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Olympiad Toolkit

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

Showing 156 toolkits
  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 Sequences 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
    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 Functions
    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 Sequences 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}
    Number of elements=last−firstd+1\text{Number of elements}=\frac{\text{last}-\text{first}}{d}+1
  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.
    Mean Inequalities
    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.
    Bounding the Fractional Part
    [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.
    Quadratic Inequalities
    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.
    Polynomial Remainders
    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

    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

    1.1.
    a∣b,  b∣c⟹a∣ca\mid b,\; b\mid c \Longrightarrow a\mid c
    2.2.
    a∣b,  a∣c⟹a∣mb+nca\mid b,\; a\mid c \Longrightarrow a\mid mb+nc
    3.3.
    gcd⁡(a,b)=1,  a∣bc⟹a∣c\gcd(a,b)=1,\; a\mid bc \Longrightarrow a\mid c
    4.4.
    p∣ab⟹p∣a or p∣b,p prime.p\mid ab \Longrightarrow p\mid a \text{ or } p\mid b,\qquad p\text{ prime.}
  47. 47.
    Modular Arithmetic

    Definition

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

    Properties

    1.1.
    a≡b(modm),  b≡c(modm)⟹a≡c(modm)a\equiv b\pmod m,\; b\equiv c\pmod m\Longrightarrow a\equiv c\pmod m
    2.2.
    a≡b(modm)⟹an≡bn(modm),n∈Z+a\equiv b\pmod m\Longrightarrow a^n\equiv b^n\pmod m,\quad n\in\mathbb Z^+
    3.3.
    ac≡bc(modm)  ⟺  a≡b(modmgcd⁡(m,c))ac\equiv bc\pmod m\iff a\equiv b\pmod{\frac{m}{\gcd(m,c)}}
    4.4.
    a≡b(modm),  d∣m⟹a≡b(modd)a\equiv b\pmod m,\; d\mid m\Longrightarrow a\equiv b\pmod d
  48. 48.
    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|
  49. 49.
    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)
  50. 50.
    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
  51. 51.
    Function Transformations
    TransformationExample
    Horizontal Shift Right 3y=f(x−3)y=f(x-3)
    Horizontal Shift Left 3y=f(x+3)y=f(x+3)
    Vertical Shift Up 3y=f(x)+3y=f(x)+3
    Vertical Shift Down 3y=f(x)−3y=f(x)-3
    Horizontal Stretch by Factor 2y=f(x/2)y=f(x/2)
    Horizontal Compression by Factor 2y=f(2x)y=f(2x)
    Vertical Stretch by Factor 2y=2f(x)y=2f(x)
    Vertical Compression by Factor 2y=12f(x)y=\frac{1}{2}f(x)
    Reflection in x-axisy=−f(x)y=-f(x)
    Reflection in y-axisy=f(−x)y=f(-x)
  52. 52.
    Area Formulas

    Triangle Area Formulas

    1.1.
    [ABC]=12Base⋅Height[ABC]=\frac12\text{Base}\cdot\text{Height}
    2.2.
    [ABC]=12bcsin⁡A[ABC]=\frac12 bc\sin A
    3.3.
    s=a+b+c2,[ABC]=s(s−a)(s−b)(s−c)s=\frac{a+b+c}{2},\qquad [ABC]=\sqrt{s(s-a)(s-b)(s-c)}
    4.4.
    [ABC]=a234(equilateral triangle)[ABC]=\frac{a^2\sqrt3}{4}\qquad\text{(equilateral triangle)}

    Quadrilateral Area Formulas

    1.1.
    [ABCD]=12AC⋅BDsin⁡α[ABCD]=\frac12 AC\cdot BD\sin\alpha
    2.2.
    [ABCD]=h(AB+CD)2(trapezoid)[ABCD]=\frac{h(AB+CD)}{2}\qquad\text{(trapezoid)}
    3.3.
    [ABCD]=bh(parallelogram)[ABCD]=bh\qquad\text{(parallelogram)}

    Circle Formulas

    1.1.
    A=πr2,C=2πrA=\pi r^2,\qquad C=2\pi r
    2.2.
    Asector=α360πr2A_{\text{sector}}=\frac{\alpha}{360}\pi r^2
    3.3.
    ℓarc=α360⋅2πr\ell_{\text{arc}}=\frac{\alpha}{360}\cdot2\pi r
  53. 53.
    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|
  54. 54.
    Gauss's Pairing Trick (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}
  55. 55.
    Trigonometric Values of Specific 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

    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}}}
  56. 56.
    Law of Cosines
    Triangle ABC with sides a, b, c
    a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A
  57. 57.
    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
  58. 58.
    Angles in Circles
    Central angle
    Inscribed angle
    Tangent-chord angle
    Interior angle
    Exterior angle
  59. 59.
    Basic 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
  60. 60.
    Trigonometric Foundations
    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}
    Unit circle showing sin alpha and cos alpha
    Unit circle showing tan alpha and cot alpha
    Unit circle showing tan alpha and cot alpha
  61. 61.
    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}
  62. 62.
    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.

