96
Appendix I: Formulas and Definitions
Formulas & Definitions Algebra Exponents Quadratic Formula Binomial Teorem Difference of Squares Zero, Rules of Probability
97
Appendix II: The “Elusive Formulas” - Part 1
Geometry riangle Pythagorean Teorem Heron’s Formula Angles
Geometry
98
Geometry Slope Formula Distance Formula Parabola
99
rigonometry Law of Cosines Law of Sines
100
rigonometry Complex Numbers Area of riangle Conics General Form Standard Form
101
Measurement
Measurement Distance Area Weight Electricity Probability Multiplication Principle Permutations
102
Definitions
Permutations of Objects not all Different Combinations Arrangements with replacement Probability, Fundamental rule of Independent Events Dependent Events Mutually Exclusive Events Complimentary Events Expected Value Binomial Probability
http://www.math.com/tables/
103
Appendix III: The “Elusive Formulas” - Part 2 2
The “Elusive Formulas”
nd
2 Edition: finalized August 1, 2001 Original Edition: finalized May 23, 2001
Section A – Symbol Table
, +
, iff
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 is equivalent to
n
a
i
a1+a2+a3+a4+a5+...+a n
i 1 n
a
i
a1•a2•a3•a4•a5•…•an
i 1
(a,b) = d [a,b] = d
(a) (a) (a) (a)
e log b a c
number of factors of a sum of the factors of a Euler Phi Function Mobius Function absolute value of a greatest integer function least integer function ratio of a to b to c ratio of a to b to c=ratio of d to e to f pi 3.141592653589793… euler number 2.718281828459… c b = a
log a c
10 = a
n! nPr
n(n–1)(n–2)(n–3)(n–4)…3×2×1 n! = n(n–1)(n–2)…(n–r+1) r!
|a| a a a:b:c a:b:c::d:e:f
nCr or
n r
c
n! r!(n r)!
n(n 1)(n 2)...(n r+1) n(n 1)(n 2)...(2)(1)
a b mod c a and b leave the same remainder when divided by c.
d is the gcd of a and b d is the lcm of a and b
Section A Algebra
Te “Elusive Formulas” pages are used with permission from: www.nysml.org/Files/formulas.pdf
104
Section B – Algebra
(a ± b)3 = a3 ± b3 iff a = 0 or b = 0 or (a±b) = 0 3 3 2 2 a ± b = (a ± b)(a ab + b ) 3 3 3 2 2 2 a + b + c – 3abc = (a + b + c)(a + b + c – ab – bc – ca) a4 + b4 + c4 – 2a2 b2 – 2b2c2 – 2c2a2 = -16s(s – a)(s – b)(s – c) when 2s = a+b+c n n n–1 n–1 n–2 n–2 a + b = (a + b)(a + b ) – ab(a + b ) n n n–1 n–2 n–3 2 n–4 3 2 n–3 n–2 n–1 n n a b + a b a b + … + a b ab a ± b = (a ± b)(a + b ) [ a + b is only true for odd n.] n n n–1 n–2 2 n–3 3 n–4 4 2 n–2 n–1 n (a ± b) = nC0a ± nC1a b + nC2a b ± nC3a b + nC4a b ± … ± nCn-2a b + nCn-1ab + nCn b 2 2 a(a+1)(a+2)(a+3) = (a +3a+1) – 1 Arithmetic Series: If a1, a2, a3, ..., an are in arithmetic series with common difference d: nth term in terms of mth term an = am + (n – m)d n n a1 a n n 2a 1 (n 1)d a Sum of an arithmetic series up to term n i 2 2 i 1 Geometric Series: If a1, a2, a3, ..., an are in geometric series with common ratio r: th
a n a1r n 1
n term of a geometric series n
Sum of a non-constant (r 1) geometric series up to term n
ai i 1
a1 (1 r n ) 1 r
a
a i 1 1 r iff |r| < 1
Sum of an infinite geometric series
i 1
n
i i 1
n(n 1) 2
n
i
2
n(n 1)(2n 1) 6
i 1
n
If P(x) = anx + an-1x
n–1
n–2
+ an-2x
+ an-3x
2
n
i
3
n (n 1) 4
i 1
n–3
n
i
4
n n 1 6n 3 9n 2 n 1
i 1
30
+ ... + a1x + a0 = 0, ai is a constant, then -a n 1
Sum of roots taken one at a time (the sum of the roots)
r i =
Sum of roots taken two at a time
ri r j = i j
Sum of roots taken p at a time
2
an a n 2 an
ri rj ...r k = (-1) p
i j... k
a n p an
Rational Root Theorem If P(x) = anx + an-1x + an-2x + an-3xn-3 + ... + a1x + a0 is a polynomial with integer coefficients and c is a rational root of the equation P(x) = 0 (where (b, c) =1), then b | a 0 and c | a n. n
n-1
n-2
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.
