mathematics competition formulas

8/10/2019 Mathematics Competition Formulas

1/17

96

ormulas & Definitionslgebraxponents

Quadratic Formulainomial Teorem

Difference of Squaresero, Rules ofrobability

Appendix

I: Formulas and Definitions


2/17


3/17

98

Geometrylope Formula

Distance Formulaarabola


4/17

rigonometryLaw of CosinesLaw of Sines


5/17

00

rigonometryomplex Numbersrea of riangleonicseneral Formandard Form


6/17

MeasurementDistance

reaWeight

lectricityrobability

Multiplication Principleermutations

Measurement


7/17

02

Permutations of Objects not all DifferentCombinations

Arrangements with replacementProbability, Fundamental rule ofIndependent EventsDependent EventsMutually Exclusive EventsComplimentary EventsExpected ValueBinomial Probability

http://www.math.com/tables/

Definitions


8/17

ection A Algebra

Te Elusive Formulas pages are used with permission from: www.nysml.org/Files/formulas.pdf

The Elusive Formulas2

2nd

Edition: finalized August 1, 2001

Original Edition: finalized May 23, 2001

Section A Symbol Table for all there exists the empty set is an element of is not an element of , + the set of natural numbers

the set of integers

the set of rational numbers

the set of real numbers

the set of complex numbers

is a subset of or and union intersection implies, iff is equivalent to

n

i

i 1

a

a1+a2+a3+a4+a5+...+ann

ii 1

a a1a2a3a4a5an

(a,b) = d d is the gcd of a and b

[a,b] = d d is the lcm of a and b

(a) number of factors of a(a) sum of the factors of a(a) Euler Phi Function(a) Mobius Function| a | absolute value of a

a greatest integer functiona least integer functiona:b:c ratio of a to b to c

a:b:c::d:e:f ratio of a to b to c=ratio of d to e to f

pi 3.141592653589793

e euler number 2.718281828459

blog a c bc= a

log a c 10c= a

n! n(n1)(n2)(n3)(n4)321

nPrn!r!

= n(n1)(n2)(nr+1)

nCrorn

r

n! n(n 1)(n 2)...(n r+1)

r!(n r)! n(n 1)(n 2)...(2)(1)

moda b c aand bleave the same remainder

when divided by c.

Appendix

III: The Elusive Formulas - Part 2


9/17

04

ection B - Algebralgebra Arithmetic Series; Geometric Series; Rational Root Teorem

Section B - AlgebraUsed with permission from:

NYSME(New York State Math League) Copyright (c) 2002 Ming Jack Po & Kevin Zheng.

Section B Algebra (a b)

3= a

3 b

3iff a = 0 or b = 0 or (ab) = 0

a3 b

3= (a b)(a

2 ab + b

2)

a3+ b

3+ c

3 3abc = (a + b + c)(a

2+ b

2+ c

2 ab bc ca)

a4+ b

4+ c

4 2a

2b

2 2b

2c

2 2c

2a

2= -16s(s a)(s b)(s c) when 2s = a+b+c

an+ b

n= (a + b)(a

n1+ b

n1) ab(a

n2+ b

n2)

an b

n= (a b)(a

n1 a

n2b + a

n3b

2 a

n4b

3+ + a

2b

n3 ab

n2+ b

n1) [ a

n+ b

nis only true for odd n.]

(a b)n

= nC0an

nC1an1

b + nC2an2

b2

nC3an3

b3

+ nC4an4

b4

nCn-2a2

bn2

+ nCn-1abn1

+ nCnbn

a(a+1)(a+2)(a+3) = (a2+3a+1)

2 1

Arithmetic Series: If a1, a2, a3, ..., anare in arithmetic serieswith common difference d:

nth

term in terms of mth

term an= am+ (n m)d

Sum of an arithmetic series up to term n n 1 n 1

i

i 1

n a a n 2a (n 1)da

2 2

Geometric Series: If a1, a2, a3, ..., anare in geometric serieswith common ratio r:

nth

term of a geometric seriesn 1

n 1a a r

Sum of a non-constant (r 1) geometric

series up to term n

nn1

ii 1

a (1 r )a

1 r

Sum of an infinite geometric series1

i

i 1

aa

1 r

iff |r| < 1

n

i 1

n(n 1)i

2

n2

i 1

n(n 1)(2n 1)i

6

2 2n3

i 1

n (n 1)i

4

3 2n4

i 1

n n 1 6n 9n n 1i

30

If P(x) = anxn+ an-1x

n1+ an-2x

n2+ an-3x

n3+ ... + a1x + a0= 0, ia is a constant, then

Sum of roots taken one at a time

(the sum of the roots)ir =

n 1

n

-a

a

Sum of roots taken two at a time i ji j

r r

= n 2n

a

a

Sum of roots taken p at a time i j ki j ... k

r r ...r

=n pp

n

a(-1)

a

Rational Root Theorem

If P(x) = anxn+ an-1x

n-1+ an-2x

n-2+ an-3x

n-3+ ... + a1x + a0is a polynomial with integer coefficients and

c is a rational root of the equation P(x)= 0 (where (b, c) =1), then b | a0and c | an.

If P(x) is a polynomial with real coefficients and P(a + bi) = 0, then P(a bi) = 0.

If P(x) is a polynomial with rational coefficients and P(a + b c ) = 0, then P(a b c ) = 0.


10/17

Section C - Number TeoryDivisibility; Modulo Congruence: Fibonacci Sequence; Farey Series; Number Teory Functions

Section C - Number TheoryUsed with permission from:

www.nysml.org/Files/Formulas.pdf

Section C Number Theory Number Theory mainly concerns and , all variables exist in unless stated otherwise

Divisibility:a,b , a0: a | b k such that ak = b1|a, a|0, a|(a) a|ba|bc a|b b|c a|ca|1a=1 a|b a|c a|(bc) a|bc (a,b) =1 a|c

a|bb|a a=b a|b c|d ab|cd a|c b|c (a,b)=1 ab|c

Modulo Congruence: a,b,m , m0: a b (mod m) m | (a-b)Suppose that ab (mod m), cd (mod m), and p is prime; then:

agcg (mod m) ab cd (mod m) (g,p)=1 gp-1 1 (mod p)agcg (mod m) ab cd (mod m) (p-1)! -1 (mod p)

(g,m)=1 g(m) 1 (mod m) hfhg (mod m) (m,h)=1 fg (mod m)

Fibonacci Sequence

Sequence of integers beginning with two 1s and each subsequent term is the sum of the previous 2 terms. 1,1,2,3,5,8,13,21,34,55,89,144, ... F(1)=F(2)=1, for n3, F(n)=F(n-1)+F(n-2)

Let= Golden Ratio = 5 12 , then F(n) =

-nn -

5

F(n)F(n+3) F(n+1)F(n+2) = (-1)n

Farey Series [Fn]

Ascending sequence of irreducible fractions between 0 and 1 inclusive whose denominator is n F3= 0 1 1 2 11 3 2 3 1, , , , ; F7=

0 3 3 5 3 5 61 1 1 1 2 1 2 1 4 2 4 11 7 6 5 4 7 3 5 7 2 7 5 3 7 4 5 6 7 1, , , , , , , , , , , , , , , , , ,

if a , cd

, and ef

are successive terms inFn, then bcad = decf = 1 andc a ed b f

Number Theory FunctionsThe following number theory functions have the property that if (a,b)=1, then f(ab)=f(a)f(b)

Tau Function: Number of factors of n: (n) = m

i

i 1

1

Sigma Function: Sum of factors of n: (n) = im

j

i

j 0i 1

p

=

i1mi

i 1 i

p 1

p 1

Euler Phi Function: Number of integers between 0 and n that are relatively prime to n

(n) = i im

1

i i

i 1

p p

=m

i 1 i

1n 1

p

Mobius Function: (n) =

0 if n is divisible by any square 1

otherwise:

1 if n is has an even number of prime factors

1 if n is has an odd number of prime factors


11/17

06

ection C - Number TeoryDivisibility Rulesection D - Logarithmsection E - Analytic Geometry

Section D - LogarithmsSection E - Analytic GeometryUsed with permission from: www.nysml.org/Files/Formulas.pdf

Copyright (c) 2002 Ming Jack Po & Kevin Zheng.