  63. 63.
    Signs Do Not Affect Parity
    ∣ ∣ ∣a∣−∣b∣+∣c∣−∣d∣ ∣ ∣≡a+b+c+d(mod2)\bigl|\,|\,|a|-|b|+|c|-|d|\,|\,\bigr|\equiv a+b+c+d\pmod{2}
  64. 64.
    Trapezoid Strategy: 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}
  65. 65.
    Circle Tangency
    Tangent and radius
    1.OP⊥tangent1.\quad OP\perp\text{tangent}
    Two tangents
    2.AB=AC2.\quad AB=AC
    Externally tangent circles
    3.O1O2=R1+R23.\quad O_1O_2=R_1+R_2
    Internally tangent circles
    4.O1O2=R2−R14.\quad O_1O_2=R_2-R_1
  66. 66.
    Approaches to Solving Square Root Problems

    1) Square it.

    2) Use a changing variable.

  67. 67.
    Geometric 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
  68. 68.
    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}
  69. 69.
    Double Counting (Different Perspectives)
    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.
  70. 70.
    Telescoping Approaches to Solving Recurrences
    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.
  71. 71.
    Triangle Method for Weighted Sums

    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}.
  72. 72.
    Try Small Examples
  73. 73.
    Approaches to Counting 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.}
  74. 74.
    Special Right Triangles

    30-60-90 Triangle

    30-60-90 right triangle
    Sides are a, a32, a2\text{Sides are } a,\ \frac{a\sqrt3}{2},\ \frac{a}{2}

    45-45-90 Triangle

    45-45-90 right isosceles triangle
    Sides are a22, a22, a\text{Sides are }\frac{a\sqrt2}{2},\ \frac{a\sqrt2}{2},\ a
  75. 75.
    When a Product Is a Square or Prime Times a Square

    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.

  76. 76.
    Greatest Common Divisor (GCD)
    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)
  77. 77.
    Circles in Coordinate Geometry

    Equation of a Circle

    Circle with center and radius
    A circle with center (x0,y0) and radius r:\text{A circle with center }(x_0,y_0)\text{ and radius }r:
    (x−x0)2+(y−y0)2=r2(x-x_0)^2+(y-y_0)^2=r^2

    Finding the Equation of a Tangent Line

    Tangent line through an external point to a circle

    Let the tangent line through the point P(x₀, y₀) have slope m.

    ℓ: y−y0=m(x−x0)\ell:\ y-y_0=m(x-x_0)
    (x−x1)2+(y−y1)2=r2(x-x_1)^2+(y-y_1)^2=r^2

    Substitute the line equation into the circle equation. Since a tangent line intersects the circle at exactly one point, set the discriminant equal to zero and solve for m.

    Δ=0\Delta=0
  78. 78.
    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
  79. 79.
    3D Geometry: 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
  80. 80.
    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.

  81. 81.
    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}
  82. 82.
    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.

  83. 83.
    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})
  84. 84.
    Speed, Distance, and Time
    v=dtv=\frac{d}{t}
    v: speedd: distancet: timev:\text{ speed}\qquad d:\text{ distance}\qquad t:\text{ time}
  85. 85.
    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
  86. 86.
    Mean, Median, Mode, and Range

    Mean

    Mean=Sum of ValuesNumber of Values\text{Mean}=\frac{\text{Sum of Values}}{\text{Number of Values}}
    Sum=Mean×Number of Values\text{Sum}=\text{Mean}\times\text{Number of Values}

    Median, Mode, and Range

    Median: the middle value after sorting

    Mode: the value that occurs most frequently

    Range=Maximum−Minimum\text{Range}=\text{Maximum}-\text{Minimum}

    Effect of Transforming Every Value

    OperationMeanMedianModeRange
    +k+k+k+k+k+k+k+kSame
    −k-k−k-k−k-k−k-kSame
    ×k\times k×k\times k×k\times k×k\times k×∣k∣\times |k|
    ÷k, k≠0\div k,\ k\ne0÷k\div k÷k\div k÷k\div k÷∣k∣\div |k|
  87. 88.
    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.
  88. 89.
    Fundamental Theorem of Algebra
    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.}
  89. 90.
    Similar Triangles
    Similar triangles
    △ABC∼△A′B′C′\triangle ABC\sim\triangle A'B'C'
    AA\text{AA}
    SAS\text{SAS}
    SSS\text{SSS}
  90. 91.
    Expected Value
    E(X)=∑xx Pr⁡(X=x)E(X)=\sum_x x\,\Pr(X=x)
  91. 92.
    Coordinate Geometry Foundations

