0}\binom{n}{k}+\binom{m}{1}\binom{n}{k-1}+\binom{m}{2}\binom{n}{k-2}+\cdots+\binom{m}{k}\binom{n}{0}=\binom{m+n}{k}
  18. 18.
    Symmetric cubic sum
    a3+b3+c3=(a+b+c)(a2+b2+c2−ab−ac−bc)+3abca^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc
  19. 19.
    Factoring quadratic by grouping
    x2+(a+b)x+ab=(x+a)(x+b)x^2+(a+b)x+ab=(x+a)(x+b)
  20. 20.
    Binomial Theorem
    (a+b)n=(n0)an+(n1)an−1b+(n2)an−2b2+⋯+(nn)bn(a+b)^n=\binom{n}{0}a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+\cdots+\binom{n}{n}b^n
  21. 21.
    Quadratic Formula
    ax2+bx+c=0  ⇒  x=−b±b2−4ac2aax^2+bx+c=0\;\Rightarrow\;x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
  22. 22.
    Approaches to Similar Equations
    Addition and Subtraction.\text{Addition and Subtraction.}
  23. 23.
    SFFT (Simon's Favorite Factoring Trick)
    xy+ax+by=c  ⇒  (x+b)(y+a)=c+abxy+ax+by=c\;\Rightarrow\;(x+b)(y+a)=c+ab
    GSFFT (General Form)\text{GSFFT (General Form)}
    axy+bx+cy=daxy+bx+cy=d
    a2xy+abx+acy=ada^2xy+abx+acy=ad
    (ax+c)(ay+b)=ad+bc(ax+c)(ay+b)=ad+bc
  24. 24.
    Square of a sum (n terms)
    (x1+x2+⋯+xn)2=x12+⋯+xn2+2x1x2+⋯+2xn−1xn(x_1+x_2+\cdots+x_n)^2=x_1^2+\cdots+x_n^2+2x_1x_2+\cdots+2x_{n-1}x_n
  25. 25.
    Arithmetic Sequence and Series
    an=a1+(n−1)da_n=a_1+(n-1)d
    Sn=a1+a2+⋯+an=n2(a1+an)S_n=a_1+a_2+\cdots+a_n=\frac{n}{2}(a_1+a_n)
    an=an−1+an+12a_n=\frac{a_{n-1}+a_{n+1}}{2}
  26. 26.
    Expansion (or factorization) of three binomials
    abc−ab−ac−bc+a+b+c−1=(a−1)(b−1)(c−1)abc-ab-ac-bc+a+b+c-1=(a-1)(b-1)(c-1)
  27. 27.
    Expansion (or factorization) of two binomials
    a2+ab+ac+bc=(a+b)(a+c)a^2+ab+ac+bc=(a+b)(a+c)
  28. 28.
    Max-QM-AM-GM-HM-Min Inequality
    x1,…,xn∈R+x_1,\dots,x_n\in\mathbb{R}^+
    max⁡{xi}≥x12+x22+⋯+xn2n≥x1+x2+⋯+xnn≥x1x2⋯xnn≥n1x1+1x2+⋯+1xn≥min⁡{xi}\max\{x_i\}\ge\sqrt{\frac{x_1^2+x_2^2+\cdots+x_n^2}{n}}\ge\frac{x_1+x_2+\cdots+x_n}{n}\ge\sqrt[n]{x_1x_2\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}\ge\min\{x_i\}
    Equality holds if and only if all xi are equal.\text{Equality holds if and only if all }x_i\text{ are equal.}
  29. 29.
    Integer and Fractional Parts
    [x]: Integer part{x}: Fractional part[x]:\text{ Integer part}\qquad\{x\}:\text{ Fractional part}
    We can write x as:\text{We can write }x\text{ as:}
    x=k+r,k∈Z,  r∈R,  0≤r<1x=k+r,\qquad k\in\mathbb{Z},\;r\in\mathbb{R},\;0\le r<1
  30. 30.
    Vieta's Formula
    ax2+bx+c=0ax^2+bx+c=0
    r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1r_2=\frac{c}{a}
    ax3+bx2+cx+d=0ax^3+bx^2+cx+d=0
    r1+r2+r3=−bar_1+r_2+r_3=-\frac{b}{a}
    r1r2+r1r3+r2r3=car_1r_2+r_1r_3+r_2r_3=\frac{c}{a}
    r1r2r3=−dar_1r_2r_3=-\frac{d}{a}
    anxn+an−1xn−1+⋯+a1x+a0=0a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0
    r1+r2+⋯+rn=−an−1anr_1+r_2+\cdots+r_n=-\frac{a_{n-1}}{a_n}
    r1r2+r1r3+⋯+rn−1rn=an−2anr_1r_2+r_1r_3+\cdots+r_{n-1}r_n=\frac{a_{n-2}}{a_n}
    ⋯\cdots
    r1r2⋯rn−1+⋯+r2r3⋯rn=(−1)n−1a1anr_1r_2\cdots r_{n-1}+\cdots+r_2r_3\cdots r_n=(-1)^{n-1}\frac{a_1}{a_n}
    r1r2⋯rn=(−1)na0anr_1r_2\cdots r_n=(-1)^n\frac{a_0}{a_n}
  31. 31.
    Signs and Roots of Quadratics
    f(x)=ax2+bx+c,Δ=b2−4acf(x)=ax^2+bx+c,\qquad \Delta=b^2-4ac
    ConditionNumber of Real RootsSign of f(x)f(x)
    Δ>0\Delta>02
    sign(a)0−sign(a)0sign(a)r₁r₂
    Δ=0\Delta=01
    sign(a)0sign(a)r
    Δ<0\Delta<00
    sign(a)
  32. 32.
    Remainder of Polynomial Division
    A(x)÷B(x):  A(x)=B(x)Q(x)+R(x),deg⁡(R(x))<deg⁡(B(x))A(x)\div B(x):\;A(x)=B(x)Q(x)+R(x),\quad \deg(R(x))<\deg(B(x))
    ⇒ If B(x)=0, then A(x)=R(x)\Rightarrow\text{ If }B(x)=0,\text{ then }A(x)=R(x)
  33. 33.
    Number of Positive Divisors
    n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
