highschool playbook

8/13/2019 Highschool Playbook

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Coach Monkss High School Playbook

This playbook is meant as a training reference for high school math competitions. Students should be familiarwith all material in Coach Monkss MathCounts Playbook (a 6-8th grade level contest) as a prerequisite tolearning this material. Learn the items marked with a first. Then once you have mastered them try to learn the

other topics.

Arithmetic! In addition to the values memorized for MathCounts, the following facts can also be useful.

n n2

21 441

22 484

23 529

24 576

25 625

26 676

27 729

28 784

29 841

30 900

n n!

1 1

2 2

3 6

4 24

5 120

6 720

7 5040

8 40320

9 362880

10 3628800

n

1 1

2 1.414

3 1.732

4 2

5 2.236

6 2.449

7 2.646

8 2.828

9 3

10 3.162

log n

log 1 0

log 2 0.301

log 3 0.477

log 4 0.602

log 5 0.699

log 6 0.778

log 7 0.845

log 8 0.903

log 9 0.954

log 10 1

this is the only value in these tables that was rounded up when rounded to the nearest thousandth

3.14159265358979and e 2.7182818284590452

Prime factorizations of recent, current, and upcoming years:

- 2002 2 7 11 13

- 2003is prime

- 2004 22 3 167

-

2005 5 401- 2006 2 17 59

- 2007 32 223

- 2008 23 251

- 2009 72 41

- 2010 2 3 5 67

Combinatorics and Probability1. Binomial Coefficient Identities

nk n!

k!nk! factorial expansion x y n

k0

nnk

xkynk binomial theorem

n

k

nnk

symmetry nk

m0

kn1m

km hockey stick

nk n

kn1k1

absorption nk

m0

nkn1m

k1 hockey stick

nk n1

k1 n1

k recursion n

k

m0

knskm

sm

Vandermonde

convolution

nm

mk n

knkmk

trinomial revision

a. Generalized Binomial Coefficients: k

12k1

k! is a well defined polynomial in and

therefore well defined for real (or even complex) values of.


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b. Generalized Binomial Theorem:1 x k0

k

xk

2. Multinomial Coefficients: nk1,k2,,km

is the number of ways of puttingn distinct objects intom categories so

that thei th category containski objects

a. nk1,k2,,km

n!k1!k2!km!

wherek1 k2 km n

b. n

k1,k2,,km

n

k1

nk1

k2

nk1k2

k3

nk1k2km1

km

c. x1 x 2 x m n

k1kmn

nk1 ,,km

x1k1x2

k2 xmkm (the multinomial theorem)

3. Catalan Numbers:Cn is the number of triangulations of a convexn 2 -gon having no internal vertices.Cnisalso the number of ways to parenthesizex 1x2xn completely into binary products.

a. Cn C0Cn1 C1Cn2 Cn2C1 Cn1C0 forn 1and C0 1

b. Cn 1n1

2nn forn 0

c. The first few values are1, 1,2, 5,14,42,132,429,1430,4862,

4. Stirling Numbers of the first kind: nk

is the number of permutations of a set withn elements having exactlyk

distinct cycles.

a. n0 0, n

1 n 1 !, n

n1 n

2 , nn 1

b. nk n 1 n1

k n1

k1 forn 1and1 k n

c. The first few values of nk :

k

1 2 3 4 5 6 7

1 1

2 1 1

3 2 3 1

4 6 11 6 1

n 5 24 50 35 10 1

6 120 274 225 85 15 1

7 720 1764 1624 735 175 21 1

5. Stirling Numbers of the second kind: nk

is the number of partitions of a set withn elements intoknon-empty

subsets.

a. n1

1, n2

2n1 1, nn1

n2

, nn 1

b. nk n1

k1 k n1

k forn 1and1 k n

c. The first few values of nk

:

k

1 2 3 4 5 6 7 8

1 1

2 1 1

3 1 3 1

4 1 7 6 1

n 5 1 15 25 10 1

6 1 31 90 65 15 1

7 1 63 301 350 140 21 1

8 1 127 966 1701 1050 266 28 1

d. nk 1

k!

i0

k

1i nk

k in

6. Partition Formula: LetPn, k the number of partitions ofn having largest summandk. Then

2003-2006 - Ken Monks & Maria Monks


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Pn, 1 Pn, n 1and

Pn, k Pn k, k Pn 1, k 1

This recursion produces the Pascal-like triangle:

k

1 2 3 4 5 6 7 8 9

1 1

2 1 1

3 1 1 1

4 1 2 1 1

n 5 1 2 2 1 1

6 1 3 3 2 1 1

7 1 3 4 3 2 1 1

8 1 4 5 5 3 2 1 1

9 1 4 7 6 5 3 2 1 1

7. Pigeonhole Principle: If you haven pigeons inkholes some hole contains at least nk

pigeons and some hole

contains at most nk

pigeons.

8. Inclusion-Exclusion Principle: Given finite sets A 1,A2, ,Anand letS1 i

|Ai |,

S2 ij

|A i Aj |, , Sn |A1 A 2 A n |. Then

|A1 A 2 A n | S1 S2 S3 S4 1n1Sn

9. Expected Value: Given a sample spaceSand a functionf : S R the expected value offon this sample spaceis

xS

PxfxwherePxis the probability ofx.

10. Van der Waerdens Theorem: Letn and kbe positive integers. Then there exists a positive integerNsuch thatif the numbers1,2,. . . ,Nare colored inkcolors, one color always contains an arithmetic progression of length n.

Graph Theory1. Euler paths: traverse every edge in a graph exactly once.

a. A connected graph has an Euler path if and only if the number of odd degree vertices is zero or two. If it iszero then the path is a cycle, and if it is two the path must begin and end at the odd degree vertices.

2. Asimplegraph has no edges from a node to itself.

3. Hamiltonian paths: traverse every vertex in a graph exactly once.

a. Diracs Theorem: If every vertex in a simple graph withv vertices has degree at leastv/2then the graph has aHamiltonian cycle.

b. Ores Theorem: A simple graph with n nodes has an Hamiltonian cycle if whenever two nodes are notconnected by an edge the sum of their degrees is at leastn.