    Slope, Line Equation, and Distance

    Two points on a line with slope, line equation, and distance formulas
    mAB=y2−y1x2−x1m_{AB}=\frac{y_2-y_1}{x_2-x_1}
    y−y1=m(x−x1)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}
    Perpendicular lines with slopes m1 and m2
    Perpendicular lines:m1m2=−1\text{Perpendicular lines:}\qquad m_1m_2=-1

    Distance from a Point to a Line

    Perpendicular distance from a point to a line
    For P(x0,y0) and ℓ:ax+by+c=0,\text{For }P(x_0,y_0)\text{ and }\ell:ax+by+c=0,
    d(P,ℓ)=∣ax0+by0+c∣a2+b2d(P,\ell)=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}

    Distance Between Two Parallel Lines

    Distance between two parallel lines
    ℓ1:ax+by+c1=0,ℓ2:ax+by+c2=0\ell_1:ax+by+c_1=0,\qquad \ell_2:ax+by+c_2=0
    d(ℓ1,ℓ2)=∣c1−c2∣a2+b2d(\ell_1,\ell_2)=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}

    Distance from a Point to a Plane

    Perpendicular distance from a point to a plane
    For P(x0,y0,z0) and α:ax+by+cz+d=0,\text{For }P(x_0,y_0,z_0)\text{ and }\alpha:ax+by+cz+d=0,
    d(P,α)=∣ax0+by0+cz0+d∣a2+b2+c2d(P,\alpha)=\frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}
  92. 93.
    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
  93. 94.
    Bounding the Answer
    L≤Answer≤UL\leq \text{Answer}\leq U
    L=U⟹Answer=LL=U\Longrightarrow\text{Answer}=L
  94. 95.
    Ellipses
    (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
  95. 96.
    Factor Theorem
    P(r)=0⟹(x−r)∣P(x)P(r)=0\Longrightarrow (x-r)\mid P(x)
  96. 97.
    Cross Sections in 3D Geometry
  97. 98.
    Number Bases
    (anan−1⋯a1a0)b=a0+a1b+a2b2+⋯+anbn(a_na_{n-1}\cdots a_1a_0)_b=a_0+a_1b+a_2b^2+\cdots+a_nb^n
  98. 99.
    Properties of Inequalities
    a<b⟹a+c<b+ca<b\Longrightarrow a+c<b+c
  99. 100.
    Derangements
    Dn=n!(10!−11!+12!−13!+⋯+(−1)nn!)D_n=n!\left(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^n}{n!}\right)
  100. 101.
    Law of Sines
    asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
  101. 102.
    Ratio Lemma
    sin⁡A1sin⁡A2=BPPC⋅bc=BPPC⋅sin⁡Bsin⁡C\frac{\sin A_1}{\sin A_2}=\frac{BP}{PC}\cdot\frac{b}{c}=\frac{BP}{PC}\cdot\frac{\sin B}{\sin C}
  102. 103.
    Triangle Proportionality Theorem
    DE∥BC⟹ADAB=AEAC=DEBC,ADAB=AEAC⟹DE∥BCDE\parallel BC\Longrightarrow\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC},\qquad \frac{AD}{AB}=\frac{AE}{AC}\Longrightarrow DE\parallel BC
  103. 104.
    Trapezoid Midsegment Theorem
    MN∥AB∥CD,MN=AB+CD2MN\parallel AB\parallel CD,\qquad MN=\frac{AB+CD}{2}
  104. 105.
    Linear Diophantine Equations
    ax+by=nax+by=n
    gcd⁡(a,b)∣n\gcd(a,b)\mid n
    x=x1+b′k,y=y1−a′kx=x_1+b'k,\qquad y=y_1-a'k
  105. 106.
    Reflection Method

    Find min{PA+PB}.

    Reflection Method
    CQ=CQ′CQ=CQ'
    BL=BL′BL=BL'
    Reflection Method in square
  106. 107.
    Linear Independence of Irrational Numbers
    a+b5=0⟹a=b=0a+b\sqrt5=0\Longrightarrow a=b=0
    a+b5=c+d5⟹a=c, b=da+b\sqrt5=c+d\sqrt5\Longrightarrow a=c,\ b=d
  107. 108.
    Solving Linear Congruences
    ax≡b(modm)ax\equiv b\pmod m

    Reduce the coefficient of x to solve.