    Number of positive divisors of n:  (α1+1)(α2+1)⋯(αk+1)\text{Number of positive divisors of }n:\;(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_k+1)
  34. 34.
    Sum of Positive Divisors
    n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
    (1+p1+p12+⋯+p1α1)(1+p2+p22+⋯+p2α2)⋯(1+pk+pk2+⋯+pkαk)(1+p_1+p_1^2+\cdots+p_1^{\alpha_1})(1+p_2+p_2^2+\cdots+p_2^{\alpha_2})\cdots(1+p_k+p_k^2+\cdots+p_k^{\alpha_k})
    =p1α1+1−1p1−1×p2α2+1−1p2−1×⋯×pkαk+1−1pk−1=\frac{p_1^{\alpha_1+1}-1}{p_1-1}\times\frac{p_2^{\alpha_2+1}-1}{p_2-1}\times\cdots\times\frac{p_k^{\alpha_k+1}-1}{p_k-1}
  35. 35.
    Product of Positive Divisors
    n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
    t=(α1+1)(α2+1)⋯(αk+1)t=(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_k+1)
    Product of positive divisors of n:  nt/2\text{Product of positive divisors of }n:\;n^{t/2}
  36. 36.
    General Solutions of Trigonometric Equations
    sin⁡α=sin⁡β  ⇒  α=β+360k,  180−β+360k,  k∈Z\sin\alpha=\sin\beta\;\Rightarrow\;\alpha=\beta+360k,\;180-\beta+360k,\;k\in\mathbb{Z}
    cos⁡α=cos⁡β  ⇒  α=±β+360k,  k∈Z\cos\alpha=\cos\beta\;\Rightarrow\;\alpha=\pm\beta+360k,\;k\in\mathbb{Z}
    tan⁡α=tan⁡β  ⇒  α=β+180k,  k∈Z\tan\alpha=\tan\beta\;\Rightarrow\;\alpha=\beta+180k,\;k\in\mathbb{Z}
    cot⁡α=cot⁡β  ⇒  α=β+180k,  k∈Z\cot\alpha=\cot\beta\;\Rightarrow\;\alpha=\beta+180k,\;k\in\mathbb{Z}
  37. 37.
    Trigonometric Transformations
    sin⁡(90∘−α)=cos⁡α\sin(90^\circ-\alpha)=\cos\alpha
    cos⁡(90∘−α)=sin⁡α\cos(90^\circ-\alpha)=\sin\alpha
    sin⁡(180∘−α)=sin⁡α\sin(180^\circ-\alpha)=\sin\alpha
    cos⁡(180∘−α)=−cos⁡α\cos(180^\circ-\alpha)=-\cos\alpha
    sin⁡(−α)=−sin⁡α\sin(-\alpha)=-\sin\alpha
    cos⁡(−α)=cos⁡α\cos(-\alpha)=\cos\alpha
    ⋯\cdots
  38. 38.
    Trigonometric Identities
    sin⁡2α+cos⁡2α=1\sin^2\alpha+\cos^2\alpha=1
    sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
    cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
    sin⁡2α=2sin⁡αcos⁡α\sin2\alpha=2\sin\alpha\cos\alpha
    cos⁡2α=2cos⁡2α−1=1−2sin⁡2α\cos2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha
    ⋯\cdots
  39. 39.
    Trigonometric Sum-to-Product Identities
    sin⁡α+sin⁡β=2sin⁡(α+β2)cos⁡(α−β2)\sin\alpha+\sin\beta=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
    sin⁡α−sin⁡β=2cos⁡(α+β2)sin⁡(α−β2)\sin\alpha-\sin\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)
    cos⁡α+cos⁡β=2cos⁡(α+β2)cos⁡(α−β2)\cos\alpha+\cos\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
    cos⁡α−cos⁡β=−2sin⁡(α+β2)sin⁡(α−β2)\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)
  40. 40.
    Trigonometric Product-to-Sum Identities
    sin⁡pcos⁡q=12(sin⁡(p+q)+sin⁡(p−q))\sin p\cos q=\frac{1}{2}(\sin(p+q)+\sin(p-q))
    cos⁡pcos⁡q=12(cos⁡(p+q)+cos⁡(p−q))\cos p\cos q=\frac{1}{2}(\cos(p+q)+\cos(p-q))
    sin⁡psin⁡q=12(cos⁡(p−q)−cos⁡(p+q))\sin p\sin q=\frac{1}{2}(\cos(p-q)-\cos(p+q))
  41. 41.
    Dual Pythagorean Theorem
    Dual Pythagorean Theorem figure
    CB2−CA2=EB2−EA2=FB2−FA2=DB2−DA2CB^2-CA^2=EB^2-EA^2=FB^2-FA^2=DB^2-DA^2
  42. 42.
    Angle Bisector Theorem
    Angle Bisector Theorem figure
    BDDC=cb\frac{BD}{DC}=\frac{c}{b}
  43. 43.
    Partial Fraction Decomposition
    5x+1(x−1)(x+2)=Ax−1+Bx+2\frac{5x+1}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}
    Method 1: Comparing coefficients\text{Method 1: Comparing coefficients}
    Method 2: Heaviside Cover-Up Method\text{Method 2: Heaviside Cover-Up Method}
  44. 44.
    Creating n Equations in n Variables for Probability Problems
  45. 45.
    Adjacency in Permutations
    Treat adjacent objects as one block\text{Treat adjacent objects as one block}
    Example: logarithm\text{Example: }\boxed{\mathrm{log}}\mathrm{arithm}
    7!×3!7!\times3!
  46. 46.
    Divisibility: Definition and Properties

