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    theWorldScholars Cup

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    DemiDec, The World Scholars Cup, Power Guide, and Cram Kit are registered trademarks of the DemiDec Corporation.

    Academic Decathlon and USAD are registered trademarks of the United States Academic Decathlon Association.

    DemiDec is not affiliated with the United States Academic Decathlon.

    MATHCRAM KIT

    I. WHAT IS A CRAM KIT?................................................................. 2II. CRAMMING FOR SUCCESS 2III. GENERAL MATH................................. 3IV. ALGEBRA........................................................ 5V. GEOMETRY...........................................................15VI. TRIGONOMETRY...................................................22VII. CALCULUS......................................................................................26VIII. CRUNCH KIT..................................................................... 33IX. ABOUT THE AUTHOR.35

    BYSTEVEN ZHUHARVARD UNIVERSITY

    FRISCO HIGH SCHOOL

    EDITED BY

    DEAN SCHAFFERSTANFORD UNIVERSITY

    TAFT HIGH SCHOOLSOPHY LEEHARVARD UNIVERSITY

    PEARLAND HIGH SCHOOL

    DEDICATED TO PYTHAGORAS,

    FOR BEING SUCH A HOMIE.

    2009 DEMIDEC

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    Math Cram Kit | 2

    WHAT IS A CRAM KIT?A Word from the Editor

    COMPETITION IS NEARING STRUCTURE OF A CRAM KIT

    The handful of days before competition can be the mostoverwhelming. You dont have enough time to revieweverything, so a strategic allocation of your resources is

    crucial. Cram Kits are designed with one goal in mind-----to provide you with the most testable and most easilyforgotten facts.

    Math. The very word strikes fear into the hearts ofmany. But dont be discouraged-----math, like any otherevent, can be mastered through studying, and perhapsmore than any other event, through test-taking.

    Sounds simple, right? Unfortunately, Decathlon math isso broad that no guide could possibly hope to cover allof nooks and crannies. This Cram Kit, then, is meant asa quick review tool to cover last-minute formulas and to

    correct minor misconceptions that may cost you pointsin competition. I advise you to go through this guidewith textbooks nearby. Doing example problems is thebest way to reinforce the concepts that you learn.

    The main body of the Cram Kit is filled with charts anddiagrams for efficient studying. Youll also find helpfulquizzes to reinforce the information as you review.

    The Crunch Kit presents the most important formulasthat you need to know for the math test. Realize,however, that knowing when to apply each formula ishalf the battle. Plugging in the numbers is often theeasiest step.

    Last, but not least, remember to relax. In the finalmoments before you open your test booklet, confidenceis your most important asset.

    Good luck and happy cramming!

    Sophy Lee

    CRAMMING FOR SUCCESSA Word from the Author

    PIECES OF THE MATH PIE

    TIME IS TICKING!

    If you have one day left, read the whole guide.

    *

    If you have one hour left, read the Crunch Kit.

    *

    If you have one minute left, scan the List of Lists

    *

    If you have one second left, good luck.

    10%

    30%

    30%

    20%

    10% General Math

    Algebra

    Geometry

    Trigonometry

    DifferentialCalculus

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    Math Cram Kit | 3

    GENERAL MATHThe Deceptively Simple and the Utterly Confusing

    INTEGERS, FRACTIONS, DECIMALS,

    AND PERCENTSBASIC COUNTING TECHNIQUES

    FRACTIONS

    Fractions must have a common denominatorbefore we can add or subtract them

    When multiplying fractions, try to cancel outcommon factors

    When dividing fractions, flip over the secondfraction and multiply it by the first one

    Step 1. Turn the second fraction upside-down(the reciprocal):

    1 4

    4 1

    Step 2. Multiply the first fraction by the reciprocal of thesecond:

    1 4 = 14 = 4

    2 1 21 2Step 3. Simplify the fraction: 2

    PERCENTAGES

    1% represents one in 100 Divide a percentage by 100 to convert it to a

    decimal

    Multiply a decimal by 100 to convert it t

    25 20$40.00 (1- ) (1- ) $24

    100 100x x o a percentage

    Formula for sale prices: x(1- )(originalprice)100

    Successive discounts do NOT have the sameeffect as a cumulative discount

    If more than one discount applies to an item, keepmultiplying the right side of the above formula by

    discount(1- )

    100

    A shirt originally priced $40.00 is markeddown by 25%. Joe uses a 20%-off coupon topurchase the shirt. How much does he haveto pay for the shirt before tax?