4. Ramseys Theorem: LetNa, bbe the smallest number such that any group ofNa, bpeople must contain eitheramutual friends orb mutual strangers.

a. Na, b Nb, a

b. Na, 2 ac. Na, b Na 1, b Na, b 1

5. Turns Theorem: LetG be a graph with n nodes which contains no complete subgraph ofknodes. Lettn, kbethe maximum number of edges of such a graph. Then

tn, k k 2k 1

n2

2

a. The graph havingtn, kedges,n nodes, and no complete k-subgraph is the completek 1 -partite graphwhich has the most evenly divided arrangement of nodes, i.e. in which the numbers of nodes each pair of thek 1 groups differ by at most1.

Sequences and Series



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1. Some Sums

a. 12 2 2 3 2 . . .n 12 n 2 nn12n1

6

b. 13 2 3 3 3 . . .n 13 n 3 nn1

2

2

c. 1 1! 2 2! 3 3! 4 4! . . .k k! k 1! 1

2. Infinite geometric series:

1 r r2 r3 11 r for 1 r 1

Many other useful series can be derived from this by substitution, differentiation, etc.

3. Generating Functions: Leta 0, a1, a2, . . . be a sequence of numbers. The generating function for this sequence is

fx

n0

anxn. It can be used to solve recurrences explicitly. Variations such as

n0

ann!

xn are sometimes useful.

a. Properties:

i. Addition and multiplication of generating functions are commutative and associative.

ii. The constant functions0and1are additive and multiplicative identities, respectively.

iii. Every generating function has an additive inverse.

iv. A generating functionAx

n0

anxn has a multiplicative inverseBx

n0

bnxn if and only ifa 0 0.

In this case,Bxis given by the recursionb 0 1a0 andb k 1a0

k

i1 a ibki.

b. Manipulating ordinary generating functions: LetAx

n0

anxn andBx

n0

bnxn

i. Ax Bxif and only ifa n bnfor alln.

ii. Ax

1x

n0

n

k0

ak xn

iii. AxBx

n0

n

k0

akbnk xn

iv. xAx

n0

nanxn

v. Ax C

n1

an1n xn

c. Manipulating exponential generating functions: LetAx

n0

a nn!

x n andBx

n0

b nn!

x n

i. AxBx

n0

k0

nk

akbnkn!

xn

ii. Ax

n0

an1n!

xn

Number Theory

Distribution of Primes

1.

The Prime Number Theorem: The approximate number of primes less than or equal to a positive integerxconverges to xlnx

asx .

2. Bertrands Postulate: For anyn 1, there is always at least one prime between n and2n.

3. Dirichlets Theorem: For any relatively prime natural numbers a,b, the arithmetic sequencea, a b, a 2b, . . .contains infinitely many primes.

Continued Fractions1. Continued Fraction Representation: Every rational numberq can be represented uniquely by the sequences

a0, a1, , an and a0, a1, , an 1, 1where

q a0 1

a1 1a2



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anda 0 qand thea i are positive integers determined by converting the reciprocal of the remainder to a mixednumber iteratively. Irrational numbers have a unique infinite representation of this form.

2. Self-similar expressions: LetX rs r

s rs

. ThenX rsX

which can be solved forX.

Diophantine Equations1. Linear Equations: For integersa 1, . . . , an, c, the equationa 1x1 . . .anxn chas integer solutions if and only if

gcda1, . . . , an c. (See also:GCDst Theoremin Number Theory - Modular Arithmetic - Other Applications -#1b below)

2. Sums of Squares

a. For any integern, the equationx 2 y 2 nhas integer solutions iff any prime factor ofnthat is congruent to3mod 4 occurs to an even power in the prime factorization ofn.

b. An odd primep can be written as the sum of two squares iffp 1 mod 4

c. A positive integern can be written as the sum of three squares iffn cannot be written in the form8k 7 4m

for any nonnegative integersk, m.

d. All positive integers can be written as the sum of four squares.

3. Pells Equation: For any non-square positive integerD, the equationx 2 Dy 2 1has infinitely many integersolutions, each of which is of the formxn,yn given by

xn y n D x1 y 1 Dn

wherex1,y1 is the solution withx 1,y 1 0and y 1 minimal, andn is a nonnegative integer.

Modular Arithmetic

Eulers Function(A.K.A.Eulers totient function)

Let n the number of positive integers less than the positive integern 1which are relatively prime ton.

1. pk pk1p 1

2. ab a bfor relatively primea, b.

3. Eulers Theorem: Leta, nbe relatively prime integers withn 1. Thenan 1mod n.

4. Fermats Little Theorem: For any nonzero integera and positive primep,ap1 1modpand ap amodp.

5. Order: Ifgcda, n 1then the order ofa modulon is the smallest positive integerksuch thata kn 1. It has the

following properties:

a. kdividesn

b. ai

n aj

if and only ifik j

6. Primitive roots: If the order ofa modulon is athena is called a primitive root ofn. It has the followingproperties:

a. ifb is relatively prime to n thenb ak for somek

b. nhas a primitive root if and only ifn 2,4,pk, or2pk for some odd primep and k 0.

Other applications

1. GCD: Letd gcda, b.

a. Euclidean Algorithm: For alla, b,gcda, b gcdb, a mod b. Iterating this formula computesgcda, bby reducing it togcdd, 0.

b. GCDst Theorem: For alla, b Z there exists integerss, tsuch thatgcda, b sa tb. One such pairs, tcan be found by the Euclidean algorithm as follows:

sia t

ib s

i t

i

a0 a 1 0

a1 b 0 1

a2 s2 t2

a3 s3 t3

gcda, b s t

wherea i ai2 mod ai1,s i si2 ai2/a i1 si1 , andti ti2 ai2/ai1 ti1. The pairS, Tis a solutionofSa Tb dif and only ifS, T s b

dk, t a

dk for somek Z. (A common situation is when



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d 1).

2. Base Conversion: To convert a whole numbern to a baseb let fx xx mod b

b . Thenn, fn, ffn. . . taken

mod b, are the digits ofn in baseb in reverse order.

3. Chinese Remainder Theorem: Ifb 0, b1, , bnare pairwise relatively prime and B b0b1bnthen the systemof congruences:

x

b0

a0, x

b1

a1 , , x

bn

an

has unique solutionmodB :

i0

n

a iuiBbi

whereu ibi

Bbi

1.