    Definition

    a∣b  ⟺  there exists an integer k s.t. b=aka\mid b\iff \text{there exists an integer }k\text{ s.t. }b=ak

    Properties

    1.a∣b,  b∣c⟹a∣c1.\quad a\mid b,\; b\mid c \Longrightarrow a\mid c
    2.a∣b,  a∣c⟹a∣mb+nc2.\quad a\mid b,\; a\mid c \Longrightarrow a\mid mb+nc
    3.gcd⁡(a,b)=1,  a∣bc⟹a∣c3.\quad \gcd(a,b)=1,\; a\mid bc \Longrightarrow a\mid c
    4.p∣ab⟹p∣a or p∣b,p prime.4.\quad p\mid ab \Longrightarrow p\mid a \text{ or } p\mid b,\qquad p\text{ prime.}
  47. 47.
    Modular Arithmetic: Definition and Properties

    Definition

    a≡b(modm)  ⟺  m∣(a−b)a\equiv b\pmod m\iff m\mid(a-b)

    Properties

    1.a≡b(modm),  b≡c(modm)⟹a≡c(modm)1.\quad a\equiv b\pmod m,\; b\equiv c\pmod m\Longrightarrow a\equiv c\pmod m
    2.a≡b(modm)⟹an≡bn(modm),n∈Z+2.\quad a\equiv b\pmod m\Longrightarrow a^n\equiv b^n\pmod m,\quad n\in\mathbb Z^+
    3.ac≡bc(modm)  ⟺  a≡b(modmgcd⁡(m,c))3.\quad ac\equiv bc\pmod m\iff a\equiv b\pmod{\frac{m}{\gcd(m,c)}}
    4.a≡b(modm),  d∣m⟹a≡b(modd)4.\quad a\equiv b\pmod m,\; d\mid m\Longrightarrow a\equiv b\pmod d
  48. 48.
    Logarithms: Definition and Properties

    Definition

    log⁡bx=y  ⟺  by=x\log_bx=y\iff b^y=x

    Domain

    x>0,b>0,b≠1x>0,\qquad b>0,\qquad b\ne1

    Properties

    1.log⁡ba+log⁡bc=log⁡b(ac)1.\quad \log_ba+\log_bc=\log_b(ac)
    2.log⁡ba−log⁡bc=log⁡b(ac)2.\quad \log_ba-\log_bc=\log_b\left(\frac ac\right)
    3.log⁡b(an)=nlog⁡ba3.\quad \log_b(a^n)=n\log_ba
    4.log⁡ba=log⁡calog⁡cb4.\quad \log_ba=\frac{\log_ca}{\log_cb}
    5.log⁡ba=ln⁡aln⁡b5.\quad \log_ba=\frac{\ln a}{\ln b}
  49. 49.
    Two Approaches to Logarithm Problems

    Approach 1

    Convert to exponential form

    Approach 2

    Use a change of variable

  50. 50.
    Principle of Inclusion–Exclusion (PIE): Fundamentals and Extensions

    Two Sets

    ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|

    Three Sets

    ∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

    General Formula

    ∣A1∪⋯∪An∣=∑i∣Ai∣−∑i<j∣Ai∩Aj∣+∑i<j<k∣Ai∩Aj∩Ak∣−⋯|A_1\cup\cdots\cup A_n|=\sum_i|A_i|-\sum_{i<j}|A_i\cap A_j|+\sum_{i<j<k}|A_i\cap A_j\cap A_k|-\cdots

    Exactly One Set

    PA1,A2=∣A1∣+∣A2∣−2∣A1∩A2∣P_{A_1,A_2}=|A_1|+|A_2|-2|A_1\cap A_2|
    PA1,A2,A3=∣A1∣+∣A2∣+∣A3∣−2∣A1∩A2∣−2∣A1∩A3∣−2∣A2∩A3∣+3∣A1∩A2∩A3∣P_{A_1,A_2,A_3}=|A_1|+|A_2|+|A_3|-2|A_1\cap A_2|-2|A_1\cap A_3|-2|A_2\cap A_3|+3|A_1\cap A_2\cap A_3|
  51. 51.
    Euler's Totient Function φ(n)
    φ(n)=∣{1≤i≤n∣gcd⁡(i,n)=1}∣\varphi(n)=|\{1\le i\le n\mid\gcd(i,n)=1\}|

    φ(n)\varphi(n) is the number of integers from 11 to nn that are relatively prime to nn.

