furman university wylie mathematics tournament ciphering...

Furman UniversityWylie Mathematics Tournament

Ciphering CompetitionMarch 19, 2005

– p. 1

House Rules

1. All answers are integers(!)

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

(

52

)

.

– p. 2

Division II Round I Ciphering

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

– p. 3

Division II Round I – Number 1

If z is a real number between π and 2π, andsin z = −2/3, what is 81 · (sin 2z)2?

– p. 4

Division I Round I – Number 2

If the binomial(2x4 − y2)8

is expanded, there is a term involving

x24y4.

What is the coefficient on this term?

– p. 5

Division II Round I – Number 3

Compute99∑

i=1

200

i(i + 1).

– p. 6

Division II Round I – Number 4

If x + 1x

= 25, then what is x2 + 1x2 =?

– p. 7

Division II Round I – Number 5

The repeating decimal

.130

is written as a rational number a

b(in lowest

terms). Find a + b.

– p. 8

Division II Round I – Number 6

Find the sum of the first 2005 positive evenintegers divided by 4,022,030.

– p. 9

Division II Round I – Number 7

The MRCA middle school soccer team has 15members, exactly two of which are mydaughters, Hannah and Darby. If the coach pickstwo players at random to be captains, what is thereciprocal of the probability that both captainsare my daughters?

– p. 10

Division II Round II Ciphering

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

– p. 11

Division II Round II – Number 1

What is the value of the following when x = 1 andy = −1?

35x5 + 5 · 34x4y + 10 · 33x3y2

+10 · 32x2y3 + 3 · 5xy4 + y5

– p. 12

Division II Round II – Number 2

Consider the real valued function

g(x) =4√

x2 − 7x + 10

2x − 10.

The complement of the domain of this function isan interval. What is the length of this interval?

– p. 13

Division II Round II – Number 3

My father-in-law has a nice circular saw and Idon’t. He won’t let me borrow it, but he will cut myboards for a small fee per cut. If he charges me$2 to cut a 12-foot board into 9 equal pieces, howmany cents would I expect to be charged if I askhim to cut a 14-foot board into 4 equal pieces?

– p. 14

Division II Round II – Number 4

What is

100 ·(√

2 +1

2 +√

2+

1√2 − 2

)

?

– p. 15

Division II Round II – Number 5

Mark and Bob start at the same time to ride theirbikes from Furman to Asheville, 60 miles away.Mark travels 4 miles an hour slower than Bob.Bob reaches Asheville and at once turns back,meeting Mark 12 miles from Asheville. What isMark’s rate, in miles per hour?

– p. 16

Division II Round II – Number 6

A certain businessman buys a number of widgetsat 3 for 10 cents, and an equal number at 5 for20 cents. He plans to sell them in lots of size 3for n cents per lot. What should n be if he is tojust break even?

– p. 17

Division II Round II – Number 7

How many of the first 2005 positive integers aredivisible by all of the first 6 positive integers?

– p. 18

Division II Round III Ciphering

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

– p. 19

Division II Round III – Number 1

What is the square of twice the distance betweenthe parallel lines y = x and y = x − 3?

– p. 20

Division II Round III – Number 2

Suppose that(

−1 +√

3i

2

)k

= 1.

What is the smallest such integer k?

– p. 21

Division II Round III – Number 3

What is

1

log9 15+

1

log25 15?

– p. 22

Division II Round III – Number 4

If the system of equations my = 2, 2x + 3y = 4,x + 2y = 3 has a unique solution, what is m?

– p. 23

Division II Round III – Number 5

The difference of the cubes of two consecutivepositive integers is 217. What is the sum of thetwo integers?

– p. 24

Division II Round III – Number 6

The sum of two numbers is 2 and the product ofthese two numbers is −8. What is the sum of thecubes of these two numbers?

– p. 25

Division II Round III – Number 7

The parabola

y = x2 + 3x + 13/4

has range {x : x ≥ k} for an integer k. What isk?

– p. 26

Division I Round I Ciphering

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

– p. 27

House Rules

1. All answers are integers(!)

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

(

52

)

.

– p. 28

Division I Round I – Number 1

Given 15 points in a plane, no three of which arecollinear, the number of lines they determine is: ?

– p. 29

Division I Round I – Number 2

Consider the region in the first quadrant boundedby the lines y = x + 1, y = 1 − x and x = 3. Whatis the area of this region, in square units?

– p. 30

Division I Round I – Number 3

If the sequence

x,

(

3x

2

)

+ 8,

(

5x

2

)

+ 1, . . .

forms an arithmetic progression, what is x?

– p. 31

Division I Round I – Number 4

From a group of green monkeys, 10 femalesleave. At this point there are 3 males for everyfemale. Then 25 males leave, at which pointthere are two males for every female. How manyfemales were there originally?

– p. 32

Division I Round I – Number 5

If

f(x) =sin(2 cos−1(x))√

1 − x2,

what is f(1/2)?

– p. 33

Division I Round I – Number 6

What is the largest positive integer k so that 5k

divides 26!?

– p. 34

Division I Round I – Number 7

The number

3

√

380 + 380 + 380

can be written in the form

3(3k)

for some integer k. What is k?

– p. 35

Division I Round II Ciphering

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

– p. 36

Division I Round II – Number 1

The equation of the line which contains the point(5, 10) and is perpendicular to the line through(5, 10) and (25, 0) has y–intercept (0, b). What isb?

– p. 37

Division I Round II – Number 2

The sum of two numbers is 20 and their productis 40. What is the reciprocal of the sum of theirreciprocals?

– p. 38

Division I Round II – Number 3

What is the smallest positive integer n so that13n leaves a remainder of 1 when divided by 11?

– p. 39

Division I Round II – Number 4

What is78(

12 − 1

3

)

15 + 1

8

?

– p. 40

Division I Round II – Number 5

The multiplicative inverse of the complex number3 + i has the form a + bi. If k = 2

a−b, what is k?

– p. 41

Division I Round II – Number 6

What is the one’s digit of the number 72005?

– p. 42

Division I Round II – Number 7

A pseudo-mathemagician says to his audience:“Pick a number but don’t tell me what it is.Subtract 9 from it; multiply your answer by 2;then add 20; then divide by 2. Now tell me youranswer.” If a volunteer replies “my answer is 32”,then the pseudo-mathemagician should say:“Your original number is:”

– p. 43

Division I Round III Ciphering

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

– p. 44

Division I Round III – Number 1

The integers listed are the first few entries of acertain row of Pascal’s Triangle. Give the nextinteger which would appear in this row.

1, 12, 66, 220, 495, . . .

– p. 45

Division I Round III – Number 2

A 100 inch wire is cut into two parts and each isbent into a square. The total area of the twosquares is 325 square inches. How many incheslonger is a side of the larger square than a sideof the smaller square?

– p. 46

Division I Round III – Number 3

The value of

5 log4

5

√

644

√

128√

32 · 16−3

4

is:

– p. 47

Division I Round III – Number 4

My entire nickel and dime collection is housed inmy cardboard box collection, which consists ofexactly 24 boxes. 13 of the boxes contain nickels,8 contain dimes and 5 contain both nickels anddimes. How many of the boxes contain neithernickels or dimes?

– p. 48

Division I Round III – Number 5

What is44

2 + 22+ 2

3

?

– p. 49

Division I Round III – Number 6

Compute the value of the largest positive integerwhich simultaneously divides each of 16660 and9350.

– p. 50

Division I Round III – Number 7

What iscsc8(27π/4)?

– p. 51

That’s All, Folks

Awards Ceremony to follow soon. Please bepatient while we tally the results.

– p. 52