Divisibility Rules

Given integer k expressed in base n 2, k = 2 30 1 2 3a a n a n a n ... = iii 0

a n

, 0 ia < n

Note: im m 1 0 in

i 0

a a ...a a n

, secondary subscript omission implies base10: im m 1 0 ii 0

a a ...a 10 a

Divisor (d) Criterion

3, 9 If 0 1 2 3 4a a a a a ... is divisible by 3 or 9

11 If 0 1 2 3 4a a a a a ... is divisible by 11

7, 13 If 2 1 0 5 4 3 8 7 6 11 10 9a a a a a a a a a a a a ... is divisible by 7 or 13

2m

, 5m If m 1 m 2 m 3 0a a a ...a is divisible by 2

mor 5

m

Basic/Specific

7Truncate rightmost digit and subtract twice the value of said digit from the remaininginteger. Repeat this process until divisibility test becomes trivial.

d | nm If m 1 m 2 m 3 m 4 0

na a a a ...a

is divisible by d

factor of

nm

1If m 1 m 2 1 0 2m 1 2m 2 m 1 m 3m 1 3m 2 2m 1 2m

n n na a ...a a a a ...a a a a ...a a ... is divisible

factor ofnm

+ 1 If m 1 m 2 1 0 2m 1 2m 2 m 1 m 3m 1 3m 2 2m 1 2mn n na a ...a a a a ...a a a a ...a a ... is divisibled = xy,

(x,y)=1( x | k and y | k ) d | k

General

d | kn1Truncate rightmost digit and add k times the value of said digit from the remaining

integer. Repeat this process until divisibility test becomes trivial.

Section D LogarithmsFor b an integer 1 ,

cblog a c b a blog b 1 blog 1 0

clog a c log a alog ba b abclog log a log b log c

a b alog b log c log c a blog b log a 1 log b log aa b

Section E Analytic GeometryDistance between line ax + by + c = 0 and

point (x0, y0) in 2D plane:

Distance between the plane ax + by + cz + d = 0

and point (x0, y0, z0) in 3D space:

0 0

2 2

| x a y b c |

a b

0 0 0

2 2 2

| x a y b z c d |

a b c

Section F Inequalities

+: the set of all positive real numbers;

: the set of all negative real numbers

a2+ b

22ab; a

2+ b

2+ c

2ab + bc + ca; 3(a

2+ b

2+ c

2+ d

2) 2(ab + bc + cd + da + ac + bd)

The quadratic-arithmetic-geometric-harmonic mean inequality: for ai> 0

1 2 3 n

2 2 2 2

1 2 3 n 1 2 3 nn1 2 3 n1 1 1 1

a a a a

a a a ... a a a a ... ana a a ...a

... n n

, with equalities holding

iff a1= a2= a3= a4= = an.

If constant k>1 and large x: 1 a, c+a>b

Basic Angle Identities A+B+C = 180, {a,b,c}(0,)Law of Cosines a

2+ b

2= c

2+ 2ab cos C

Law of Tangents tan(A)tan(B)tan(C) = tan(A)+tan(B)+tan(C)

Assorted Identities

a b b c c ar r r r r r = s2

D2= R

2 2Rr 4mc

2= 2a

2+ 2b

2+ c

2a b cr r r r = 4R

r =A B2 2

C2

c sin sin

cos c

K

r s c

r

2

=

(s a)(s b)(s c)

s

a b c

1 1 1 1

r r r r

C2

s a s bsin

ab

C2

rtan

s c

C2

s a s btan

s s c

C2

s s ccos

ab

tc= 2 a b s s c

a b

tc=

C2

2abcos

a ba b cm m m3 1

4 a b c

A B2

A B2

tana b

a b tan


15/17

10

ection I - Euclidean G. II (Quadril.)General Quadrilateral Diagonals; General Quadrilateral Midpoints; Circumscribed Quadrilateral; Cyclic Quadrilateral; Parallelogram; Rectangle

Section I - Euclidean G. II (Quadrilateral)Used with permission from: www.nysml.org/Files/Formulas.pdf

Copyright (c) 2002 Ming Jack Po & Kevin Zheng.