4. Wilsons Theorem: For any integern greater than one,n is prime if and only ifn 1 !n 1.

5. Wolstenholmes Theorem: For any primep 5and any nonnegative integersa and b,p 3 ap

bp a

b .

6. Roots of Unity modp n: Letp be an odd prime,n . Thenxp 1 mod pn iffx 1 mod pn1 .

7. Quadratic Reciprocity: In the following letp be an odd prime andgcda,p gcdb,p 1.

a. Quadratic residue:a is a quadratic residuemod pif and only if there is an integerx such thatx 2p a

b. Legendre Symbol: ofa on p is ap 1 ifa is a quadratic residuemod p

1 otherwise.

c. Eulers Criterion: ap p ap1/2

d. Properties of the Legendre symbol:

i. ifap bthen ap

bp

ii. a2

p 1

iii. a bp ap

bp

iv. 1p 1and 1

p 1p1/2

v. 2p 1 ifp

8

1orp8

1

1 otherwise

e. Law of Quadratic Reciprocity: Ifp, qare distinct odd primes thenpq 1

p1

2

q1

2qp

i. Corollary: pq

qp ifp

4

1orq4

1

qp ifp

4

q4

3

Algebra

Basics1. Absolute value:

a. Geometric interpretation:|x|is the distancex is from the origin.

b. Algebraic interpretation:|x| x ifx 0

x ifx 0.

c. x2 |x|

2. Adding Proportions: Leta, b,x,y, rbe real numbers. Thenxy

ab r x a

y b x a

y b r

3. Useful Factorizations:

a. For any positive integern,xn y n x y xn1 x n2y x n2y2 xy n2 y n1



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b. For odd positive integersn,xn y n x y xn1 x n2y x n2y2 1n1yn1

c. a2 b 2 c2 d2 ac bd2 ad bc 2 ac bd2 ad bc2

- Thus the product of two sums of squares is a sum of squares.

d. x4 4y4 x2 2xy 2y2 x2 2xy 2y2

- This partially alleviates the problem of not being able to factor sums of squares.

4. Simplifying Nested Radicals: It is sometimes possible to simplify nested radicals with the denesting equation

X Y X X2 Y2

2

X X2 Y2

2

Number systems1. Fields: Q,R,C and Zpfor a primep are all examples of fields. These sets with their usual operations of,satisfy

the properties:,are commutative and associative, is distributive over, there is an additive and multiplicativeidentity, every element has an additive inverse and every nonzero element has a multiplicative inverse.

Linear Algebra1. Vectors: a real vector is finite sequence of real numbers denoted a1, a2, , an . The set of all such vectors is

denoted Rn.

a. Vector addition:a1, a2, , an b1, b2, , bn a1 b 1, a2 b 2, , an b n

b.

Scalar multiplication: Ifris a real number thenra1, a2, , an ra1, ra2, , ran c. Dot Product:a1, a2, , an b1, b2, , bn a1b1 a 2b2 a nbn

d. Cross Product:a1, a2, a3 b1, b2, b3 e,f,gwheree deta2 a3

b2 b3,f det

a1 a3

b1 b3,

andg deta1 a2

b1 b2.

e. (See also:Analytic Geometrybelow)

2. Matrix: An arrayMconsisting ofm rows andn columns of complex numbers is called an m n matrix. Theentry in thei th row andj th column is denotedMi,j

3. Matrix multiplication: IfMis anm n matrix andNis ann p matrix theMNis them p matrix such thatMNi,jis the dot product of the i th row ofMwith thej th column ofN.

4. Determinant:det a b

c d ad bc and

det

a b c

d e f

g h i

a dete f

h ib det

d f

g ic det

d e

g h

aei bfg cdh afh bdi ceg

5. Identity matrix: is annxn matrixIn havingIni,j 1 ifi j

0 otherwise. For anyn n matrixMwe have

MIn InM M.

6. Inverse matrix: Twon n matricesM,Nare inverses if and only ifMN NM In

. A matrixMhas an inverseif and only ifdetM 0.

a. Inverse of a 2 2 Matrix: LetA a b

c d. Then

A1 1detA

d b

c a

7. Systems of linear equations: the system



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a1x b 1y c 1z d1

a2x b 2y c 2z d2

a3x b 3y c 3z d3

has a unique solutionx,y,zif and only ifdet

a1 b1 c1

a2 b2 c2

a3 b3 c3

0.

8. Cramers Rule: If the system of linear equations

a1x b 1y c 1z d1

a2x b 2y c 2z d2

a3x b 3y c 3z d3

has solutionx,y,zthen

x

det

d1 b1 c1

d2 b2 c2

d3 b3 c3

D ,y

det

a1 d1 c1

a2 d2 c2

a3 d3 c3

D ,z

det

a1 b1 d1

a2 b2 d2

a3 b3 d3

D ,

whereD det

a1 b1 c1

a2 b2 c2

a3 b3 c3

. This generalizes to any number of equations and unknowns.

Polynomials1. Polynomial: LetFbe either a field orZ, anda 0, a1, , an F. Thenpx anxn a n1xn1 a 0with

an 0is called a polynomial of degree n with coefficients in F. The set of all such polynomials is denotedFx.

Quotients,Remainders,and Factorization

1. Division algorithm: LetFbe a field andfx,gx Fxwithgx 0. Then there exist unique polynomialsqx, rxsuch that

fx qxgx rxand rx 0or degrx deggx .

As with integersqxis called the quotient andrxthe remainder whenfxis divided bygx.2. Euclidean algorithm: the Euclidean algorithm and GCDst theorem can be applied to two polynomials with real

coefficients (see Number Theory - Modular Arithmetic - Other Applications - #1 above)

3. Remainder Theorem: LetFbe a field,px Fx, anda F. Then there existsqx Fxsuch that

px x a qx pa

i.e. the remainder whenpxis divided byx a is pa.

4. Factor Theorem: LetFbe a field,px Fx,px 0, anda F. Thenx a is a factor ofpxif and onlyifpa 0.

5. Fundamental Theorem of Algebra: Letpx anxn a n1xn1 . . .a0 Cxwitha n 0. Thenpxfactorsuniquely (up to reordering the factors) as:

px anx r1 x r2 x rn

for somer1, r2, , rn C.

