Mad Hatter 11-12. Part I.

Math Field Day. California State University, Fresno.April 17th, 2010.

Question 1.

Which of the following numbers is a perfect square?

(a) 98!99!

(b) 98!100!

(c) 99!100!

(d) 99!101!

(e) 100!101!

Question 2.

A square has sides of length 10, and a circle centered at one of its verticeshas radius 10. What is the area of the union of the regions enclosed by thesquare and the circle?

(a) 100 + 100π

(b) 100 + 75π

(c) 100 + 50π

(d) 100 + 25π

(e) 100

Question 3.

The equation 32x+2 − 3x+3 − 3x + 3 = 0 has

(a) No solution

(b) One solution

(c) Two solutions

(d) Three solutions

(e) Infinitely many solutions

Question 4.

Suppose K , L, and M are on a line `, with L between K and M. Whichterm best describes the set of all points P such that L is between M andP ? (assume that L could be equal to P)

(a) LM

(b) A ray containing K and M

(c) KM

(d) A ray containing L and M

(e) A ray containing K and L

Question 5.

How many positive angles less than 360◦ are there in the figure?

(a) 4

(b) 6

(c) 7

(d) 8

(e) More than 8

Question 6.

If in a regular polygon you double the length of its sides, then whathappens with the area of the polygon?

(a) It stays the same.

(b) It doubles.

(c) It quadruples.

(d) It cannot be determined.

(e) None of the above.

Question 7.What is the perimeter of the figure below? (All angles are right)

24

28

(a) 24 + 28

(b) 2(24 + 28)

(c)2

3(24 + 28)

(d) It cannot be determined.

(e) None of the above.

Question 8.

The radius of a sphere is tripled, by what number is its volume multiplied?

(a) It stays the same.

(b) It triples.

(c) It increases by a factor of 9.

(d) It increases by a factor of 27.

(e) It cannot be determined.

Question 9.

A cube and a 3× 8× 9 rectangular box have the same volume. What isthe area of a side of the cube?

(a) 3

(b) 6

(c) 8

(d) 9

(e) 36

Question 10.The figure below represents a dart-board in which ABCD and ECFG arerectangles. Assume that E is the midpoint of BC and that F is twice asfar from D than it is from C . What is the probability that a dart that hitsthe board lands in the shaded area?

G

FD

A B

C

E

(a) 16

(b) 15

(c) 25

(d) 35

(e) 56

Question 11.In the following figure ABCD is a square with area 9. EFCG is a squarewith side equal to a third of the length of BC . HIJC is a square with sideequal to a third of the length of EC . If this pattern is continued, what isthe area of the tenth square?

D CJ

B

H

EF

G

I

A

(a) 9−12

(b) 9−11

(c) 9−10

(d) 9−9

(e) 9−8

Question 12.

The mean of a group of 20 numbers is 9. If one number is removed, themean of the remaining numbers is 7. What is the value of the removednumber?

(a) 16

(b) 25

(c) 40

(d) 47

(e) It cannot be determined.

Question 13.

What is the area of the circle with equation x2 + 8x + y2 + 12y + 3 = 0?

(a) 98π

(b) 64π

(c) 49π

(d) 14π

(e) None of the above.

Question 14.

The edge of a cube is 2 in. What is the distance between two oppositecorners of the cube?

(a) 2√

6 in

(b) 2√

5 in

(c) 2√

3 in

(d) 2√

2 in

(e) None of the above.

Question 15.

Ten people are to be seated in a row with 10 seats in a movie theater.Two of the 10 people do not want to sit in either of the two end seats ofthis row. In how many accommodating ways can all ten people be seated?

(a) 8!

(b) 2 · 8!

(c) 7 · 8 · 8!

(d) 10!

(e) None of the above.

Question 16.

Which of the following triples of numbers could represent the lengths ofthe sides of a triangle?

(a) 2, 2, 5

(b) 3, 3, 5

(c) 4, 4, 8

(d) 5, 5, 15

(e) None of the above.

Question 17.