    If n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k},

    φ(n)=n(1−1p1)(1−1p2)⋯(1−1pk)\varphi(n)=n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right)\cdots\left(1-\frac{1}{p_k}\right)
  52. 52.
    Euler's Theorem

    Let nn be a positive integer. If gcd⁡(n,a)=1\gcd(n,a)=1,

    aφ(n)≡1(modn)a^{\varphi(n)}\equiv1\pmod n
  53. 53.
    Graphing Functions: Transformations and Absolute Value Functions

    Graph of ∣x∣+∣y∣=5|x|+|y|=5:

    Graph of |x|+|y|=5

    Graph Transformations

    TransformationExample
    Horizontal Shifty=f(x−a)y=f(x-a)
    Vertical Shifty=f(x)+ay=f(x)+a
    Horizontal Stretchy=f(x/a)y=f(x/a)
    Vertical Stretchy=af(x)y=af(x)
    Reflection in x-axisy=−f(x)y=-f(x)
    Reflection in y-axisy=f(−x)y=f(-x)
  54. 54.
    Area Formulas and Important Geometry Formulas

    Triangle Area Formulas

    1.[ABC]=12Base⋅Height1.\quad [ABC]=\frac12\text{Base}\cdot\text{Height}
    2.[ABC]=12bcsin⁡A2.\quad [ABC]=\frac12 bc\sin A
    3.s=a+b+c2,[ABC]=s(s−a)(s−b)(s−c)3.\quad s=\frac{a+b+c}{2},\qquad [ABC]=\sqrt{s(s-a)(s-b)(s-c)}
    4.[ABC]=a234(equilateral triangle)4.\quad [ABC]=\frac{a^2\sqrt3}{4}\qquad\text{(equilateral triangle)}

    Quadrilateral Area Formulas

    1.[ABCD]=12AC⋅BDsin⁡α1.\quad [ABCD]=\frac12 AC\cdot BD\sin\alpha
    2.[ABCD]=h(AB+CD)2(trapezoid)2.\quad [ABCD]=\frac{h(AB+CD)}{2}\qquad\text{(trapezoid)}
    3.[ABCD]=bh(parallelogram)3.\quad [ABCD]=bh\qquad\text{(parallelogram)}

    Circle Formulas

    1.A=πr2,C=2πr1.\quad A=\pi r^2,\qquad C=2\pi r
    2.Asector=α360πr22.\quad A_{\text{sector}}=\frac{\alpha}{360}\pi r^2
    3.ℓarc=α360⋅2πr3.\quad \ell_{\text{arc}}=\frac{\alpha}{360}\cdot2\pi r

    Special Right Triangles

    30∘ ⁣−!60∘ ⁣−!90∘:1:3:230^\circ\!-!60^\circ\!-!90^\circ:\qquad 1:\sqrt3:2
    45∘ ⁣−!45∘ ⁣−!90∘:1:1:245^\circ\!-!45^\circ\!-!90^\circ:\qquad 1:1:\sqrt2
  55. 55.
    Shoelace Theorem
    Polygon with vertices A_1 through A_n
    Area=12∣x1y2+x2y3+⋯+xn−1yn+xny1−y1x2−y2x3−⋯−yn−1xn−ynx1∣\text{Area}=\tfrac{1}{2}\bigl|x_1y_2+x_2y_3+\cdots+x_{n-1}y_n+x_ny_1-y_1x_2-y_2x_3-\cdots-y_{n-1}x_n-y_nx_1\bigr|
  56. 56.
    Gauss' Idea (Rainbow Idea)

    Pair up the elements:

    Gauss pairing 1+100, 2+99, ... equals 50 x 101

    or

    S=1+2+3+⋯+98+99+100S=1+2+3+\cdots+98+99+100
    S=100+99+98+⋯+3+2+1S=100+99+98+\cdots+3+2+1
    2S=101+101+⋯+1012S=101+101+\cdots+101
    2S=100×1012S=100\times101

    Therefore:

    S=100×1012S=\frac{100\times101}{2}
  57. 57.
    Famous Trigonometric Values

    For the following important angles:

    θ\thetasin⁡θ\sin\thetacos⁡θ\cos\thetatan⁡θ\tan\thetacot⁡θ\cot\theta
    0∘0^\circ001100Undefined
    30∘30^\circ12\tfrac{1}{2}32\tfrac{\sqrt3}{2}33\tfrac{\sqrt3}{3}3\sqrt3
    45∘45^\circ22\tfrac{\sqrt2}{2}22\tfrac{\sqrt2}{2}1111
    60∘60^\circ32\tfrac{\sqrt3}{2}12\tfrac{1}{2}3\sqrt333\tfrac{\sqrt3}{3}
    90∘90^\circ1100Undefined00
    180∘180^\circ00−1-100Undefined
    270∘270^\circ−1-100Undefined00
  58. 58.
    Law of Cosines
    Triangle ABC with sides a, b, c
    a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A
  59. 59.
    Ptolemy's Theorem
    Cyclic quadrilateral ABCD with diagonals

    A quadrilateral ABCDABCD is cyclic if and only if:

    AB⋅CD+AD⋅BC=AC⋅BDAB\cdot CD+AD\cdot BC=AC\cdot BD
  60. 60.
    Angles in Circles
    Central angle
    Inscribed angle
    Tangent-chord angle
    Interior angle
    Exterior angle
  61. 61.
    4 Rules of Cyclic Quadrilaterals
    Opposite angles
    Opposite angles
    1.∠A+∠C=180∘1.\quad \angle A+\angle C=180^\circ
    Equal angles
    2.∠A1=∠C12.\quad \angle A_1=\angle C_1
    Intersecting chords
    3.PA⋅PC=PB⋅PD3.\quad PA\cdot PC=PB\cdot PD
    Power of a point
    4.PA⋅PD=PB⋅PC4.\quad PA\cdot PD=PB\cdot PC
  62. 62.
    Consider the Unit Circle in Trigonometry
    Unit circle showing sin alpha and cos alpha
    Unit circle showing tan alpha and cot alpha
  63. 63.
    Derivative
    Derivative
    f′(x0)=lim⁡h→0f(x0+h)−f(x0)hf'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}
    Critical points
    f′(x)=0(critical points)f'(x)=0\qquad\text{(critical points)}
    Increasing and decreasing
    f′(x)>0⇒f(x)↑f′(x)<0⇒f(x)↓f'(x)>0\Rightarrow f(x)\uparrow\qquad f'(x)<0\Rightarrow f(x)\downarrow
  64. 64.
    Remainder Rules
    3,9:n≡(sum of digits)3,9:\quad n\equiv\text{(sum of digits)}
    11:n≡(alternating sum of digits)11:\quad n\equiv\text{(alternating sum of digits)}
    2,5,10:last digit2,5,10:\quad \text{last digit}
    4,25,100:last 2 digits4,25,100:\quad \text{last 2 digits}
    8,125,1000:last 3 digits8,125,1000:\quad \text{last 3 digits}
  65. 65.
    Casework: Start with the Most Limiting Case