6. Irreducible Polynomials: Every polynomial with real coefficients factors as a product of irreducible linear andquadratic polyomials.

7. Gausss Theorem: Ifpx Zxand pxcan be factored over the rationals, then it can be factored over theintegers.

Synthetic Division and Substitution

1. Synthetic Division/Substitution: To compute the quotient and remainder whenpx anxn a n1xn1 . . .a0is divided byx rwe can use synthetic substitution:

r an an1 an2 a0

an anr a n1 anr a n1 r a n2 pr



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2. Upper and Lower Bounds Theorem: If the numbers in the second row of the synthetic division all have the samesign or are zero thenris an upper bound for the roots ofp. If the numbers in the second row of the syntheticdivision have alternating signs thenris a lower bound for the roots ofp.

Symmetric Polyonomials

1. Coefficients vs.Roots: Letpx xn a n1xn1 . . .a0 Cxand r1, r2, , rnits (not necessarily distinct)roots. Then for all0 i n

ai 1k1kin

1nirk1 rk2 rki

In particular,a 0 1nr1r2rnand a n1 r1 r2 rn.

2. Reduction Algorithm for Symmetric Polynomials: Definespx1, ,xn 1k1kpnxk1xk2 xkp to be the

elementary symmetric polynomial inn variables of degreep (and definespx1, ,xn 0ifp n). These form abasis for the algebra of all symmetric polynomials. Define the height of a monomialx 1

e1x2e2 xn

en to bee1 2e2 ne n and the height of a polynomial to be the maximum height of any of its monomial terms andzero for the zero polynomial. Iffis a symmetric polynomial whose maximal height term is cx1

e1x2e2 xn

en , then thepolynomialg f cs 1

enen1s2en1en2 sn1

e2e1sne1 has strictly lower height thanfso that iterating gives an expression

forfas a polynomial in the elementary symmetric polynomials.

3. Newton-Girard Identities: Define

Nps1, ,sp det

s1 1 0 0 0

2s2 s1 1 0 0

3s3 s2 s1 1 0

4s4 s3 s2 s1 0

1

psp sp1 sp2 sp3 s1

wheres i are the elemetary symmetric polynomials inx 1,x2, ,xnthen expandingNgives

Np x1px 2

p x n

p

The first few values ofNpare

N1 s1

N2 s12 2s2

N3 s13 3s1s2 3s3

N4 s14 4s1

2s2 2s22 4s1s3 4s4

These satisfy the recurrence

Nn s1Nn1 s 2Nn2 s 3Nn3 1n1snN0

Roots

1. Descartess Rule of Signs: Ifpx Rxthen the number of positive roots ofpxis equal toN 2kfor somek Z, whereNis the number of sign changes in the sequencea 0, a1, , an. The number of negative roots ofpxequals the number of positive roots ofpx.

2. Rational Root Theorem: Ifpx anxn a n1xn1 a 0 Zxand rs (in reduced form) is a rational root

ofpx, thenrdividesa 0 and s dividesa n.

3. Complex Conjugate Roots: Ifpx Rxand a bi is a complex root ofp then so isa bi.

4.

Irrational Conjugate Roots: Ifpx Qxand a b c is a root ofp wherea, b, c Q and c R Q,thena b c is also a root ofp.

5. Eisensteins Irreducibility Criterion: Ifpx Zxand if there exists a primeq that divides each of thecoefficients excepta n and q

2 does not dividea 0, thenpxis irreducible over the rationals.

6. Lagrange Interpolation: Letxi,yii0n be a set of points. Then then th-degree polynomial

px i0

n

yi ji

x xj

xi xj

is the unique polynomial of degree at most n passing through each of the points.

Partial Fractions

1. Equality of Polynomial functions: Ifpx, qx Rxthen the functionsp and q are equal if and only if the



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10.............................................................................................................................................................................

polynomialspxand qxhave the same degree and their corresponding coefficients are equal.

2. Partial Fraction Decomposition: Ifpx Rxhas degree less thank 2mand l1xlkx Rxareirreducible linear polynomials and q 1xqmx Rxare irreducible quadratic polynomials then there exist realnumbersA 1, ,Ak,B1, ,Bk, C1, , Cksuch that

px

l1xlkxq1xqmx

A1l1x

Aklkx

B1x C1

q1x

Bmx Cmqmx

Synthetic Geometry

Terminology and Common Notation1. Definitions: The following notation is somewhat standard and will be used in this document.

a. Cevian: Any segment from a vertex of a triangle to a point on the opposite side

b. Median: A cevian which bisects the opposite side

c. Altitude: The perpendicular from a vertex to the opposite side of a triangle

d. Centroid(G): The point of intersection of the medians of a triangle (they are concurrent)

e. Orthocenter(H): The point of intersection of the altitudes of a triangle (they are concurrent)

f. Circumcircle,Circumcenter(O),Circumradius(R): Every triangle can be circumscribed by a uniquecircle whose center is the intersection of the perpendicular bisectors of the three sides.

g. Incircle,Incenter(I),Inradius(r): Every triangle circumscribes a unique circle whose center is the

intersection of the angle bisectors.

h. Excircles,Excenters(IA,IB,IC),Exradii(rA, rB, rC): Any of the three centers of the excircles (tangent toone side and the extensions of the other two) of a triangle; also the intersection of the external bisectors.

i. Semiperimeter(s): half of the perimeter

j. Medial Triangle: The triangle whose vertices are the midpoints of the sides of a given triangle. Itsubdivides the triangle into four congruent sub-triangles.

k. Orthic Triangle: The triangle whose vertices are the feet of the altitudes of a given triangle. It is thetriangle with minimum perimeter of all triangles whose vertices are on the three sides.

l. Euler Line: the line containing the orthocenter, centroid, and circumcenter of a triangle (HGO). In anytriangle|HG | 2|GO|and9OH2 a2 b 2 c 2 wherea,b,c are the sides of the triangle.

m. Gergonne Point: the point of intersection of the cevians through the points of tangency of the incircle to thesides of a triangle

n. Nagle Point: the point of intersection of the cevians through the points of tangency of the excircles to thesides of a triangle

o. Fermat Point: the pointFin an acute triangleABCfor which|FA | |FB| |FC|is minimal. In any acutetriangleAFB BFC CFA 120.