Section B - Algebra Algebra Arithmetic Series; Geometric Series; Rational Root Teorem
Section B - Algebra Used with permission from: NYSME(New York State Math League)
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
105
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 = b 1|a, a|0, a|(±a) a|b a|bc a|b b|c a|c a|1 a=±1 a|b a|c a|(b±c) a|bc (a,b) =1 a|c a|b b|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 a b (mod m), cd (mod m), and p is prime; then: p-1 a±g c±g (mod m) a±b c±d (mod m) (g,p)=1 g 1 (mod p) ag cg (mod m) ab cd (mod m) (p-1)! -1 (mod p) (m) (g,m)=1 g hf hg (mod m) (m,h)=1 f g (mod m) 1 (mod m)
Fibonacci Sequence Sequence of integers beginning with two 1’s 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 =
, then F(n) =
5 1 2
F(n)•F(n+3) – F(n+1)•F(n+2) = (-1)
n
-
-n
5
n
Farey Series [F n] Ascending sequence of irreducible fractions between 0 and 1 inclusive whose denominator is n F 3 = 01 , 31 , 21 , 32 , 11 ; F 7 = 10 , 71 , 61 , 51 , 41 , 72 , 31 , 52 , 73 , 21 , 74 , 53 ,32 ,75 ,43 ,54 ,65 ,76 ,11
if
a
, dc , and
e f
are successive terms in F n, then bc–ad = de–cf = 1 and
c d
ba f e
Number Theory Functions The following number theory functions have the property that if (a,b)=1, then f(a×b)=f(a)×f(b) m
1
(n) =
Tau Function: Number of factors of n:
i
i 1
j (n) = pi = i 1 j 0 m
Sigma Function: Sum of factors of n:
i
pi1 1 i 1 pi 1 m
i
Euler Phi Function: Number of integers between 0 and n that are relatively prime to n
i 1 i 1 if n is divisible by any square 1
(n) =
Mobius Function:
(n) =
m
p
i
i
Used with permission from: www.nysml.org/Files/Formulas.pdf
m
i
= n 1 p1
0 otherwise: 1 if n is has an even number of prime factors 1 if n is has an odd number of prime factors
Section C - Number Teory Divisibility; Modulo Congruence: Fibonacci Sequence; Farey Series; Number Teory Functions
Section C - Number Theory
pi
i 1
106
Divisibility Rules
2
ai n , 0
3
i
Given integer k expressed in base n 2, k = a 0 a1n a 2 n a3 n ... =
ai < n
i 0
Note: a m a m 1 ...a 0 Divisor (d) 3, 9
c i f i c e p S / c i s a B
n
m
am 1...a 0 10i ai
i 0
i 0
Criterion If a 0 a 1 a 2 a 3 a 4 ... is divisible by 3 or 9
If a 2 a1a 0 a 5 a 4 a3 a8 a7 a6 a11 a10 a9 ... is divisible by 7 or 13
7, 13 2 ,5
m
m
If a m 1a m 2 a m 3 ...a 0 is divisible by 2 or 5
m
Truncate rightmost digit and subtract twice the value of said digit from the remaining integer. Repeat this process until divisibility test becomes trivial.