  108. 109.
    Solving Systems of Congruences
    a≡r(modm),a≡r(modn)a\equiv r\pmod m,\qquad a\equiv r\pmod n
    ⟺a≡r(modlcm⁡(m,n))\Longleftrightarrow a\equiv r\pmod{\operatorname{lcm}(m,n)}

    Combine congruences by listing values or substituting one congruence into the other.

    a≡2(mod5),a≡3(mod11)⟹a≡47(mod55)a\equiv2\pmod5,\qquad a\equiv3\pmod{11}\Longrightarrow a\equiv47\pmod{55}
    a≡5(mod19),a≡11(mod47)⟹a≡575(mod893)a\equiv5\pmod{19},\qquad a\equiv11\pmod{47}\Longrightarrow a\equiv575\pmod{893}
  109. 110.
    Finding Last Digits

    Find the last 3 digits of 23578 × 16327

    Finding last digits of multiplication
  110. 111.
    Modulo Prime Powers
    2341≡  ?(mod125)2^{341}\equiv\; ?\pmod{125}
    2341=2×(5−1)1702^{341} = 2\times(5-1)^{170}
  111. 112.
    Minimum and Maximum Distances with Circles
    Minimum and Maximum Distances with Circles
    min⁡{PA}=PB,max⁡{PA}=PC\min\{PA\}=PB,\qquad \max\{PA\}=PC
    Distance between two circles
    min⁡{PQ}=BC,max⁡{PQ}=AD\min\{PQ\}=BC,\qquad \max\{PQ\}=AD
  112. 113.
    Trigonometric Function Graphs
    Graph of sine function
    Graph of cosine function
    Graph of tangent function
    Graph of cotangent function
  113. 114.
    Absolute Value

    Graph of ∣x∣+∣y∣=5|x|+|y|=5:

    Graph of |x|+|y|=5
  114. 115.
    Reciprocal Functions

    Graph of y=12xy=\dfrac{1}{2x}:

    Graph of y = 1/(2x)
  115. 116.
    Radical Axis
    Radical axis
  116. 117.
    Functional Graphs

    A functional graph represents a function

    f:X→Xf:X\to X

    by drawing a directed edge

    x→f(x)x\to f(x)

    For example, assume X={1,2,3,…,14}X=\{1,2,3,\ldots,14\} and

    Example of a functional graph for a function on the set X = {1, 2, 3, ..., 14}
  117. 118.
    Counting Set Partitions

    A set partition divides a set of distinct elements into nonempty, disjoint groups whose union is the entire set.

    For example, the number of ways to divide 1515 people as shown is

    Partition of 15 people into three groups of 2, one group of 4, and one group of 5
    (152)(132)(112)(94)(55)3!=15!2!3 4! 5! 3!\frac{\binom{15}{2}\binom{13}{2}\binom{11}{2}\binom{9}{4}\binom{5}{5}}{3!}=\frac{15!}{2!^3\,4!\,5!\,3!}
  118. 119.
    Conditional Probability

    Definition

    Probability of AA given BB is

    P(A∣B)=P(A∩B)P(B)P(A\mid B)=\frac{P(A\cap B)}{P(B)}
    Venn diagram of events A and B with their intersection highlighted

    Bayes' Theorem

    P(A∣B)=P(A)P(B∣A)P(B)P(A\mid B)=\frac{P(A)P(B\mid A)}{P(B)}
  119. 120.
    Altitude Projection Formula

    Let BD=xBD=x, then

    x=a2+c2−b22ax=\frac{a^2+c^2-b^2}{2a}
    Triangle ABC with altitude AD to BC, where BD = x, BC = a, AC = b, and AB = c
  120. 121.
    Rotation in Coordinate Geometry
    (x′y′)=(hk)+(cos⁡θ−sin⁡θsin⁡θcos⁡θ)(x−hy−k) \begin{pmatrix} x'\\ y' \end{pmatrix} = \begin{pmatrix} h\\ k \end{pmatrix} + \begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x-h\\ y-k \end{pmatrix}
    OP=OP′,∠POP′=θ OP=OP', \qquad \angle POP'=\theta
    Rotation of point P(x,y) about O(h,k) through angle theta to P'(x',y')
  121. 122.
    Coloring Strategies

    Coloring can reveal patterns or invariants in problems involving boards, tilings, and moves.

    Checkerboard Coloring

    Color the squares alternately black and white, so squares sharing a side have different colors.

    An 8 by 8 checkerboard with alternating black and white squares and rows and columns numbered from 1 to 8

    A square in row rr and column cc can be colored according to the parity of

    r+cr+c

    • r+cr+c even → white
    • r+cr+c odd → black

  122. 123.
    Split into Groups

    When we want to prove that some selected objects must have a certain relationship, it can be useful to split all possible objects into carefully chosen groups.

    Key Idea

    1. Determine the relationship you want to force.
    2. Split the objects into groups based on that relationship.

  123. 124.
    Consider the Complement

    For example, this applies to:

    • a subset and its complement
    • a graph and its complement
    • selected and unselected objects
    • an event and its complementary event

  124. 125.
    Centroid of a Triangle

    The centroid of a triangle is the point where its three medians intersect.