TrianglesInABCwe definea |BC|, b |AC|, c |AB|, and abbreviate the three angles as A, B, andC.

1. Pythagorean triples: A right triangle has relatively prime integer length sides if and only if it has legs2uvandu2 v 2 and hypotenuseu2 v 2 for some relatively prime, opposite parity, positive integers u, vwithu v.

2. Area: Let|ABC|denote the area ofABC. Then

|ABC| rs

rAs a rBs b rCs c

abc

4R 1

2ab sin C 1

2bc sinA 1

2ac sinB

ss a s b s c

a. Area from coordinates: SupposeA,B, Care vectors in R2. Defineu u1, u2 A Candv v1, v2 B C. Then



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11.............................................................................................................................................................................

|ABC| 12

detu1 v1

u2 v2

12

|u v |

i. Shoelace Theorem: Ifx1,y1 , , xn,yn are the vertices of an n-gon, the area of then-gon is

Area 12

x1 y1

x2 y2

xn1 yn1

xn yn

12

xny1 k1

n1

xkyk1 x1yn k1

n1

xk1yk

b. (See also:Picks Theoremin Geometry - Polygons - #1 below)

3. Eulers Theorem: Letdbe the distance between the incenterIand circumcenterO of triangleABC

d2 RR 2r

a. Corollary:d2 R 2 2rRis the power of the point Iwith respect to the circumcircle (see also Geometry -Circles - #2 below)

b. Eulers Inequality: In any triangleR 2r.

Similarity

1. Basic proportionality: A segment connecting points on two sides of a triangle is parallel to the third side if andonly if the segments it cuts off are proportional to the sides. (see also Algebra - Basics - #2 above)

Cevians

1. Angle Bisector Theorem: IfD is the point where the angle bisector ofAin ABCmeetsBCthen

|BD|

|BA|

|CD|

|CA |

D

C

A B

2. Cevas Theorem: Three ceviansAX,BY, CZofABCare concurrent if and only if

|BX|

|XC|

|CY|

|YA |

|AZ|

|ZB | 1

Z

X

Y

AB

C

3. Menelaus Theorem: LetD,E,Fbe three points on, respectively, the linesBC, AC, andAB containing the sides ofABC. ThenD,E,Fare collinear if and only if

AF

BF

BD

CD

CE

AE 1

whereXYis the signed length of the directed segment XY.



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C

F

DB

A

E

4. Stewarts Theorem: LetABCbe a triangle with cevian AXof lengthp and letm XBand n XC. Thenap2 mn b2m c 2n.

bc

nm

p

B

A

CX

5. Inradius in terms of altitudes: Leth a,h b,h cbe the lengths of the altitudes andrbe the inradius ofABC. Then

1r

1ha

1hb

1hc

6. Nine Point Circle: The circle whose center is the midpoint of the Euler Line (N) ofABCwith radius R2

passes

through the feet of the altitudes, the midpoints of the sides, and the midpoints ofHA,HB, andHC.

O

H

A

B C

7. Feuerbachs Theorem: The nine point circle of a triangle is tangent to the incircle and to the three excircles.

8. Brocard Points: There is exactly one pointPin ABCsuch that |PAB| |PBC| |PCA |which is thepoint where the circle throughA tangent toBCatB intersects the circle throughCtangent toAB at A. This pointand its isogonal conjugateP (the point making |PBA | |PCB | |PAC|) are called the Brocard

points of the triangle. Since is always equal to

, this angle is called the Brocard angle of the triangle. It isgiven by the formula:

cot cotA cotB cot C

Trigonometry

1. Triangle solvers: Assume a triangle with anglesA,B, andC, that respectively intercept sidesa,b, andc, havingcircumradiusR.

a. Extended Law of Sines:

asinA

bsinB

csinC

2R

b. Law of Cosines:

a2 b2 c 2 2bc cosA

b2 a2 c 2 2ac cosB

c2 a2 b 2 2ab cosC

c. Law of Tangents:

a btan AB

2

a btan AB

2

2. Identities:

a. Pythagorean Identity:sin 2x cos 2x 1

b. Angle Addition:

i. sinx y sinx cosy cosx siny

ii. cosx y cosx cosy sinx siny



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iii. tanx y tanxtany

1tanx tany

c. Double Angle:

i. sin2x 2sinx cosx

ii. cos2x cos2x sin 2x

iii. tan2x 2tanx

1tan 2x

d. Triple Angle:i. sin3x 4sin3x 3 sinx

ii. cos3x 4cos3x 3cosx

e. Half Angle:

i. sin2 x2

1cosx

2

ii. cos2 x2

1cosx

2

iii. tan2 x2

1cosx

1cosx

iv. tan x2

sinx

1cosx

1cosx

sinx

f. Sum to Product:

i. sinx siny 2sin xy

2 cos

xy

2

ii. cosx cosy 2cos xy

2

cos xy

2

iii. cosx cosy 2sin xy

2 sin

x y

2

iv. tanx tany sinxy

cosx cosy

g. Product to Sum:

i. sinx cosy sinxysinxy

2

ii. cosx cosy cosxycosxy

2

iii. sinx siny cosxycosxy

2

iv. tanx tany cosxycosxy

cosxycosxy

h. InABC:

i. tan A2

|ABC|

ssa

sbsc

ssa

ii. tanA tanB tanC tanA tanB tanCiii. c a cosB b cosA

i. Miscellaneous:

i. Difference of two squares for sine:sin 2x sin 2y sinx y sinx y

ii. cosx sinx 2 cos 4

x

iii. sin15 14

6 2 andcos15 14

6 2

Quadrilaterals1. Ptolemys Theorem

A

B

C

D

In any cyclic quadrilateralABCD, AB CD BC AD AC BD.

i.e. The sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals.

2. Ptolemys inequality: In any quadrilateralABCD,AB CD BC AD AC BD.

i.e. The sum of the products of the opposite sides of any quadrilateral is greater than or equal to the product of thediagonals.