7 d|n
i
If a 0 a 1 a 2 a 3 a 4 ... is divisible by 11
11 m
a n , secondary subscript omission implies base10: a
i
m
If a If a
If a m 1a m 2 a m 3 a m 4 ...a 0
factor of m l n – 1 a r factor of e n nm + 1 e G d = xy, (x,y)=1 d | kn±1
m 1 m 2
a
...a1a 0
a
...a1a 0
m 1 m 2
a a n
n
n
is divisible by d
2m 1 2m 2
a
...am1 am
a
...am1 am
2m 1 2m 2
a a n
3m 1 3m 2
a
...a2m1 a2m
a
...a2m1 a2m
3m 1 3m 2
n
n
n
... is divisible ... is divisible
( x | k and y | k ) d | k Truncate 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 –Logarithms For b an integer 1 , log b a c bc a
log b b 1
log a c c log a log a b
a
logb c loga c
log a b
log a b
log b 1 0 log abc log a log b log c
b
log b a 1
a
log b
b
log a
Section E – Analytic Geometry Distance between line ax + by + c = 0 and point (x0, y0) in 2D plane: | x 0a y0 b c |
Distance between the plane ax + by + cz + d = 0 and point (x0, y0, z0) in 3D space: | x 0 a y0 b z0 c d |
a 2 b2
a 2 b 2 c2
Section F – Inequalities
+
–
: the set of all positive real numbers; : the set of all negative real numbers 2 2 2 2 2 2 2 2 2 a + b 2ab; a + b + c ab + bc + ca; 3(a + b + c + d ) 2(ab + bc + cd + da + ac + bd) The “quadratic-arithmetic-geometric-harmonic mean inequality:” for ai > 0
n 1 1 1 a1 Teory a 2 a3 Number
...
Section C Divisibility Rules iff a1 = a2 = Section D - Logarithms Section E - Analytic Geometry
1 an
n
a1a 2 a 3 ...a n
a1 a 2 a 3 ... a n n
2
2
2
a1 a 2 a3 ... an
2
, with equalities holding
n
a3 = a4 = … = an.
k>1 and large x: 1 < k If Section Dconstant - Logarithms
Section E - Analytic Geometry
1
x
1
x
< log(x) < x
1
k
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
Used with permission from: www.nysml.org/Files/Formulas.pdf
k
log(x)
< x < x log(x) < x < x
x
x
< k < x! < x
107
nd
2
2
2
2
2
Cauchy-Schwarz Inequality- For 2 degree: (a1 b1+a2 b2) (a1 +a2 )(b1 +b2 ) with equality holding iff a1:a2::b1:b2. In general, for any 2 sequences of real numbers, ai and bi, each of length n: 2 2 2 2 2 2 2 2 2 (a1 b1+a2 b2+a3 b3+…+an bn) (a1 +a2 +a3 +…+an )(b1 +b2 +b3 +…+bn ) with equality holding iff a1:a2:a3: … :an::b1:b2:b3: … :bn. Chebyshev’s Inequality- If 0a1a2a3 … an, 0 b1 b2 b3 … bn, then: (a1+a2+a3+…+an) (b1+b2+b3+…+bn) n•(a1 b1+a2 b2+a3 b3+…+an bn) a a 2 ... a n . More Jensen’s Inequality- For a convex function f(x): f(a1)+f(a2)+f(a3)+…+f(an) n• f 1 n generally, if b1+b2+…+bn=1 and bi>0, then: b1f(a1)+b2f(a2)+b3f(a3)+…+bnf(an) f(b1a1+b2a2+b3a3+…+an)
Section G – Number Systems
Algebraic numbers: numbers that can be solutions to polynomial equations with integer coefficients: 5
5
23,
23 7 5 , ...
Transcendental numbers: numbers that cannot be solutions to polynomials: e, , ... is the ratio of the length of the circumference to the length of the diameter of a circle o o
= natural numbers: 1, 2, 3, 4, 5, ...