    If G is the centroid of △ABC and M is the midpoint of BC, then

    Triangle ABC with medians AM and BN intersecting at G
    AG:GM=2:1AG:GM=2:1
    AG=23AMGM=13AMAG=\frac23 AM \qquad GM=\frac13 AM

    The three medians divide the triangle into six equal-area triangles.

    G=(x1+x2+x33,y1+y2+y33)G=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)
  125. 126.
    Section Formula
    Section Formula
    ACCB=mn\frac{AC}{CB}=\frac{m}{n}
    C=mB+nAm+nC=\frac{mB+nA}{m+n}
    A=(x1,y1),B=(x2,y2)A=(x_1,y_1),\qquad B=(x_2,y_2)
    C=(nx1+mx2m+n,ny1+my2m+n)C=\left(\frac{nx_1+mx_2}{m+n},\frac{ny_1+my_2}{m+n}\right)

    Midpoint Formula

    C=A+B2C=\frac{A+B}{2}
    C=(x1+x22,y1+y22)C=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
  126. 127.
    Projection and Reflection in Coordinate Geometry
    Projection and reflection of a point across a line

    Projection onto a Line

    H=(x0−a(ax0+by0+c)a2+b2,y0−b(ax0+by0+c)a2+b2)H=\left(x_0-\frac{a(ax_0+by_0+c)}{a^2+b^2},y_0-\frac{b(ax_0+by_0+c)}{a^2+b^2}\right)

    Reflection Across a Line

    P′=(x0−2a(ax0+by0+c)a2+b2,y0−2b(ax0+by0+c)a2+b2)P'=\left(x_0-\frac{2a(ax_0+by_0+c)}{a^2+b^2},y_0-\frac{2b(ax_0+by_0+c)}{a^2+b^2}\right)

    Special Cases

    (x,y)⟶(x,−y)across the x-axis(x,y)\longrightarrow(x,-y)\qquad \text{across the }x\text{-axis}
    (x,y)⟶(−x,y)across the y-axis(x,y)\longrightarrow(-x,y)\qquad \text{across the }y\text{-axis}
    (x,y)⟶(y,x)across y=x(x,y)\longrightarrow(y,x)\qquad \text{across }y=x
    (x,y)⟶(−y,−x)across y=−x(x,y)\longrightarrow(-y,-x)\qquad \text{across }y=-x
    (x,y)⟶(2h−x,y)across x=h(x,y)\longrightarrow(2h-x,y)\qquad \text{across }x=h
    (x,y)⟶(x,2k−y)across y=k(x,y)\longrightarrow(x,2k-y)\qquad \text{across }y=k
  127. 128.
    Equation of a Plane

    Equation of a Plane

    ax+by+cz=dax+by+cz=d
    n=(a,b,c)\mathbf{n}=(a,b,c)
    A plane with a normal vector

    Point-Normal Form

    a(x−x0)+b(y−y0)+c(z−z0)=0a(x-x_0)+b(y-y_0)+c(z-z_0)=0

    Plane Through Three Points

    AB→=B−A,AC→=C−A\overrightarrow{AB}=B-A,\qquad \overrightarrow{AC}=C-A
    n=AB→×AC→\mathbf{n}=\overrightarrow{AB}\times\overrightarrow{AC}

    Parallel and Perpendicular Planes

    (a1,b1,c1)=k(a2,b2,c2)(a_1,b_1,c_1)=k(a_2,b_2,c_2)
    a1a2+b1b2+c1c2=0a_1a_2+b_1b_2+c_1c_2=0

    Angle Between Two Planes

    cos⁡θ=∣n1⋅n2∣∣n1∣∣n2∣\cos\theta=\frac{|\mathbf{n}_1\cdot\mathbf{n}_2|}{|\mathbf{n}_1||\mathbf{n}_2|}

    Intercept Form

    xp+yq+zr=1\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1
  128. 129.
    Angle Bisector Plane

    Angle Bisector Plane

    Angle bisector planes of two intersecting planes
    ∣a1x+b1y+c1z+d1∣a12+b12+c12=∣a2x+b2y+c2z+d2∣a22+b22+c22\frac{|a_1x+b_1y+c_1z+d_1|}{\sqrt{a_1^2+b_1^2+c_1^2}}=\frac{|a_2x+b_2y+c_2z+d_2|}{\sqrt{a_2^2+b_2^2+c_2^2}}
    a1x+b1y+c1z+d1a12+b12+c12=±a2x+b2y+c2z+d2a22+b22+c22\frac{a_1x+b_1y+c_1z+d_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\pm\frac{a_2x+b_2y+c_2z+d_2}{\sqrt{a_2^2+b_2^2+c_2^2}}
  129. 130.
    Dot Product