3. Inscribed circle: A quadrilateralABCDhas an inscribed circle if and only ifAB CD AC BD.



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4. Midline of diagonals: In a quadrilateral with side lengthsa,b,c, anddand diagonalse and f, letXand Ybe themidpoints of the diagonals. Then

4|XY|2 a2 b 2 c 2 d2 e 2 f2

a. Corollary: In a parallelogram, the sum of the squares of the sides is equal to the sum of the squares of thediagonals.

5. Ways to prove a quadrilateral is cyclic:

a. The converse of Ptolemys Theorem is true.

b. If a pair of opposite angles of a quadrilateral are supplementary, the quadrilateral is cyclic.

c. If one side of the quadrilateral subtends equal angles with the other two vertices, the quadrilateral is cyclic.(see Circles, 1b)

Polygons1. Picks Theorem: The area of any closed polygon whose vertices have integer coordinates is i b/2 1wherei

is the number of points with integer coordinates in the interior of the figure andb is the number of points on theboundary of the figure.

Circles1. Angles on a circle

a. Star Trek Lemma: An inscribed angle has one half as many degrees as the intercepted arc (Cor: Any angle

that intercepts a diameter is a right angle).

2

2

2

b. Different inscribed angles intercepting the same arcs are equal.

c. An angle formed by two chords intersecting within a circle has one-half as many degrees as the sum of theintercepted arcs.

+

2

d. Any angle formed by two secants, a secant and a tangent, or two tangents is equal to half the difference ofthe intercepted arcs.



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-

2

-

2

180-

e. An angle formed by a chord and a tangent to a circle has one-half as many degrees as the intercepted arc. 1

2 arcAT ATB

/2

2. Power of a Point:

a. In both of the following:PA PB PX PY

B

P

Y

A

X

P

X

Y

AB

b. AssumingPA is a secant and PTis a tangent,PA PB PT PT

A

B

T

P

c. Using Eulers Theorem, we find that the power of the incenterIof a triangle with respect to the circumcircleis2rR

3. The Radical Axis

a. The Radical Axis of two circles is the locus of points with equal power to both circles.

b. The Radical Axis is a straight line, perpendicular to the line connecting the centers of the circles. If the circlesintersect, it passes through the two points of intersection. If the circles are tangent to each other, the axis istheir common tangent.

c. Given three circles, either the three radical axes between each pair are parallel, or they are concurrent.

4. Other circle facts:



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a. The Butterfly: Given a circle and a chordAB of the circle whose midpoint is M, letXYand ZWbe twochords passing throughM, and letXWand YZintersectAB atPand Q, respectively. ThenMis also themidpoint ofPQ.

b. The formula for the graph of a circle centered ath, kwith radiusris:

x h2 y k2 r2

c. Brahmaguptas Formula: The area of a quadrilateral inscribed in a circle with side lengthsa, b, c, dis

s as bs cs d

wheres abcd2

.

d. The coordinates of the center of a circle that is inscribed in a triangle whose legs are on the positivex and yaxes area, awherea

xyz

2 given thatx and y are the lengths of the legs and zis the length of the

hypothenuse.

Transformational Geometry(See also:Complex Transformationsin Complex Numbers)

Inversive1. Definitions and Notation:

a. Inversive plane: P2 2 PwhereP R2 is called thepoint at infinity.

b. Figure: in the inversive plane is a set of points F P2.c. Cline: A figure that is either a circle in 2 or a figurel Pwherelis a line in 2.

d. Inverse: Given any pointA and any circle with centerO and radiusk, theinverseofA with respect to isthe pointB on rayOA satisfyingOA OB k2. The inverse ofA with respect to a linelis the reflection ofAaboutl.

e. The inverse of a pointA is denotedA . Similarly, the inverse image of a figureFis denotedF.

2. Properties of Inversion:

a. O P and P O, whereO is the center of the circle of inversion.

b. For any point or figureA,A A.

c. Clines invert to clines

i. , where is the circle of inversion.

ii. For any circle passing throughO, is the radical axis of and .

iii. For any circle not passing through O, is a circle.iv. For any line lpassing throughO,l l.

v. For any line lnot passing through O,l is a circle passing throughO.

d. Conformal(angle-preserving): If two clines and intersect at an angle, then and intersect at anangle.

3. Inversive Distance Formula: LetA,B be two points in the inversive plane. Then

AB k2 AB

OA OB

Projective1. Definitions:

a. Pencil

i. Apencil of parallel linesis the set of all lines in R2 parallel to a given line, together with the line itself.

ii. Apencil of concurrent linesis the set of all lines passing through a given point.

b. Projective plane: 2 l wherel consists of an infinite set of points, one for each pencil of parallel lines in2.

c. Perspective from a point: TrianglesA 1A2A3 and B 1B2B3are perspective from point Ciff the lines A 1B1,A2B2,A 3B3are concurrent atC.

d. Perspective from a line: TrianglesA 1A2A3 and B 1B2B3are perspective from lineliff the pointsA1A2 B 1B2,A 2A3 B 2B3, andA 3A1 B 3B1lie onl.

2. Duality of projective theorems: IfPis a theorem about the projective plane, then the dualofPis the statementobtained by interchanging "point" with "line", "collinear" with "concurrent", etc. The dual ofPis always atheorem as well.



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3. Desarguess Theorem: If two triangles are perspective from a point, then they are perspective from a line.

4. Pappuss Theorem: IfA 1,A 2, andA 3are collinear and B 1,B 2, andB 3 are collinear, thenA 1B1 A 3B2,A2B1 A 3B3, andA 2B2 A 1B3 are collinear.

A1 A3

B1

B3

A2

B2

5. Pascals Theorem(Dual of Brianchon): If a hexagon is inscribed in a conic, the points of intersection of pairs ofopposite sides are collinear.

6. Brianchons Theorem(Dual of Pascal): If a hexagon is circumscribed about a conic, its three diagonals (joiningpairs of opposite vertices) are concurrent.