Simplify1

x−1 −1

x−21

x−2 −1

x−3

(a) x−3x−1

(b) − x−3x−1

(c) 1

(d) 35

(e) None of the above.

Question 18.

What is the largest power of 5 dividing 125!?

(a) 55

(b) 511

(c) 525

(d) 531

(e) 535

Question 19.

Assume that log3 x = logy 5 = 2. Then (xy)2 =

(a) 75

(b) 45

(c) 225

(d) 405

(e) 1875

Question 20.

How many functions can be defined from a domain D = {0, 1, 2} onto arange R = {3, 4}?

(a) 5

(b) 6

(c) 7

(d) 8

(e) 9

Question 21.

Ten liters of a 30% acid solution are obtained by mixing a 25% solutionand a 50% solution. How many liters of the 25% solution must be used?

(a) 1

(b) 1.5

(c) 2

(d) 5

(e) 8

Question 22.

In a group of 50 students, 28 are taking Calculus, 36 are taking NumberTheory, and 22 are taking both Calculus and Number Theory. How manyof the 50 students are taking neither Calculus nor Number Theory?

(a) 8

(b) 6

(c) 14

(d) 28

(e) 30

Question 23.

Oscar’s mother is 20 years older than Oscar. Ten years ago, Oscar’smother was twice as old as Oscar. What is the mother’s present age?

(a) 30

(b) 40

(c) 25

(d) 51

(e) 50

Question 24.

Suppose that a, b, c , and d are positive real numbers. If d = c n√

ab, thenn equals

(a)log d − log c

log a + log b

(b)log a + log b

log d − log c

(c)log a log b

log dlog c

(d)

log dlog c

log a log b

(e)log (a + b)

log (d − c)

Question 25.

If an integer between 1 and 200, inclusive, is randomly selected, what isthe probability that it is a perfect square?

(a) 0.5

(b) 0.25

(c) 0.1

(d) 0.07

(e) 0.05

Question 26.

Oscar watches a spherical soap bubble of radius r land on a flat mat andform a hemisphere. Assuming that the volume remains the same, what isthe radius of the hemisphere?

(a) 2r

(b) r√

2

(c) 2√

33 r

(d) r 3√

2

(e) r

Question 27.

A cable stretches from the top of a vertical pole to a point on the ground15 feet away from the base of the pole. If the length of the cable is onefoot more than the height of the pole, then how tall is the pole?

(a) 20 ft

(b) 65 ft

(c) 112 ft

(d) 210 ft

(e) 441 ft

Question 28.

If f (x) = f (x + 1)− x and f (0) = 5, find f (3).

(a) 2

(b) 3

(c) 4

(d) 6

(e) 8

Question 29.

How many numbers are there between 60 and 360 that are divisible by 17?

(a) 14

(b) 15

(c) 16

(d) 17

(e) 18

Question 30.

If f (x) = 3−x and g(x) = 3x − 1, then (f ◦ g)(1) equals

(a) −6

(b) −9

(c) 16

(d) 19

(e) 181

Question 31.

Simplify(√√

27−√

11)(√√

27 +√

11)

(a) 2

(b) 4√

38

(c) 16

(d) 6

(e) 4

Question 32.

The number 1221 is a numerical palindrome because it reads the sameforwards and backwards. How many four digit numbers are palindromic?

(a) 90

(b) 900

(c) 1000

(d) 1111

(e) 9000

Question 33.

What is the sum of the prime factors of 2010?

(a) 23

(b) 77

(c) 208

(d) 211

(e) None of the above.

Question 34.

If the figure in the middle is a square, then what is the ratio of the area ofcircumscribed circle to the area of inscribed circle?

(a) 1 : 2

(b) 2 : 1

(c) 1 :√

2

(d)√

2 : 1

(e) None of the above.

Question 35.

A bag contains 30 balls; 18 of them are red. What are the odds that if aball is drawn then it will be red?

(a)2

5

(b)3

5

(c)12

18

(d)18

12

(e)6

30

Question 36.Let A,B, and C be squares of equal area. If the sum of the areas of thesquares is 432 cm2, what is the area of the triangle?

