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Olympiad Toolkit

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

Showing 101–150 of 156 toolkits
  1. 101.
    Law of Sines
    asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
  2. 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}
  3. 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
  4. 104.
    Trapezoid Midsegment Theorem
    MN∥AB∥CD,MN=AB+CD2MN\parallel AB\parallel CD,\qquad MN=\frac{AB+CD}{2}
  5. 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
  6. 106.
    Reflection Method

    Find min{PA+PB}.

    Reflection Method
    CQ=CQ′CQ=CQ'
    BL=BL′BL=BL'
    Reflection Method in square
  7. 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
  8. 108.
    Solving Linear Congruences
    ax≡b(modm)ax\equiv b\pmod m

    Reduce the coefficient of x to solve.

  9. 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}
  10. 110.
    Finding Last Digits

    Find the last 3 digits of 23578 × 16327

    Finding last digits of multiplication
  11. 111.
    Modulo Prime Powers
    2341≡  ?(mod125)2^{341}\equiv\; ?\pmod{125}
    2341=2×(5−1)1702^{341} = 2\times(5-1)^{170}
  12. 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
  13. 113.
    Trigonometric Function Graphs
    Graph of sine function
    Graph of cosine function
    Graph of tangent function
    Graph of cotangent function
  14. 114.
    Absolute Value

    Graph of ∣x∣+∣y∣=5|x|+|y|=5:

    Graph of |x|+|y|=5
  15. 115.
    Reciprocal Functions

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

    Graph of y = 1/(2x)
  16. 116.
    Radical Axis
    Radical axis
  17. 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}
  18. 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!}
  19. 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)}
  20. 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
  21. 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')
  22. 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

  23. 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.

  24. 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

  25. 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)
  26. 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)
  27. 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
  28. 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
  29. 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}}
  30. 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}
  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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
  37. 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}.
  38. 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)
  39. 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.
  40. 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
  41. 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}
  42. 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!}
  43. 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
  44. 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)
  45. 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.
  46. 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}}
  47. 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}}
  48. 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}
  49. 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}}
  50. 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