e = lim 1 x
1 x
x
if we define the square root of –1 to be i, then: = complex numbers = a+bi, where a,bR o n r i a s l r o e P b & m u r a N l x u e g l n p t a m c o e C R
2
2
2
a + b = r ; tan = a ; a = r • cos ; b = r • sin Z = a + bi = r • cis (polar form of a complex number) The magnitude of Z, represented by i
e = cos + i sin = cis n
n
n
(a+bi) = (r cis ) = r • cis(n)
|a+bi| = a 2 b2 cis (+) = cis • cis cis (-) =
Section F - Inequalities Section G - Number Systems
Section F - Inequalities Section G - Number Systems Used with permission from: www.nysml.org/Files/Formulas.pdf
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
cis cis
2,
108
Section H – Euclidean Geometry I (The Triangle) Stewart’s Theorem
Angle Bisector
man + dad = bmb +cnc
bm = cn;
Ceva’s Theorem
Circumcenter ( bisectors)
b
c
= 2R
sin A sin B sin C Extended Law of Sines The 4-5-6 Triangle
A = 2B;
K=
d² = bc - mn
Centroid (medians)
AFBDCE AEBFCD VD VE VF 1 AD BE CF
a
Menelaus’ Theorem
AM
BM
CM
2
MD ME MF K AFM = K FBM = K BDM = K DCM = K CEM = K EAM = 16 K ABC
ADBECF DBECFA Orthocenter (altitudes)
AFC ~ AEB ~ OEC ~ OFB BDA ~ BFC ~ OFA ~ ODC CEB ~ CDA ~ ODB ~ OEA
Nagel Point
Joins semi-perimeter points to vertices
Golden Triangle
ABC ~ DAB; =
CD
K = 84; R = 658 ; r = 4
4
5 1
Trisectors of the largest angle has length 6
Section H - Euclidean G. I (riangle) Stewart’s Teorem; Angle Bisector; Menelaus’ Teorem; Ceva’s Teorem; Orthocenter; Circumcenter Nagel Point Golden riangle
Section H - Euclidean G. I (Triangle) Used with permission from: www.nysml.org/Files/Formulas.pdf
BC CD
2 BC CD The 8-8-11 Triangle
The 13-14-15 Triangle
15 7
36° = /5
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
109
A Triangle and Its Circles ABC has
sides a, b and c and angles A, B, and C. The radius of the inscribed circle is r. The radius of the circumscribed circle is R. The area of the triangle is K. The semi-perimeter of the triangle is s. The altitude to sides a, b, c are ha, h b, hc respectively. The angle bisectors to angles A, B, C are ta, t b, tc respectively. The medians to side a, b, c are ma, m b, mc respectively.
The circles tangent to each line AB , BC ,
CA and directly next to sides a, b, c are called excircles Ia, I b, Ic respectively. The radii to ex-circles Ia, I b, Ic are ra , rb, rc respectively. The distance from I to circumcenter is d.
Area Formulas of the Triangle c hc absinC abc c 2 sin A sin B K= K= K= K= s(s a)(s b)(s c) K=rs K= 2 2 4R 2sinC For planar triangle with vertices P1(x1, y1), P2(x2, y2), P3(x3 y3) Coordinates of the centroid are x1 y1 1 1 x1 x2 x3 y1 y2 y3 K x 2 y 2 1 , 2 3 3 x 3 y3 1
a+b>c, b+c>a, c+a>b A+B+C = 180°, {a,b,c} (0,) 2 2 2 a + b = c + 2ab cos C tan(A)tan(B)tan(C) = tan(A)+tan(B)+tan(C)
Basic Edge Inequalities Basic Angle Identities Law of Cosines Law of Tangents
ra rb rb rc rc ra = s
2
r=
sin C2
tc =
c sin A2 sin B2 cos C2
s a s b ab
2 a bs s c ab
Assorted Identities 2 2 2 2 D = R – 2Rr 4mc = 2a + 2b + c 2
2
r c
K sc
tan C2 tc =
r sc
2
r =
(s a)(s b)(s c)
1
s
r
tan C2
2abcos C2
3
ab
4
s a s b s s c
m a m b mc a bc
1
Area Formulas of the riangle; riangle Assorted Identities; riangle & its Circles
Used with permission from: www.nysml.org/Files/Formulas.pdf
ra rb r c – r = 4R
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
1 ra
cos C2 a b ab
1 rb
1 rc
s s c ab tan A2 B tan A 2 B
110