    Constraints can be based on the number of elements, on the size of the elements, or on other conditions.

    But usually, if you base your answer on the most limiting case, it will make the problem easier to solve.

  66. 66.
    Signs Have No Effect on the Parity
    ∣ ∣ ∣a∣−∣b∣+∣c∣−∣d∣ ∣ ∣≡a+b+c+d(mod2)\bigl|\,|\,|a|-|b|+|c|-|d|\,|\,\bigr|\equiv a+b+c+d\pmod{2}
  67. 67.
    When We Have a Trapezoid in a Problem, Draw the Altitudes
    Trapezoid ABCD with altitudes drawn from A and B to base DC

    When working with a trapezoid, drawing the altitudes creates right triangles and often simplifies the problem.

    Isosceles trapezoid with altitudes and equal side lengths
    x=CD−AB2x=\frac{CD-AB}{2}
  68. 68.
    Circles, Tangents, and Radical Axis
    Tangent and radius
    1.OP⊥tangent1.\quad OP\perp\text{tangent}
    Two tangents
    2.AB=AC2.\quad AB=AC
    Radical axis
    Externally tangent circles
    4.O1O2=R1+R24.\quad O_1O_2=R_1+R_2
    Internally tangent circles
    5.O1O2=R2−R15.\quad O_1O_2=R_2-R_1
  69. 69.
    Two Approaches for Square Roots

    1) Square it.

    2) Use a changing variable.

  70. 70.
    Geometric Approach in Probability

    1 dimension

    0≤x≤10\le x\le 1
    P ⁣(13<x<12)=12−13=16P\!\left(\tfrac{1}{3}<x<\tfrac{1}{2}\right)=\tfrac{1}{2}-\tfrac{1}{3}=\tfrac{1}{6}
    Number line from 0 to 1 with segment between 1/3 and 1/2 highlighted

    2 dimensions

    0≤x,y≤10\le x,y\le 1
    P(∣x−y∣≥12)=2×12×122=14P(|x-y|\ge \tfrac{1}{2})=2\times\frac{\tfrac{1}{2}\times\tfrac{1}{2}}{2}=\tfrac{1}{4}
    Unit square with two shaded triangles representing |x-y| >= 1/2

    3 dimensions

    0≤x,y,z≤10\le x,y,z\le 1
    P(x+y+z≤1)=12×12×123=16P(x+y+z\le 1)=\frac{\tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2}}{3}=\tfrac{1}{6}
    Unit cube with tetrahedron representing x+y+z <= 1 shaded
  71. 71.
    Stars and Bars
    Stars and Bars
    x1+x2+⋯+xk=n,xi≥1x_1+x_2+\cdots+x_k=n,\qquad x_i\ge1
    (n−1k−1)\binom{n-1}{k-1}
    x1+x2+⋯+xk=n,xi≥aix_1+x_2+\cdots+x_k=n,\qquad x_i\ge a_i
    xi′=xi−ai+1x_i'=x_i-a_i+1
    (n−∑ai+k−1k−1)\binom{n-\sum a_i+k-1}{k-1}
    x1+x2+⋯+xk≤n,xi≥1x_1+x_2+\cdots+x_k\le n,\qquad x_i\ge1
    (nk)\binom{n}{k}
    xi≤M  ⇒  Total−Badx_i\le M\;\Rightarrow\;\text{Total}-\text{Bad}
  72. 72.
    Double Counting (Consider from Different Aspects)
    A,B⊆{1,2,…,n}A,B\subseteq\{1,2,\ldots,n\}
    A∩B=∅A\cap B=\varnothing

    Instead of considering the subsets, we should consider the elements.

    Each element has 3 possible choices for AA and BB:

    AB✓✓✓××✓\begin{array}{c|c} A & B\\ \hline \checkmark & \checkmark\\ \checkmark & \times\\ \times & \checkmark \end{array}

    Therefore,

    3⋅3⋅3⋯3⏟n times=3n.\underbrace{3\cdot 3\cdot 3\cdots 3}_{n\text{ times}}=3^n.
  73. 73.
    3 Approaches for Finding the Closed Form of ana_n
    1.an=an−1+f(n)1.\quad a_n=a_{n-1}+f(n)
    Write an−an−1=f(n) and telescope.\text{Write }a_n-a_{n-1}=f(n)\text{ and telescope.}
    2.an=can−1+f(n)2.\quad a_n=ca_{n-1}+f(n)
    bn=ancnb_n=\frac{a_n}{c^n}
    Then telescope bn.\text{Then telescope }b_n.
    3.an=nan−1+f(n)3.\quad a_n=na_{n-1}+f(n)
    bn=ann!b_n=\frac{a_n}{n!}
    Then telescope bn.\text{Then telescope }b_n.
  74. 74.
    Triangle Method for Identities with Coefficients

    Start with:

    1⋅21+2⋅22+3⋅23+⋯+(n−1)2n−1+n 2n.1\cdot 2^1+2\cdot 2^2+3\cdot 2^3+\cdots+(n-1)2^{n-1}+n\,2^n.