A

B

C

(a) 144 cm2

(b) 36 cm2

(c) 72 cm2

(d) 18√

3 cm2

(e) 36√

3 cm2

Question 37.

Three horses enter a race. If horse A is twice as likely to win as horse B,and horse B is 5 times as likely to win as horse C , what is the probabilitythat horse A wins the race?

(a)1

16

(b)5

16

(c)7

16

(d)5

8

(e)1

7

Question 38.

Find the sum of the terms of the finite sequence

7, 14, 21, · · · , 1995, 2002, 2009

(a) 89 · 177

(b) 7 · 89 · 177

(c)7 · 2009

2

(d)2009 · 2010

2

(e)7

2(2009 · 2010)

Question 39.

Steve wants to cover a football field with sod. The field is 360 feet longand 160 feet wide. Sod can be purchased in squares in 1 foot incrementsfrom 1 foot wide up to 7 feet wide. What is the largest size squares Stevecan purchase with which he can cover the field completely without anygaps or overhang?

(a) 3 feet wide.

(b) 4 feet wide.

(c) 5 feet wide.

(d) 6 feet wide.

(e) 7 feet wide.

Question 40.

What is the perimeter of a rhombus with diagonals measuring 32 in and24 in ?

(a) 20 in

(b) 48 in

(c) 80 in

(d) 96 in

(e) 192 in

Mad Hatter 11-12. Part II.

Math Field Day. California State University, Fresno.April 17th, 2010.

Question 41.

Exactly one of the following numbers is prime. Which one?

(a) 999, 991

(b) 999, 973

(c) 999, 983

(d) 1, 000, 001

(e) 7, 999, 973

Question 42.

Suppose that f and g are two polynomials such that

f (g(0)) = g(f (0)) = f (f (0)) = 0.

Then g(g(0)) =

(a) 0

(b) 1

(c) 2

(d) −1

(e) Cannot be determined.

Question 43.

What is the largest positive integer y such that there exists a positiveinteger x satisfying √

x +√

y =√

550?

(a) 22

(b) 88

(c) 352

(d) 441

(e) 500

Question 44.

On Wednesday, four trucks drove in a line (one directly behind the other).None of the trucks passed any of the others, so the order of the trucksnever changed. How many ways are there to rearrange the order of thetrucks so that on the next day, no truck is directly behind a truck that itwas directly behind the day before?

(a) 4

(b) 8

(c) 11

(d) 12

(e) 16

Question 45.

For how many integers x in the set {1, 2, 3, . . . , 99, 100} is x3 − 2x2 thesquare of an integer?

(a) 7

(b) 8

(c) 9

(d) 10

(e) 11

Question 46.The black star in the figure below is obtained by rotating an equilateraltriangle about its center 60 degrees counterclockwise. The ratio of thestriped area to the solid area is

(a) 9 : 11

(b) 2 : 3

(c) 1 : 1

(d) 4 : 5

(e) 5 : 4

Question 47.

Suppose p(x) = ax2 + bx + c , and q(x) = 2010x2 − πx +√−1 are two

polynomials (with complex coefficients). If p(0) = q(0), p(1) = q(1) andp(2) = q(2), then b =

(a) 2010

(b) −π(c)√−1

(d) 5

(e) Cannot be determined.

Question 48.

Chuck leaves the Buymore in Burbank at 8:00 a.m. Hannah is 300 milesaway, and is late leaving for their meeting - she departs at 10 a.m. If shedrives twice as fast as he does, and they arrive at the meeting place halfway between their original location at the same time, what time do theymeet?

(a) 11:00 a.m.

(b) 11:30 a.m.

(c) 12:00 p.m.

(d) 12:30 p.m.

(e) 1:00 p.m.

Question 49.

An isosceles triangle has sides 5, 5 and x . What is the height of such atriangle with largest area?

(a) 2√5

(b) 5√2

(c) 3

(d) 4

(e) None of the above.

Question 50.

How many zeros are at the end of the base 6 representation of 100! ?

(a) 35

(b) 97

(c) 84

(d) 48

(e) 52

Question 51.