Analytic Geometry

Basics1. Distance Formula: IfP1 x1,x2, ,xn and P2 y1,y2, ,yn then



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|P1P2 | i1

n

yi x i 2

Lines1. Midpoint Formula: The midpoint of a segment whose endpoints arex1,x2, ,xnand y1,y2, ,ynis given

by

x1y1

2 ,

x2y2

2 , ,

xnyn

2 .2. Forms of Lines:

a. Point-Slope:y y 1 mx x 1

b. Slope-Intercept:y mx b

c. General:Ax By C 0, whereA 2 B 2 0

3. Slope Formula: m y2y1

x2x1

4. Perpendicular lines: Two nonvertical lines are perpendicular if and only if the product of their slopes is1.

5. Distance from a Point to a Line: the distance from pointp, qto the line ax by cwherea 2 b 2 1(which

can always be obtained by dividing both sides of the equation by a2 b 2 if necessary) is

|ap bq c |

Conic Sections

Standard Equations1. Circle: a circle is the set of all points a fixed distance rfrom a point a, bcalled the center.

x a 2 y b 2 r2

2. Parabolas: a parabola is the set of points equidistant from a given point (the focus) and a given line (thedirectrix). For focus0,pand directrixy p

x2 4py

3. Ellipses: an ellipse is the set of all points such that the sum of the distances to two fixed foci is constant. For focic, 0and c, 0and semimajor axisa and semiminor axis b witha b

x2

a2

y2

b2 1

Note that b 2 c 2 a2.

a. The area of an ellipse isab, wherea and b are the semimajor and semiminor axes.

4. Hyperbolas: a hyperbola is the set of all points such that the difference of the distances to two fixed foci isconstant. For focic, 0and c, 0and x-interceptsa and a

x2

a2

y2

b2 1

wherea 2 b 2 c2.

General Equations

1. General Equations for conic sections: can be obtained from the standard equations by applying the appropriatetransformations

a. Translation byh, k:x x h and y y k

b. Rotation about the origin by angle:x x cos y sinand y x sin y cos

c. Reflection across the x-Axes:x xand y y

d. (See also:Transformationin Complex Numbers - Complex Transformations - #1a,b,c)

2. Removal of xy-term: Thexy-term can be removed fromAx 2 Bxy Cy 2 Dx Ey F 0, B 0by a rotationof the axes if is selected so thatcot 2 AC

B .

Other Representations1. Polar-Rectangular Relations:

x rcos, y rsin

r2 x2 y 2, tan yx

2. Parametric Equations: The set of equationsx ft, y gtare parametric equations of the relation

x,y : x ftand y gtand tis in the intersection of the domains offand g .



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Vector GeometryIn the following ifv R2 then v v1, v2 .

1. Equal Vectors: two vectors are equal if and only if their corresponding coordinates are equal.

2. Vector Addition: a b a1 b 1, a2 b 2

3. Length: v v12 v 2

2 (aunit vectorhas length one)

4. Direction of Vector: the unit vector in the direction ofv is

u v

v

v1

v,

v2

v cos,sin

where arctan v2v1

is the angle between uand the positivex-axis in standard position

5. Multiplication by Scalar:kv kv1, kv2 for anyk R

6. Dot Product: a b a1b1 a 2b2 a b cos where, are the direction angles fora ,b respectively.

a. two vectors are perpendicular if and only if their dot product is zero

b. a b is the length of the projection ofa onto the line containing b ifb is a unit vector

Analysis

Real Analysis

Inequalities

1. The Arithmetic-Geometric-Harmonic Mean Theorem

a. For all nonnegative real numbers x 1, . . . ,xn,n

1x1 1x2 . . .

1xn

n x1x2. . .xn x1 x 2. . .xn

n

x1x2...xnn is called the Arithmetic Mean, or average, ofx1, . . . ,xn.

nx 1x2. . .xn is called the Geometric Mean ofx 1, . . . ,xn.n

1x1 1x2

... 1xnis called the Harmonic Mean ofx 1, . . . ,xn.

b. (Weighted)Power Mean Theorem: For all nonnegative real numbersx 1, . . . ,xn, 1, . . . ,n with

1 2. . .n 1, and all real numbersp 0definemp i1

n

ixip

1/p

andm 0 limp0

mp. Then

i. For all real numbersr,swithr swe havem r ms (this is the generalization of the AM-GM-HMtheorem)

ii. An important special case is1 2 . . . n 1

n; in this casem 0is the geometric mean,m 1is thearithmetic mean, andm 1is the harmonic mean. In general,mp is called thepth power mean.

iii. m0 x11x2

2 . . .xnn is the weighted geometric mean.

2. Triangle Inequality: For all real or complex numbersx 1, . . . ,xn,

|x1 x 2 . . .xn | |x1 | |x2 |. . .|xn |

a. Minkowskis Inequality: is the generalization of the triangle inequality to higher dimensions. Given realnumbersa 1, a2, , an and b 1, b2, , bn

i1

n

ai b i 2

i1

n

ai2

i1

n

bi2

3. Geometric Mean Machine: IfACis a diameter of a circle through pointB, andD is the foot of theperpendicular throughB to ACthenBD is the geometric mean ofAD and DC.



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xy

yx DA

B

C

a. Symmetry-Product Principle: As the distance between two positive numbers decreases their productincreases if their sum remains constant.

4. Cauchy-Schwartz Inequality: For any two sequences of real numbersx 1, . . . ,xn and y 1, . . . ,yn,

x1y1 x 2y2 . . .xnyn 2 x1

2 . . .xn2y1

2 . . .yn2

5. Rearrangement: For any sequences of real numbers a 1 a2 anand b 1 b2 bn and anypermutation of1 , 2 , . . . ,n,

a1bn a 2bn1. . .anb1 a1b1 a 2b2. . .anbn a1b1 a 2b2. . .anbn

6. Chebyshevs Inequality: For any sequences of real numbersa 1 a2 anand b 1 b2 bn,

1

n k1

n

akbnk1

1

n k1

n

ak1

n k1

n

bk

1

n k1

n

akbk

7. Jensens Inequality: Iffis a continuous real valued function that is concave upwards on the closed intervala. . b(e.g.fx x2) then for all1,2, ,n in 0..1such that1 2 n 1and for allx1,x2, ,xn a. . b

f1x1 2x2 nxn 1fx1 2fx2 nfxn

If the function is concave downwards the inequality is reversed. An important special case is where each k 1

n.