    Write the following equations in a triangular arrangement:

    21+22+23+⋯+2n−1+2n ⟶ 2n+1−21  22+23+⋯+2n−1+2n ⟶ 2n+1−22  23+⋯+2n−1+2n ⟶ 2n+1−23  ⋮  2n−1+2n ⟶ 2n+1−2n−1 2n ⟶ 2n+1−2n\begin{array}{l} 2^1+2^2+2^3+\cdots+2^{n-1}+2^n\ \longrightarrow\ 2^{n+1}-2^1\\[2pt] \qquad\ \ 2^2+2^3+\cdots+2^{n-1}+2^n\ \longrightarrow\ 2^{n+1}-2^2\\[2pt] \qquad\qquad\ \ 2^3+\cdots+2^{n-1}+2^n\ \longrightarrow\ 2^{n+1}-2^3\\[2pt] \qquad\qquad\qquad\qquad\ \ \vdots\\[2pt] \qquad\qquad\qquad\qquad\qquad\ \ 2^{n-1}+2^n\ \longrightarrow\ 2^{n+1}-2^{n-1}\\[2pt] \qquad\qquad\qquad\qquad\qquad\qquad\qquad\ 2^n\ \longrightarrow\ 2^{n+1}-2^n \end{array}

    Then add them:

    =n 2n+1−(21+22+23+⋯+2n−1+2n).=n\,2^{n+1}-\left(2^1+2^2+2^3+\cdots+2^{n-1}+2^n\right).

    Since

    21+22+⋯+2n−1+2n=2n+1−21,2^1+2^2+\cdots+2^{n-1}+2^n=2^{n+1}-2^1,

    we get

    =n 2n+1−(2n+1−21)=n\,2^{n+1}-(2^{n+1}-2^1)
    =(n−1)2n+1+21.=\boxed{(n-1)2^{n+1}+2^1}.
  75. 75.
    Try Small Examples
  76. 76.
    3 Ideas in Paths
    Placing numbers
    1.Each point =sum of its predecessor(s).1.\quad \text{Each point }=\text{sum of its predecessor(s).}
    Permutation
    2.#paths=(R+U)!R! U!2.\quad \#\text{paths}=\frac{(R+U)!}{R!\,U!}
    Levels
    3.Choose the transition point at each level.3.\quad \text{Choose the transition point at each level.}
  77. 77.
    sin, cos, tan, cot — Special Angles

    15° and 75°

    Anglesin⁡θ\sin\thetacos⁡θ\cos\thetatan⁡θ\tan\thetacot⁡θ\cot\theta
    15∘15^\circ6−24\dfrac{\sqrt{6}-\sqrt{2}}{4}6+24\dfrac{\sqrt{6}+\sqrt{2}}{4}2−32-\sqrt{3}2+32+\sqrt{3}
    75∘75^\circ6+24\dfrac{\sqrt{6}+\sqrt{2}}{4}6−24\dfrac{\sqrt{6}-\sqrt{2}}{4}2+32+\sqrt{3}2−32-\sqrt{3}

    18°, 36°, 54°, and 72°

    Anglesin⁡θ\sin\thetacos⁡θ\cos\thetatan⁡θ\tan\thetacot⁡θ\cot\theta
    18∘18^\circ5−14\dfrac{\sqrt{5}-1}{4}10+254\dfrac{\sqrt{10+2\sqrt{5}}}{4}15+25\dfrac{1}{\sqrt{5+2\sqrt{5}}}5+25\sqrt{5+2\sqrt{5}}
    36∘36^\circ10−254\dfrac{\sqrt{10-2\sqrt{5}}}{4}1+54\dfrac{1+\sqrt{5}}{4}5−25\sqrt{5-2\sqrt{5}}15−25\dfrac{1}{\sqrt{5-2\sqrt{5}}}
    54∘54^\circ1+54\dfrac{1+\sqrt{5}}{4}10−254\dfrac{\sqrt{10-2\sqrt{5}}}{4}15−25\dfrac{1}{\sqrt{5-2\sqrt{5}}}5−25\sqrt{5-2\sqrt{5}}
    72∘72^\circ10+254\dfrac{\sqrt{10+2\sqrt{5}}}{4}5−14\dfrac{\sqrt{5}-1}{4}5+25\sqrt{5+2\sqrt{5}}15+25\dfrac{1}{\sqrt{5+2\sqrt{5}}}
  78. 78.
    When ab = x² or ab = px²

    aa and bb are positive integers.

    78.1

    If ab=x2ab=x^2 and gcd⁡(a,b)=1\gcd(a,b)=1, then a=y2, b=z2a=y^2,\ b=z^2.

    78.2

    If ab=x2ab=x^2 and gcd⁡(a,b)=d\gcd(a,b)=d, then a=dy2, b=dz2a=dy^2,\ b=dz^2.

    pp is a prime number.

    78.3

    If ab=px2ab=px^2 and gcd⁡(a,b)=1\gcd(a,b)=1, then a=py2, b=z2a=py^2,\ b=z^2 or a=y2, b=pz2a=y^2,\ b=pz^2.

    78.4

    If ab=px2ab=px^2 and gcd⁡(a,b)=d\gcd(a,b)=d, then a=dpy2, b=dz2a=dpy^2,\ b=dz^2 or a=dy2, b=dpz2a=dy^2,\ b=dpz^2.