    Dot Product

    Geometric interpretation of dot product
    u→⋅v→=u1v1+u2v2+⋯+unvn\overrightarrow{u}\cdot\overrightarrow{v}=u_1v_1+u_2v_2+\cdots+u_nv_n
    u→⋅v→=∣u→∣∣v→∣cos⁡θ\overrightarrow{u}\cdot\overrightarrow{v}=|\overrightarrow{u}||\overrightarrow{v}|\cos\theta
    u→⊥v→  ⟺  u→⋅v→=0\overrightarrow{u}\perp\overrightarrow{v}\iff\overrightarrow{u}\cdot\overrightarrow{v}=0
    proj⁡v→u→=u→⋅v→∣v→∣2v→\operatorname{proj}_{\overrightarrow{v}}\overrightarrow{u}=\frac{\overrightarrow{u}\cdot\overrightarrow{v}}{|\overrightarrow{v}|^2}\overrightarrow{v}
  130. 131.
    Rational Root Theorem

    Rational Root Theorem

    P(x)=anxn+an−1xn−1+⋯+a1x+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0
    r=pq,gcd⁡(p,q)=1r=\frac pq,\qquad \gcd(p,q)=1
    p∣a0,q∣anp\mid a_0,\qquad q\mid a_n

    Key Idea

    Use divisibility conditions to reduce the possible rational roots of a polynomial

  131. 132.
    Homogeneity and Ratio Substitution

    Homogeneous Expressions

    An expression is homogeneous if every term has the same total degree

    2ax2+x2b=b2a2ax^2+x^2b=b^2a

    When all terms have the same degree, divide by a suitable power of one variable and introduce ratios

    r=ax,t=bxr=\frac{a}{x},\qquad t=\frac{b}{x}

    Key Idea

    When all terms have the same total degree:

    1. Divide by a suitable power of one nonzero variable
    2. Replace ratios of variables by new variables
    3. Solve the simpler problem involving fewer variables

  132. 133.
    Symmetry and Fixed Cases Principle

    Symmetry and Fixed Cases Principle

    When a set of cases has a symmetry, we can often pair each case with another case that behaves in the opposite or corresponding way.

    The cases that are unchanged by the symmetry are called fixed cases. These must be considered separately.

    Key Idea

    Pair cases that are transformed into each other by a symmetry. Handle fixed cases separately.

    Applications

    Counting problems with opposite pairs, sign choices, and symmetry arguments

  133. 134.
    Binomial Coefficient Identities

    Sum of Binomial Coefficients

    (n0)+(n1)+(n2)+⋯+(nn)=2n\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots+\binom{n}{n}=2^n

    Sums of Even and Odd Binomial Coefficients

    (n0)+(n2)+(n4)+⋯=(n1)+(n3)+(n5)+⋯=2n−1\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\cdots=2^{n-1}

    Applications

    Simplifying binomial sums and solving combinatorial counting problems

  134. 135.
    Cross Product

    Definition

    u→×v→=(u2v3−u3v2, u3v1−u1v3, u1v2−u2v1)\overrightarrow{u}\times\overrightarrow{v}=(u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1)

    Geometric Meaning

    ∣u→×v→∣=∣u→∣∣v→∣sin⁡θ|\overrightarrow{u}\times\overrightarrow{v}|=|\overrightarrow{u}||\overrightarrow{v}|\sin\theta

    The cross product produces a vector perpendicular to both vectors

    Right-Hand Rule

    To find the direction of A→×B→\overrightarrow{A}\times\overrightarrow{B}, point the fingers of your right hand in the direction of A→\overrightarrow{A} and curl them toward B→\overrightarrow{B}. Your thumb points in the direction of A→×B→\overrightarrow{A}\times\overrightarrow{B}.

    Right-hand rule for the cross product showing vectors A and B and the perpendicular vector C equals A cross B

    Applications

    Finding areas, volumes, normal vectors, and checking whether vectors are parallel

  135. 136.
    Functions

    Definition

    A function assigns exactly one output to each input.

    f:A→Bf:A\to B
    Function mapping from domain A to codomain B

    Domain and Range

    Domain⁡(f)={x:f(x) is defined}\operatorname{Domain}(f)=\{x:f(x)\text{ is defined}\}
    Range⁡(f)={f(x):x∈Domain⁡(f)}\operatorname{Range}(f)=\{f(x):x\in\operatorname{Domain}(f)\}

    One-to-One Functions

    f(x1)=f(x2)⟹x1=x2f(x_1)=f(x_2)\Longrightarrow x_1=x_2

    Inverse Functions

    f(a)=b⟺f−1(b)=af(a)=b\Longleftrightarrow f^{-1}(b)=a
  136. 137.
    Rates

    A rate compares how much of something is completed with the amount of time required.