8. Hlders Inequality

a. Leta11, . . . , a1n , a21, . . . , a2n , , ak1, . . . , akn be sequences of nonnegative real numbers and 1, ,knonnegative reals satisfying1 k 1. Then

a111 a21

2 ak1k a12

1 a222 ak2

k a1n1 a2n

2 aknk a11 a 1n

1 a21 a 2n 2 ak1 a kn

k

i.e. given a matrix of nonnegative real numbers

a11 a1n

a21 a2n

ak1 akn

the arithmetic mean of the (weighted) geometric means of the columns is less than or equal to the (weighted)geometric mean of the arithmetic means of the rows.

b. (Generalized Minkowskis and Hlders) In the matrix above, for any realsr s, thesth power mean of therth power means of the columns is less than or equal to the rth power means of of thesth power means of therows.

9. Bernoullis Inequality: For any nonzero real numberx 1and integern 1

1 nx 1 xn

10. Nesbitts Inequality: For all positive realsa, b, c

32

ab c

ba c ca b

11. Schurs Inequality: Given positive real numbers a, b, cand any real numberr

0 ara b a c b rb a b c c rc a c b

12. Muirheads Inequality: Let0 s1 snand 0 t1 tn be real numbers such that i1

n

si i1

n

ti

and i1

k

si i1

k

ti (k 1,, n 1). Then for any nonnegative numbersx 1,,xn,



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x1t1

xntn

x1s1

xnsn

where the sums run over all permutations of1,2,, n.

13. (See also:Eulers Inequalityin Geometry - Triangles - #3b;Ptolemys Inequalityin Geometry - Quadrilaterals- #2)

Logarithms

Ifb 0,b 1, andx 0then

1. logbx log by logbxyfory 0

2. logbxy y logbxfor ally

3. logyx log bx

log by fory 0and y 1

4. For any functionsfand g,flng glnf

Note:ln loge wheree n0 1

n!.

Cauchys Functional Equations

1. Iffis a continuous function from R to R then

a. Iffx y fx fyfor everyx,ythenfx mxfor somem.

Complex NumbersBasics

Let C R2. For eachx,y C we formally writex,y x yi.Letx yi, a bi C, then:

1. Complex conjugate:x yi x yi

2. Complex norm:|x yi | x2 y 2

3. Argument:Argx yi the angle in02ofx,yin polar form (not defined forx y 0)

4. Real part:Rex yi x

5. Imaginary part:Imx yi y

6. Addition:x yi a bi x a y bi

7. Multiplication:x yia bi xa yb ya xbi

8. Complex exponential: Let R. Thene i cos i sin

9. Standard polar form: ofx yi isre i wherer |x yi |and Argx yi

10. Distance: between complex numbersz, wis|z w |

Properties1. ei 1 0

2. Let , R

a. eiei ei

b. |ei | 1

c. ei ei

3. Let z,z1,z2 C. Then:

a. |z1z2 | |z1 ||z2 |

b. z1z2 z1 z2

c. z1 z2 z1 z2

d. z z |z|2

e. |z| |z|

f. Ifz rei in polar form, then z rei

Complex Transformations1. Transformation: of a set Sis a bijection fromSto S. (a.k.a. apermutationofS)

Letw C and, k R.

a. Translation by w:Tz z w

b. Rotation by radians counterclockwise about the origin:Tz eiz



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c. Reflection across the x-axis:Tz z

d. Homothety by positive factorkwith respect to the origin:Tz kz

e. Inversion* with respect to the unit circle:Tz 1z

(*Inversion is a transformation of the extended

complex plane C Cwith 10 and 1 0.)

Strategies &Tactics

GeneralPaul Zietz, in his bookThe Art and Craft of Problem Solvingsuggests the following strategies and tactics forapproaching any problem.

1. Get oriented.

2. Consider the penultimate step.

3. Get your hands dirty.

4. Impose or look for symmetry.

5. Use wishful thinking.

6. Consider a simpler case.

7. Use peripheral vision.

8. Consider the extreme cases.

9. Find an invariant.

10. Draw a picture.

Arithmetic1. Be careful!

Combinatorics and Probability1. Use combinatorial arguments to solve binomial coefficient identities.

2. Make a bijection and count something easier, or count the complement.

3. Try recursion - to solve the recursion explicitly:

a. Conjecture and induct

b. Use generating functions

c. Algebraic manipulation

4. Ways to think of binomial coefficients

a. Coefficients ofx yn

b. The number of ways to choosekthings fromnthings

c. The elements of Pascals triangle

5. Use inclusion-exclusion.

Number TheoryMelanie Wood suggested the following ways to approach a number theory problem in her 2005 MOP lecturesand notes.

1. Plug in simple values. (See also Strategies & Tactics - General #3 and #6)

2. Check values modulom, wherem is carefully chosen.

3. Divide the problem into multiple cases.

4. Consider the orders of values modulo some integer.

5.

Dont be afraid to use the quadratic formula.6. Use infinite descent.

7. Build large numbers with the properties you want, for example using the Chinese Remainder Theorem.

8. Keep in mind that

a. Consecutive numbers are relatively prime.

b. Ifp aand p b, thenp a b .

c. a b mod n if and only ifn a b.

9. Ways to tell that a number is an integer:

a. It is rational and the root of a monic polynomial with integer coefficients.

b. It is the answer to a counting problem.



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c. It is a term in a recursive sequence with integer initial values and a recursion that is closed for integer values.

Algebra1. Factor!

2. Make a substitution.

3. Find a telescoping sum.

4.

Every polynomial can be thought ofa. as a function.

b. in terms of its coefficients.

c. in terms of its roots.

Geometry1. Draw very accurate and large diagrams.

2. Make cyclic quadrilaterals and parallel lines.

3. Melanie Wood suggested in her 2005 MOP notes that a problem solver should consider using inversion when theproblem contains:

a. circles.

b. a busy point (with many circles and lines passing through it).

c. weird angle conditions.

d. products of lengths.e. reciprocals of lengths.

f. tangencies and orthogonalities.

4. Use trigonometry for both angle-chasing and side-chasing.

5. Find a useful transformation.

6. Consider area.

7. Collinearity and Concurrency:

a. Find the special point of intersection or line of concurrence (e.g. orthocenter, Euler line, etc.)

b. Proof by contradiction.

c. Try to use Menelaus, Ceva, all projective theorems.

highschool playbook

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