  79. 79.
    Computing gcd(a,b)
    gcd⁡(a+kb,b)=gcd⁡(a,b)\gcd(a+kb,b)=\gcd(a,b)
    gcd⁡(a,b)=gcd⁡(a mod b,b)\gcd(a,b)=\gcd(a\bmod b,b)
  80. 80.
    Trigonometric Ratios in a Right Triangle
    Right triangle ABC with right angle at A and angle alpha at C
    sin⁡α=Opposite SideHypotenuse=ABBC\sin\alpha=\frac{\text{Opposite Side}}{\text{Hypotenuse}}=\frac{AB}{BC}
    cos⁡α=Adjacent SideHypotenuse=ACBC\cos\alpha=\frac{\text{Adjacent Side}}{\text{Hypotenuse}}=\frac{AC}{BC}
    tan⁡α=Opposite SideAdjacent Side=ABAC\tan\alpha=\frac{\text{Opposite Side}}{\text{Adjacent Side}}=\frac{AB}{AC}
    cot⁡α=Adjacent SideOpposite Side=ACAB\cot\alpha=\frac{\text{Adjacent Side}}{\text{Opposite Side}}=\frac{AC}{AB}
  81. 81.
    Substitution

    Example

    3ab+2=6b,3bc+2=5c,3ca+2=4a3ab+2=6b,\qquad 3bc+2=5c,\qquad 3ca+2=4a

    Express two variables in terms of the third:

    c=4a−23a,b=26−3ac=\frac{4a-2}{3a},\qquad b=\frac{2}{6-3a}

    Substitute into the remaining equation:

    (7a−8)(a−1)=0(7a-8)(a-1)=0

    Substitute the solutions back:

    a=1⇒abc=49a=1\Rightarrow abc=\frac49
    a=87⇒abc=23a=\frac87\Rightarrow abc=\frac23
  82. 82.
    3D Shapes: Surface Area and Volume

    Cube

    Cube
    TSA=6a2\mathrm{TSA}=6a^2
    V=a3V=a^3

    Rectangular Prism

    Cuboid
    TSA=2(ab+ac+bc)\mathrm{TSA}=2(ab+ac+bc)
    V=abcV=abc

    Prism

    Prism
    V=h⋅AreabaseV=h\cdot\mathrm{Area}_{\mathrm{base}}

    Sphere

    Sphere
    TSA=4πr2\mathrm{TSA}=4\pi r^2
    V=43πr3V=\frac43\pi r^3

    Cylinder

    Cylinder
    TSA=2πr(r+h)\mathrm{TSA}=2\pi r(r+h)
    V=πr2hV=\pi r^2h

    Cone

    Cone
    TSA=πr(r+l)\mathrm{TSA}=\pi r(r+l)
    l=r2+h2l=\sqrt{r^2+h^2}
    V=13πr2hV=\frac13\pi r^2h

    Pyramid

    Pyramid
    V=13Areabase⋅hV=\frac13\mathrm{Area}_{\mathrm{base}}\cdot h

    Regular Tetrahedron

    Regular tetrahedron
    TSA=a23\mathrm{TSA}=a^2\sqrt3
    V=a3212V=\frac{a^3\sqrt2}{12}

    Triangular Prism

    Triangular prism with triangular base and height
    V=12Areabase×LV=\frac12\text{Area}_{\text{base}}\times L
  83. 83.
    Diagonal of a Grid
    Diagonal of an m by n grid crossing lattice lines

    If you draw the diagonal of an m×nm\times n grid, then the diagonal is split into m+n−gcd⁡(m,n)m+n-\gcd(m,n) parts.

  84. 84.
    Ratios of Areas
    Triangle ABC with cevian AP
    [APB][APC]=BPPC\frac{[APB]}{[APC]}=\frac{BP}{PC}
    Triangle ABC with point L on cevian AP
    [ALB][ALC]=BPPC\frac{[ALB]}{[ALC]}=\frac{BP}{PC}
  85. 85.
    Menelaus' Theorem
    Menelaus configuration with transversal cutting triangle ABC
    Menelaus configuration with transversal cutting extensions
    DBDC⋅ECEA⋅FAFB=1\frac{DB}{DC}\cdot\frac{EC}{EA}\cdot\frac{FA}{FB}=1

    if and only if D,E,FD,E,F are collinear.

  86. 86.
    Regular Polygons

    For an nn-sided polygon,

    Sum of interior angles=180∘(n−2)\text{Sum of interior angles}=180^\circ(n-2)

    Regular Hexagon

    For a regular hexagon with side length aa:

    a)

    ∠A=180∘(6−2)6=120∘\angle A=\frac{180^\circ(6-2)}{6}=120^\circ

    b)

    Regular hexagon divided into six equilateral triangles
    [ABCDEF]=6(a234)=3a232[ABCDEF]=6\left(\frac{a^2\sqrt{3}}{4}\right)=\frac{3a^2\sqrt{3}}{2}

    c)

    Regular hexagon with diagonal AC of length a√3
    AC=a3AC=a\sqrt{3}

    d)

    Regular hexagon with long diagonal AD of length 2a
    AD=2aAD=2a

    Regular Octagon

    For a regular octagon with side length aa:

    a)

    ∠A=180∘(8−2)8=135∘\angle A=\frac{180^\circ(8-2)}{8}=135^\circ

    b)