Section I Euclidean Geometry II (The Quadrilateral)General Quadrilateral Diagonals

E and F are midpoints of AC and BD

KGAB KGCD= KGBC KGDA

K = 12

AC BD sin AGB 2 2 2 2 2 2 2

AB BC CD DA AC BD 4EF

General Quadrilateral Midpoints

IfAH DG CF BE

HD GC FB EA = n

Then:2

EFGH

2

ABCD

K n 1

K (n 1)

Circumscribed Quadrilateral

AB CD BC AD = s; KABCD= rs

If QuadABCDis also cyclic, then

K = AB CD BC AD

Cyclic Quadrilateral

A + C = B + D = 180

KABCD= s AB s BC s CD s DA

BC AD AB CD BD AC

AC BC CD DA AB BD AB BC CD DA Parallelogram

2 2 2 2

2(BC BA ) BD AC

Rectangle

For all point P:2 2 2 2

PA PC PB PD

Three Pole Problem

if a || b || c, then1 1 1

a b c

Quadrilateral with Diagonals

AC BD K = 12AC BD

2 2 2 2

AB CD BC DA

Ptolemys Theorem:

In any QuadABCD, BD AC BC AD AB CD , with equality holding iff QuadABCDis cyclic.


16/17

ction J - Euclidean G. III (Circle)ction K - rigonometrythagorean: Odd-Even Functions; Summation of Angles, Multiple Angles

ection J - Euclidean G. III (Circle)ection K - Trigonometrysed with permission from:

www.nysml.org/Files/Formulas.pdf Copyright (c) 2002 Ming Jack Po & Kevin Zheng.

Section J Euclidean Geometry III (The Circle)

Circles

12AEC BED AC BD Power of the point: AE BE CE DE

Circles 2

AB AG ; 12DAF DF CE 2

AB AD AC AF AE

Section K Trigonometry

sin= cb

; cos = ab

; tan = ca sin= AB; cos = OA ; tan = BC

15 18 30 36 45 54 60 75

sin6 2

4

5 1

4

12

4

552 22

5 1

4

32

6 2

4

cos6 2

4

4

552 32

5 1

4

22

4

552 12

6 2

4

tan 2 3

552

215

3

3

15

552

1

552

215

3 2 3

Pythagorean Odd-Even Functions Summation of Angles

sin2+ cos

2= 1 sin(-) = -sin() sin ( ) = sin()cos() cos()sin()

1 + tan2= sec

2 cos(-) = cos() cos ( ) = cos()cos() sin()sin()

1 + cot2= csc

2 tan(-) = -tan() tan ( ) =

tan tan

1 tan tan

sin 2= 2 sincos sin 3= 3sin 4sin3 sin 4= 4sincos(cos

3sin)

cos 2= cos sin cos 3= 4cos3 3cos cos 4= sin

4+ cos

4 6cos

2sin

2

Multiple

Angles

tan 2=2

2tan

1 tan

tan 3=

3

2

tan 3tan

3tan 1

tan 4=

24 2

4 tan 1 tan

tan 6 tan 1


17/17

12

rigonometryum to Product; Product to Sum; Square Identities; Cube Identities;

Used with permission from

www nysml org/ Copyright (c) 2002 Ming Jack Po & Kevin Zheng

Sum to Product Product to Sum

sin sin = 2sin cos2 2

sin sin = 1

2[cos() cos(+)]

cos+ cos = 2cos cos2 2

cos cos = 12

[cos() + cos(+)]

cos cos = 2sin sin2 2

sin cos = 12

[sin() + sin(+)]

tan tan = sin

cos cos

tan tan =

cos cos

cos cos

Square Identities Cube Identities Angle Identities tan (/2) Identities

sin2= 1

2(1-cos2) sin

3=

3sin sin 3

4

2

1 cossin

2

sin= 22

2

2tan

1 tan

cos2= 1

2(1+cos2) cos

3=

3cos cos3

4

2

1 coscos

2

cos=

2

2

2

2

1 tan

1 tan

tan2= 1 cos 21 cos 2

tan3= 3sin sin 33cos cos3

2

1 costan1 cos

Authors:Ming Jack Po (Johns Hopkins University)

Kevin Zheng (Carnegie Mellon University)

Proof Readers:Jan Siwanowicz (City College of New York)

Jeff Amlin (Harvard University)

Kamaldeep Gandhi (Brooklyn Polytechnic University)

Joel Lewis (Harvard University)Seth Kleinerman (Harvard University)

Programs Used:Math Type 4, 5

CadKey 5

Geometers Sketchpad 3, 4

Microsoft Word XPMathematica 4.1

References:IMSA Noah Sheets

Bronx Science High School Formula Sheets, Math Bulletin

mathematics competition formulas

Documents