Furman University Wylie Mathematics Tournament Ciphering...

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Furman UniversityWylie Mathematics Tournament

Ciphering CompetitionMarch 10, 2007

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House Rules

1. All answers are integers(!)

2. All answers must be written in standard form.For example, 8 not 23, and 10, not

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Division II Round I Ciphering

Participants in Round I ciphering from Division IIschools should now make their way to the front.

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Division II Round I – Number 1

What is the coefficient of the term involving x4y3

in the expansion of (3x2 − 2y)5?

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The function f(x) = x2 + 5x + 9

4has range

{x : x ≥ k} for an integer k. What is k?

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If cos(x) = 5

13what is 144 cot2(x)?

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Find a positive value x such that the distancebetween (x, 3) and (2,−1) is 5.

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The sum of two numbers is 15 while their ratio is1.5:1. What is their product?

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What is the area in the first quadrant bounded bythe x-axis and the lines 4x + 8y = 16 and2x + y = 2?

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Of the nine contestants in a contest, three willreceive black VW Beetles, three will receive Sonyplasma TV sets, and three will receive 20gigabyte video ipods. In how many different wayscan the prizes be awarded?

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Division II Round II Ciphering

Participants in Round II ciphering from Division IIschools should now make their way to the front.

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Division II Round II – Number 1

Bob’s age is 4 more than 3 times John’s age. Iftheir combined age is 100, how old is Bob?

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A person playing a certain lottery can win $100,$10, $1, break even or can lose $10. Theprobability of winning the $100 is 1/50. Theprobability of losing $10 is 1/2. The other optionsare equally likely. If the probability the personplays the lottery and finishes with more moneythan he or she started with is k/100, what is k?

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If (x, y) represents the coordinates of themidpoint between (−5,−3) and (9, 3), what is

y2 − x2y

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If n is the largest integer so that 6n divides 10!,what is n?

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If .639639639 · · · is expressed as a rationalnumber in reduced form, then the value of thenumerator is what?

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Mark has a cylindrical storage unit with radius 2ft and height of 3 ft. He has two sphericalbasketballs that he stores there both with 6 inchradii. If the volume that remains in the storage iskπ

3, what is k?

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A ball is dropped vertically from a height of 16feet and each time it bounces it rebounds backvertically one-half the height it had previouslyfallen. What is the total distance the ball travels?

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Division II Round III Ciphering

Participants in Round III ciphering from DivisionII schools should now make their way to the front.

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Division II Round III – Number 1

Five houses in a row are each to be painted withthe colors red, blue, and green. In how manydifferent ways can the houses be painted so thatno two adjacent houses are of the same color?

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In the figure below, AB ⊥ BC, BC ⊥ CD,AB = 8, BC = 5, and CD = 4. What is theshortest distance from A to D?

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What is the unit’s digit of the number 32007?

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The line L with equation 3x + y = 12 passesthrough the point (3, 3). What is the y-intercept ofthe line perpendicular to L also passing throughthe point (3, 3)?

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If log10 x3 = log10 8 + 3. What is x?

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A movie theater sold tickets for three hundredseats and the box office receipts of $2400 camefrom $9 adult tickets and $6 child tickets. Howmany more adult tickets than children ticketswere sold?

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How many diagonals does a hexagon have?

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Division I Round I Ciphering

Participants in Round I ciphering from Division Ischools should now make their way to the front.

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House Rules

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Division I Round I – Number 1

If you were to roll a die until you rolled a six, thechance you would stop after the third roll is k

What is k?

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How many ways are there to spell FURMANU inthe following diagram? Your movement in thediagram is restricted to always be southeast orsouthwest beginning at F.

NNA A A

M M M MR R R

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Last week we added a new bag of 100 M&M’s toour existing stash. Each day of that week ourdepartment’s stash of M&M’s diminished by 40%.At the end of the third day, 54 M&M’s were left.How many were in the jar at the beginning of theweek before we added more?

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A Furman student dropped by the bookstore tobuy a pencil before class. The bill is $0.73, andshe gives the cashier $1.00. The cashier hasavailable 1 quarter, 2 dimes, 3 nickels, and 2pennies. In how many ways can the cashiermake change?

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If (a1, a2) is the center and r is the radius of thecircle with equation

(x + 3)2 + y2 − 3y =7

what is −64(a1+a2

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What is the radius of the circle whose generalequation is given by

4x2 + 4y2 + 20x − 16y + 37 = 0?

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What is√

8 ·√√

8 ·√

√√8 · · ·?

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Division I Round II Ciphering

Participants in Round II ciphering from Division Ischools should now make their way to the front.

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Division I Round II – Number 1

200 students took a 3-question math exam. 114students answered the first question correctly, 50the second, and 41 the third. Moreover, 14answered the first two correctly, 15 answered thesecond and the third correctly and 11 the firstand third. 5 answered all three correctly. Howmany students answered no question correctly?

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Evaluate

−27(log8 2 + log4

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If x, y are in [0, π

2] and sin(x) = 3

5and sec(y) = 5

evaluate 25 sin(x + y).

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A trapezoid has two parallel sides of length b1

and b2. It has a height of 3 and an area of 18. Ifone of the sides, b1 or b2, is twice the length ofthe other, what is the length of the larger of b1

and b2?

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Two angles are supplementary and one, x, is 20o

more than three times the other, y. Find x − y.

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If g(x) = 2x

x+3and f(g(x)) = −x, then what is

−3f(−2)?

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If the operation ◦ is defined by the equationx ◦ y = 2x + y, what is the value of a in theequation 2 ◦ a = a ◦ 3?

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Division I Round III Ciphering

Participants in Round III ciphering from Division Ischools should now make their way to the front.

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Division I Round III – Number 1

When x = 1 and y = −1, what is the value of

24x4 + 4 · 23x3y + 6 · 22x2y2 + 4 · 2xy3 + y4?

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Evaluate

16 ∗

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The vertices of triangle ABC are (4, 3), (4, 7),and (8, 3). What is the sum of the area of ABC

and (2 −√

2) times the perimeter of ABC?

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A company is considering hiring someemployees. Currently it takes 10 employees,working 8 hours a day, all at the same rate, toproduce 400 widgets. If two people are hired,and they work at the same rate as the currentemployees, how many more widgets can thecompany produce per eight hour day?

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There are 200 questions on a 3-hourexamination. Among these questions are 50mathematics problems. It is suggested that twiceas much time be allowed for each mathematicsproblem as for each of the other questions. Howmany minutes should be spent on themathematics problems combined?

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Two triangles are similar. The sides of one are 9,12, 15. If the perimeter of the second triangle is24, find the product of the sides of the secondtriangle.

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Consider the following sequence of terms:

18, A,B,C, · · · }

What is the sum of the numerator anddenominator in the term C?

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That’s All, FolksAwards Ceremony to follow soon. Please bepatient while we tally the results.

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Furman University Wylie Mathematics Tournament Ciphering...

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