    Regular octagon inscribed in a square of side a+a√2
    [ABCDEFGH]=(a+a2)2−4(a2/22)[ABCDEFGH]=(a+a\sqrt{2})^2-4\left(\frac{a^2/2}{2}\right)
    =a2+2a2+2a22−a2=2a2(1+2)=a^2+2a^2+2a^2\sqrt{2}-a^2=2a^2(1+\sqrt{2})

    c)

    Regular octagon with diagonal AC and 135° angle at B
    AC=a2+2AC=a\sqrt{2+\sqrt{2}}
    AC2=a2+a2−2a2cos⁡135∘=2a2+2a222AC^2=a^2+a^2-2a^2\cos 135^\circ=2a^2+2a^2\frac{\sqrt{2}}{2}
    =2a2+a22=a2(2+2)=2a^2+a^2\sqrt{2}=a^2(2+\sqrt{2})

    d)

    Regular octagon with diagonal AD of length a(√2+1)
    AD=a2+a=a(2+1)AD=a\sqrt{2}+a=a(\sqrt{2}+1)

    e)

    Regular octagon with diagonal AE and right triangle ADE
    AE=a4+22AE=a\sqrt{4+2\sqrt{2}}
    AE2=a2+AD2=a2+(a(2+1))2AE^2=a^2+AD^2=a^2+\left(a(\sqrt{2}+1)\right)^2
    =a2+a2(3+22)=a2(4+22)=a^2+a^2(3+2\sqrt{2})=a^2(4+2\sqrt{2})
  87. 87.
    Speed Formula
    v=dtv=\frac{d}{t}
    v: speedd: distancet: timev:\text{ speed}\qquad d:\text{ distance}\qquad t:\text{ time}
  88. 88.
    Triangle Inequality
    Triangle ABC with side lengths a, b, and c
    a<b+ca<b+c
    b<a+cb<a+c
    c<a+bc<a+b
  89. 89.
    Mean → Think Sum
    Mean  ⟹  Sum\text{Mean}\;\Longrightarrow\;\text{Sum}
  90. 91.
    Intermediate Value Theorem (IVT)
    If f is continuous on [a,b] and f(a)f(b)<0, then ∃ c∈(a,b) such that f(c)=0.\text{If }f\text{ is continuous on }[a,b]\text{ and }f(a)f(b)<0,\text{ then }\exists\,c\in(a,b)\text{ such that }f(c)=0.
  91. 92.
    Fundamental Theorem of Algebra (Real Consequence)
    A polynomial of degree n with real coefficients has at most n real roots.\text{A polynomial of degree }n\text{ with real coefficients has at most }n\text{ real roots.}
  92. 93.
    Similar Triangles
    Similar triangles
    △ABC∼△A′B′C′\triangle ABC\sim\triangle A'B'C'
    AA\text{AA}
    SAS\text{SAS}
    SSS\text{SSS}
  93. 94.
    Expected Value
    E(X)=∑xx Pr⁡(X=x)E(X)=\sum_x x\,\Pr(X=x)
  94. 95.
    Coordinate Geometry
    Slope, equation, and distance formulas for two points
    m=y2−y1x2−x1,y−y1=m(x−x1)m=\frac{y_2-y_1}{x_2-x_1},\qquad y-y_1=m(x-x_1)
    AB=(x2−x1)2+(y2−y1)2AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
  95. 96.
    Ratio Manipulation
    ab=cd\frac{a}{b}=\frac{c}{d}
    a+bb=c+dd\frac{a+b}{b}=\frac{c+d}{d}
    aa+b=cc+d\frac{a}{a+b}=\frac{c}{c+d}
    ab−a=cd−c\frac{a}{b-a}=\frac{c}{d-c}
    a−ba+b=c−dc+d\frac{a-b}{a+b}=\frac{c-d}{c+d}
    a1b1=a2b2=⋯=anbn=k⟹λ1a1+λ2a2+⋯+λnanλ1b1+λ2b2+⋯+λnbn=k\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_n}{b_n}=k\quad\Longrightarrow\quad\frac{\lambda_1a_1+\lambda_2a_2+\cdots+\lambda_na_n}{\lambda_1b_1+\lambda_2b_2+\cdots+\lambda_nb_n}=k
  96. 97.
    Basic Complex Numbers
    Complex number on the complex plane
    z=a+bi=reiθz=a+bi=re^{i\theta}
    r=∣z∣=a2+b2r=|z|=\sqrt{a^2+b^2}
    a=rcos⁡θ,b=rsin⁡θa=r\cos\theta,\qquad b=r\sin\theta
    (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di)=(ac-bd)+(ad+bc)i
    (reiθ)n=rneinθ(re^{i\theta})^n=r^ne^{in\theta}
  97. 98.
    Bounding the Answer
    L≤Answer≤UL\leq \text{Answer}\leq U
    L=U⟹Answer=LL=U\Longrightarrow\text{Answer}=L
  98. 99.
    Ellipse
    (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1
    c2=a2−b2c^2=a^2-b^2
    e=cae=\frac ca
    Area=πab\text{Area}=\pi ab
  99. 100.
    Factor Theorem
    P(r)=0⟹(x−r)∣P(x)P(r)=0\Longrightarrow (x-r)\mid P(x)
  100. 101.
    Cross Sections in 3D Geometry
  101. 102.
    Basic Integrals
    ∫abf(x) dx\int_a^b f(x)\,dx
    ∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C
    ∫ex dx=ex+C\int e^x\,dx=e^x+C
    ∫sin⁡x dx=−cos⁡x+C\int \sin x\,dx=-\cos x+C
  102. 103.
    Applications of Integrals
    A=∬R1 dA,V=∭D1 dVA=\iint_R 1\,dA,\qquad V=\iiint_D 1\,dV