    MULTIPLICATION PRINCIPLE

    Helps us find the total number of possibilities whenwe are choosing one item from each of severalgroups

    Multiply the number of choices from each group If Sally can choose an outfit from 4 pairs of jeans,

    5 shirts, and 3 pairs of shoes, she has 4 x 5 x 3 =60 outfit choices

    FACTORIALS, PERMUTATIONS, AND COMBINATIONS

    FACTORIALS

    ! denotes a factorial (50! 50 49 48... 2 1) x x x x

    PERMUTATIONS

    Arrangements of a set of objects in which ordermatters

    When arranging r objects out of a set of n totalobjects, the number of permutations is n r

    n!P

    (n-r)!

    A club of 12 people wants to elect a president, avice-president, and a treasurer. How manydifferent results can this election have?

    The three positions are different, so order matters 12 3 12! 12!P 12 x 11x 10 1320

    (n-3)! 9!

    COMBINATIONS

    Arrangements of a set of objects in which order doesNOT matter

    When arranging r objects out of a set of n totalobjects, the number of combinations is

    n rn!

    C(r!)(n-r)!

    A club of 12 people wants to elect three people toa committee. How many different results can thiselection have?

    The three seats on the committee are the same,so order does not matter

    12 3 12! 12! 12 x 11x 10C 2203!(12-3)! 3!9! 3 x 2 x 1

    TRY THIS MNEMONIC!

    Permutations = Prizes (order matters) Combinations = Committees (order doesnt matter)

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    GENERAL MATHMore Counting; Vegas

    BASIC COUNTING TECHNIQUES (PT. 2) PROBABILITY OF EQUALLY LIKELY EVENTS

    ARRANGEMENT RULES

    ARRANGEMENT PRINCIPLE

    When a set has two or more identical objects, weneed to take away the redundant arrangementscaused by the identical objects

    To arrange the letters in CALIFORNIA, weneed to find the number of permutations anddivide by the factorials of the identical letters

    CALIFORNIA has two As and two Is possible arrangements

    ARRANGING OBJECTS IN CIRCLES

    When we arrange objects in circles, we need tomake sure that each arrangement represents adistinct ordering of objects, not a mere rotation ofanother arrangement

    Number of possible circular arrangements = hk

    We have to keep one object in place to mark thebeginning of the arrangements

    How many different ways can four people sitaround a circular table?

    Keep one person in place and rearrange theother three

    53

    When arranging keys on a keychain, we mustdivide the result by 2 since we can flip thekeychain over, which makes arrangements thatare mirror images of each other identical

    In how many different ways can 4 keys bearranged on a keychain?

    53

    PROBABILITY

    The chance that an event will happen

    RULES

    The probability that event A happens is P(A) csc 6x 2 8 The probability that independent, unrelated events A

    and B will occur is P(A+B) = P(A) x P(B)

    If events A and B are not mutually exclusive, theprobability of one or the other occurring isP(A or B) = P(A) + P(B) --- P(A+B)

    USEFUL FACTS

    A standard poker deck has 52 cards Such a deck has 4 suits (2 red and 2 black) of 13

    cards each

    A decks face cards are the Jack, Queen, and King ofeach suit (12 face cards total in a standard pokerdeck)

    A standard die has 6 facesEXAMPLES

    What is the probability of rolling a sum of 9 with twodice?

    We have 4 outcomes with a sum of 9 (3-6, 6-3,4-5, 5-4) The total possible number of outcomes is 6 x 6 =

    36

    The probability of rolling a sum of 9 is Asin(kx h) b or Acos(kx h) b

    What is the probability of drawing a red Queen froma standard deck of cards?

    A 52-card deck has four Queens, two of whichare red

    r2 1

    P(Q ) 52 26

    What is the probability of a coin landing heads fourtosses in a row?

    For each toss, the chance of landing heads is 12

    The tosses are independent events, since eachtoss does not affect the result of any other toss

    41 1

    P(4H) P(H) P(H) P(H) P(H)2 16

    CALCULATOR USE

    When dealing with permutations and combinations,use the built-in functions on your scientific or

    graphing calculator to avoid typing in the formulas.Master these (and other calculator techniques)

    beforethe test!

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    ALGEBRA

    Separate but Equal

    SOLVING POLYNOMIAL EQUATIONS

    (THE BASICS)

    SOLVING POLYNOMIAL EQUATIONS (LINEAR)

    EQUATION

    A mathematical statement that two expressions areequal

    Examples 3 + 7 = 14 --- 4 4x + 5 = 2y

    POLYNOMIAL

    An expression containing variables 4 25x x 23

    9

    The variables cannot be contained in fractiondenominators

    The variables also cannot be contained inexponents

    Polynomials with only one term are calledmonomials

    212x y is an example of a monomial Even though the expres