    If aa units are completed in bb units of time, the rate is

    ab.\frac{a}{b}.

    Combined Rates

    When several people or machines work at the same time, their rates can be added:

    rcombined=r1+r2+⋯+rn.r_{\text{combined}}=r_1+r_2+\cdots+r_n.

    Amount Completed

    amount=rate×time.\text{amount}=\text{rate}\times\text{time}.
  137. 138.
    Geometric Transformations

    Geometric transformations map points in the plane to new points.

    Translation

    (x,y)⟼(x+a,y+b)(x,y)\longmapsto(x+a,y+b)

    Rotation

    (x,y)⟼(xcos⁡θ−ysin⁡θ, xsin⁡θ+ycos⁡θ)(x,y)\longmapsto(x\cos\theta-y\sin\theta,\ x\sin\theta+y\cos\theta)
    Counterclockwise rotation of a point about the origin

    Reflection

    (x,y)⟼(y,x)across y=x(x,y)\longmapsto(y,x)\qquad\text{across }y=x

    Dilation

    (x,y)⟼(kx,ky)(x,y)\longmapsto(kx,ky)
  138. 139.
    Vectors

    A vector is a quantity that has both magnitude and direction.

    If

    A=(x1,y1),B=(x2,y2),A=(x_1,y_1),\qquad B=(x_2,y_2),

    then

    AB→=(x2−x1,y2−y1).\overrightarrow{AB}=(x_2-x_1,y_2-y_1).

    Magnitude

    v=(a,b)⟹∣v∣=a2+b2.\mathbf{v}=(a,b)\qquad\Longrightarrow\qquad |\mathbf{v}|=\sqrt{a^2+b^2}.

    Vector Addition

    (a,b)+(c,d)=(a+c,b+d).(a,b)+(c,d)=(a+c,b+d).
    Geometric interpretation of vector addition

    Scalar Multiplication

    k(a,b)=(ka,kb),∣kv∣=∣k∣ ∣v∣.k(a,b)=(ka,kb),\qquad |k\mathbf{v}|=|k|\,|\mathbf{v}|.

    Points on a Segment

    p=(1−t)a+tb,0≤t≤1.\mathbf{p}=(1-t)\mathbf{a}+t\mathbf{b},\qquad 0\le t\le1.
  139. 140.
    Floor and Ceiling Functions

    Floor Function

    The floor of x is the greatest integer less than or equal to x.

    ⌊x⌋≤x<⌊x⌋+1\boxed{\lfloor x\rfloor\le x<\lfloor x\rfloor+1}
    Graph of the floor function

    Ceiling Function

    The ceiling of x is the smallest integer greater than or equal to x.

    ⌈x⌉−1<x≤⌈x⌉\boxed{\lceil x\rceil-1<x\le\lceil x\rceil}
    Graph of the ceiling function
    x∉Z⟹⌈x⌉=⌊x⌋+1x\notin\mathbb Z\Longrightarrow\lceil x\rceil=\lfloor x\rfloor+1
  140. 141.
    Counting Principles

    Addition Principle

    For mutually exclusive cases, add the numbers of possibilities.

    n1+n2+⋯+nkn_1+n_2+\cdots+n_k

    Multiplication Principle

    For successive choices, multiply the numbers of possibilities.

    n1n2⋯nkn_1n_2\cdots n_k
    Counting choices using the Multiplication Principle

    Complement Principle

    Desired=Total−Undesired\text{Desired}=\text{Total}-\text{Undesired}
  141. 142.
    Permutations

    A permutation is an arrangement of objects in which order matters.

    Arranging All Objects

    n!n!

    Arranging Some of the Objects

    P(n,r)=n!(n−r)!P(n,r)=\frac{n!}{(n-r)!}

    Identical Objects

    n!n1!n2!⋯nk!\frac{n!}{n_1!n_2!\cdots n_k!}
  142. 143.
    Combinations

    A combination is a selection of objects in which order does not matter.

    (nr)=n!r!(n−r)!\binom{n}{r}=\frac{n!}{r!(n-r)!}

    The notation (nr)\binom{n}{r} is read as “n choose r”.

    Symmetry Identity

    (nr)=(nn−r)\binom{n}{r}=\binom{n}{n-r}

    Choosing the objects to include is equivalent to choosing the objects to exclude.

    Pascal's Identity

    (nr)=(n−1r)+(n−1r−1)\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}

    Faster Calculation

    (107)=(103)=10⋅9⋅83⋅2⋅1=120\binom{10}{7}=\binom{10}{3}=\frac{10\cdot9\cdot8}{3\cdot2\cdot1}=120
  143. 144.
    Probability

    Probability measures how likely an event is to occur.