Section I – Euclidean Geometry II (The Quadrilateral) General Quadrilateral Diagonals
General Quadrilateral Midpoints
E and F are midpoints of AC and BD K GAB • K GCD = K GBC • K GDA 1 2
K= AB
2
2
If
ACBDsin AGB 2
2
BC CD DA
2
AC
BD
2
AH
HD Then:
2
4EF
Circumscribed Quadrilateral
DG
CF
BE
GC FB EA K EFGH n2 1
K ABCD
=n
(n 1)2
Cyclic Quadrilateral
A + C = B + D = 180° K ABCD =
AB CD BC AD = s; K ABCD = rs If QuadABCD is also cyclic, then K=
s ABs BC s CD s DA BCAD ABCD BDAC
AC BCCD DAAB
ABCDBCAD Parallelogram
2(BC
2
BA
2
)
BD
2
Rectangle
AC
2
if a || b || c, then
a
1 b
For all point P: PA
2
2
PC
2
PB
2
PD
Quadrilateral with Diagonals
Three Pole Problem
1
BD ABBC CDDA
1 c
AC BD AB
2
K 2
CD
=
BC
Section I - Euclidean G. II (Quadril.) Ptolemy’s Theorem: General Quadrilateral Diagonals; General Quadrilateral Midpoints; Circumscribed Quadrilateral; Cyclic Quadrilateral; Parallelogram; Rectangle
1 2 2
ACBD DA
2
Section I - In Euclidean G. II (Quadrilateral) any QuadABCD , BD AC BC AD AB CD , with equality holding iff QuadABCD is cyclic. Used with permission from: www.nysml.org/Files/Formulas.pdf
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
111
Section J – Euclidean Geometry III (The Circle) Circles
Circles 2
1 2
AEC BED
Power of the point: AE
AC BD
BE
AB AG ;
CE DE
2
AB AD
AC
DF CE
AF
AE
DAF
1 2
Section K – Trigonometry
sin = c ; cos = a ; tan = c b b a
15°
18°
30°
5 1
1
6 2
sin
4
cos
tan
4
6 2
25 5
4
2 3
4
3
5 1 2
3
2 5 5
Pythagorean 2 2 sin + cos = 1 2 2 1 + tan = sec 2
2
sin = AB; cos = OA ; tan = BC 36°
2 5 5
45°
2
4 5 1
2
2
4
2 5 5
1
3
5 1 Odd-Even Functions sin(-) = -sin() cos(-) = cos()
2
1 + cot = csc
3
Section K - rigonometry Pythagorean: Odd-Even Functions; Summation of Angles, Multiple Angles
5 1 2
2 5 5 2
4
5 1 2
2 5 5
1
75°
6 2 2
4 6 2
2
4
3
2 3
tan 3 3 tan 3 tan 2 1
tan tan 1 tan tan 3
sin 4 = 4•sin•cos•(cos –sin) 4 4 2 2 cos 4 = sin + cos – 6cos •sin tan 4 =
Section J - Euclidean G. III (Circle) Section K - Trigonometry Used with permission from: www.nysml.org/Files/Formulas.pdf
3
4
tan ( ) =
sin 3 = 3sin – 4sin 3 cos 3 = 4cos – 3cos tan 3 =
60°
Summation of Angles sin ( ) = sin()cos() ± cos()sin() cos ( ) = cos()cos() sin()sin()
tan(-) = -tan()
sin 2 = 2 sin cos e s l e cos 2 = cos² – sin² p l i t l g 2tan u n A tan 2 = MG. III (Circle) Section J - Euclidean 1 tan 2
54°
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
4 tan 1 tan 2 tan 4 6 tan 2 1
112
Sum to Product
Product to Sum
cos 2 2 cos cos + cos = 2 cos 2 2 cos – cos = 2sin 2 sin 2 sin tan ± tan = cos cos sin ± sin = 2 sin
Square Identities 2
sin =
1 2
2
1 2
(1-cos2)
cos = (1+cos2) 2
tan =
1 cos 2 1 cos 2
3
3
cos = 3
tan =
1 2
[cos( – ) – cos(+)]
cos • cos =
1 2
[cos( – ) + cos(+)]
sin • cos =
1 2
[sin( – ) + sin(+)]
tan • tan =
Cube Identities
sin =
sin • sin =
cos cos cos cos
½ Angle Identities
3sin sin 3 4 3cos cos 3 4 3sin sin 3 3cos cos 3
sin 2 cos
2
tan 2
1 cos 2 1 cos 2
tan ( /2) Identities 2tan 2 sin = 1 tan 2 2
cos =
1 cos 1 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 Geometer’s Sketchpad 3, 4 Microsoft Word XP Mathematica 4.1
References: IMSA – Noah Sheets Bronx Science High School – Formula Sheets, Math Bulletin
rigonometry Sum to Product; Product to Sum; Square Identities; Cube Identities;
Used with permission from www.nysml.org/
Copyright (c) 2002 Ming Jack Po & Kevin Zheng.
1 tan 2
1 tan
2
2 2