What is the smallest positive integer n such that 1 + 2 + · · ·+ n > 100?

(a) 10

(b) 12

(c) 13

(d) 14

(e) 15

Question 52.

Suppose that a and x are two positive real numbers for whichloga x + logx a = 3. What is the value of (loga x)2 + (logx a)2 ?

(a) 2

(b) 5

(c) 7

(d) 9

(e) 11

Question 53.

In a group of five friends, the sums of the ages of each group of four ofthem are 124, 128, 130, 136 and 142. What is the age of the youngest ofthe friends?

(a) 18

(b) 21

(c) 23

(d) 25

(e) 34

Question 54.

The surface area of a right rectangular prism (a box) is 48 square feet, andthe sum of its length, width, and height is 13 feet. What is the length ofthe longest diagonal connecting two corners of the box?

(a) 8 feet

(b) 9 feet

(c) 10 feet

(d) 11 feet

(e) 12 feet

Question 55.

How many positive integers n have the property that the measures (indegrees) of the interior angles of a regular n-gon are integers?

(a) 4

(b) 22

(c) 30

(d) 32

(e) 64

Question 56.

Eleven teams play in a soccer tournament. Each team must play each ofthe other teams exactly once. If a game ends in a tie, each team gets 1point. For the games that do not end in a tie, the winning team gets 5points and the losing team gets 0 points. Which of the following is apossible value for the total number of points earned by the 11 teams bythe end of the tournament?

(a) 92

(b) 196

(c) 257

(d) 290

(e) None of the above

Question 57.

From a point 2 units from a circle, a tangent is drawn. If the radius of thecircle is 8 units, find the length of the tangent segment.

(a) 6 units

(b) 8 units

(c) 10 units

(d) 14 units

(e) None of the above.

Question 58.

How many integers between 1 and 1000 have exactly 27 positive divisors?

(a) 0

(b) 1

(c) 2

(d) 27

(e) 28

Question 59.

What is the coefficient of x50 in the expansion of the following product?

(1 + 2x + 3x2 + 4x3 + · · ·+ 101x100) · (1 + x + x2 + x3 + · · ·+ x25)

(a) 50

(b) 125

(c) 501

(d) 923

(e) 1001

Question 60.

Let x =√

7 + 2√

6 +√

7− 2√

6. Which of the following intervalscontains x?

(a) (3.9, 4)

(b) (4.8, 4.9)

(c) (4.9, 5)

(d) (5, 5.1)

(e) (5.2, 5.3)

Question 61.

Let R be the circle x2 + (y + 2)2 = 9. Let S be the set of all circles in theplane such that for each circle C in S , we have:

(i) C lies in the first quadrant outside R.

(ii) C is tangent to R and to the x-axis.

On what geometric object must the centers of the circles in S lie?

(a) an ellipse that is not a circle

(b) a circle

(c) a straight line

(d) a hyperbola

(e) a parabola

Question 62.

What is the remainder when 100101102103104105106107108 is divided by999?

(a) 0

(b) 27

(c) 522

(d) 936

(e) 990

Question 63.

Suppose that the sum of k consecutive integers is an even integer divisibleby k and that the smallest of the k consecutive integers is even. Which ofthe following must be true about k?

(a) k + 3 is divisible by 4

(b) k + 2 is divisible by 4

(c) k + 1 is divisible by 4

(d) k is divisible by 4

(e) None of the above must be true.

Question 64.

How many pairs of integers (x , y) satisfy the equation y =x + 12

2x − 1?

(a) 2

(b) 4

(c) 6

(d) 8

(e) Infinite

Question 65.

For x2 + 2x + 5 to be a factor of x4 + px2 + q, the value of q must be

(a) 5

(b) 10

(c) 15

(d) 20

(e) 25

Question 66.

If the real numbers x and y satisfy (x + 5)2 + (y − 12)2 = 142, then theminimum value of x2 + y2 is

(a) 1

(b) 2

(c)√

3

(d)√

2

(e) Cannot be determined

Question 67.