    0≤P(A)≤10\le P(A)\le1

    Sample Space and Events

    The sample space SS is the set of all possible outcomes, and an event AA is a collection of outcomes from the sample space.

    Equally Likely Outcomes

    P(A)=∣A∣∣S∣P(A)=\frac{|A|}{|S|}

    Complement

    P(Ac)=1−P(A)P(A^c)=1-P(A)

    Probability of A or B

    P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)

    Independent Events

    P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)
  144. 145.
    Square-Root Long-Division Algorithm

    A step-by-step algorithm for finding square roots by grouping digits into pairs, subtracting squares, and determining each successive digit.

    Square-root long-division algorithm illustrated with 4047 and 954.
  145. 146.
    Optimization by Equalization

    For two nonnegative numbers with a fixed sum, their product increases and the sum of their squares decreases as the numbers become closer together.

    Moving two numbers closer together while keeping their sum fixed.
    a+b=S⟹ab≤S24\boxed{a+b=S\Longrightarrow ab\le\frac{S^2}{4}}
  146. 147.
    Sperner's Theorem

    The maximum size of an antichain of subsets of an n-element set is

    (n⌊n/2⌋)\boxed{\binom{n}{\lfloor n/2\rfloor}}
  147. 148.
    Radical Expressions

    If x and y are nonnegative integers, d is squarefree, and

    x+y=kd,\sqrt{x}+\sqrt{y}=k\sqrt{d},

    then for some nonnegative integers a and b,

    x=da2,y=db2,a+b=k\boxed{x=da^2,\quad y=db^2,\quad a+b=k}
  148. 149.
    Invariants

    An invariant is a quantity or property that remains unchanged throughout a process.

    I(S0)=I(S1)=I(S2)=⋯I(S_0)=I(S_1)=I(S_2)=\cdots

    If the initial and target configurations have different invariant values, the target is impossible to reach.

    I(initial)≠I(target)⟹impossible\boxed{I(\text{initial})\ne I(\text{target})\Longrightarrow\text{impossible}}
  149. 150.
    Rational and Irrational Numbers

    A rational number can be written as

    ab,a,b∈Z,b≠0.\frac ab,\qquad a,b\in\mathbb Z,\quad b\ne0.

    A real number that cannot be written in this form is irrational.

    37, 0, 0.125⏟Rational2, π⏟Irrational\underbrace{\frac37,\ 0,\ 0.125}_{\text{Rational}}\qquad\underbrace{\sqrt2,\ \pi}_{\text{Irrational}}
    xxyyx+yx+yxyxy
    RationalRationalRationalRational
    RationalIrrationalIrrationalMay be either
    IrrationalIrrationalMay be eitherMay be either
  150. 151.
    Mathematical Induction

    Mathematical induction proves statements for every integer in a specified range using a base case and an inductive step.

    Base Case

    P(n0)P(n_0)

    Inductive Step

    P(k)⟹P(k+1)P(k)\Longrightarrow P(k+1)

    Extensions include strong induction, step-size induction, binary induction, and Cauchy induction.

    Four quadrants with the original missing cell and a central L-shaped tromino (16.3.png)
  151. 152.
    Fibonacci Numbers

    The Fibonacci sequence starts with 0 and 1, and each subsequent term is the sum of the previous two.

    F0=0,F1=1,Fn+2=Fn+1+FnF_0=0,\quad F_1=1,\quad F_{n+2}=F_{n+1}+F_n
    0, 1, 1, 2, 3, 5, 8, 13, 21,…0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ldots
    nn0011223344556677
    FnF_n001111223355881313
  152. 153.
    Nonnegative Squares

    Every real square is nonnegative. This helps prove inequalities and find minimum values.

    x2≥0(x∈R)x^2\ge0\qquad(x\in\mathbb{R})
    a2+b2−2ab=(a−b)2≥0a^2+b^2-2ab=(a-b)^2\ge0
    x2−6x+13=(x−3)2+4≥4x^2-6x+13=(x-3)^2+4\ge4
  153. 154.
    Stewart's Theorem
    Stewart's Theorem in triangle ABC, where AD = d, BD = m, DC = n, AB = c, and AC = b.
    a(d2+mn)=mb2+nc2\boxed{a(d^2+mn)=mb^2+nc^2}
  154. 155.
    Incenter of a Triangle

    The incenter is the intersection of the three internal angle bisectors of a triangle.

    Triangle ABC with incenter I and its incircle of radius r.
    r=[ABC]s\boxed{r=\frac{[ABC]}{s}}
  155. 156.
    Approximations and Bounds
    π=3.14159…\pi=3.14159\ldots
    2=1.41421…\sqrt{2}=1.41421\ldots
    3=1.73205…\sqrt{3}=1.73205\ldots
    e=2.71828…e=2.71828\ldots