How many of the solutions to the following equation are negative?

x4 − 6x2 + 9 = 3x3 + 5x

(a) 0

(b) 1

(c) 2

(d) 3

(e) 4

Question 68.

Find the maximum value of k for which the inequality√

x +√

6− x ≥ khas a solution.

(a)√

6−√

3

(b)√

3

(c)√

6 +√

3

(d)√

6

(e) 2√

3

Question 69.

Suppose ω is a complex number satisfying ω2 + 2ω + 4 = 0. Then ω6 =?

(a) 1

(b) 2

(c) 8

(d) 32

(e) 64

Question 70.

It snowed on exactly 11 days during Janes holiday trip to the East Coast.On each snowy day it snowed either in the morning or in the afternoon butnot both. There are exactly 13 afternoons when it did not snow andexactly 16 mornings when it did not snow. How many days did the triplast?

(a) 18

(b) 19

(c) 20

(d) 21

(e) 22

Question 71.

Given f (x) + f

(1

1− x

)=

1

x, for x 6= 0, 1, the value of f (2) is

(a) 0

(b)1

4

(c)7

4(d) 1

(e) Cannot be determined

Question 72.

Suppose that f (x) = ax + b where a and b are real numbers. Given thatf (f (f (x))) = 8x + 21, what is the value of a + b ?

(a) 2

(b) 3

(c) 4

(d) 5

(e) 6

Question 73.

A box of coins contains a total of $26.00 in nickels, dimes, and quarters. Ifthere is the same number of nickels as dimes, but twice as many quartersas nickels, how many dimes are in the box?

(a) 26

(b) 30

(c) 36

(d) 40

(e) 50

Question 74.

How many positive real solutions does the equation√

x = |x4 − 1| have ?

(a) 0

(b) 1

(c) 2

(d) 3

(e) 4

Question 75.

What is the value of the following product?

tan 5◦ tan 15◦ tan 25◦ tan 35◦ tan 45◦ tan 55◦ tan 65◦ tan 75◦ tan 85◦

(a) 2

(b)√

3

(c)

√3

3(d) 1

(e)

√1

2

Question 76.

If x2 +1

x2= 3, what is the value of x4 +

1

x4= ?

(a) 3

(b) 4

(c) 5

(d) 6

(e) 7

Question 77.

If i =√−1, what is the value of

(1 + i

1− i

)2010

?

(a) 0

(b) 1

(c) −i

(d) -1

(e) i

Question 78.

How many points do the graphs of 4x2 − 9y2 = 36 and x2 − 2x + y2 = 15have in common?

(a) 0

(b) 1

(c) 2

(d) 3

(e) 4

Question 79.

Suppose that a is a non-zero real number for which sin x + sin y = a andcos x + cos y = 2a. What is the value of cos(x − y) ?

(a)a2 − 2

2

(b)3a2 − 2

2

(c)5a2 − 2

2

(d)7a2 − 2

2

(e)9a2 − 2

2

Question 80.The triangle 4ABE is inscribed within square ABCD and has a height of6 cm. Assuming that the little triangle inside 4ABE is formed bymidlines, what is the area of the shaded region?

E

A

D C

B

(a) 13.5

(b) 18

(c) 22.5

(d) 36

(e) It cannot be determined.

Answer Key: Part I.

(1) c (2) b (3) c (4) e (5) e (6) c (7) b (8) d

(9) e (10) a (11) e (12) d (13) c (14) c (15) c (16) b

(17) a (18) d (19) d (20) b (21) e (22) a (23) e (24) b

(25) d (26) d (27) c (28) e (29) d (30) d (31) e (32) a

(33) b (34) b (35) b (36) d (37) d (38) b (39) c (40) c

Answer Key: Part II.

(41) c (42) e (43) c (44) c (45) d (46) a (47) b (48) c

(49) b (50) d (51) d (52) c (53) c (54) d (55) b (56) c

(57) a (58) b (59) e (60) b (61) e (62) d (63) a (64) c

(65) e (66) a (67) a (68) e (69) e (70) c (71) c (72) d

(73) d (74) c (75) d (76) e (77) d (78) d (79) c (80) c