2008 iTest Problems
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 Problem 51
- 52 Problem 52
- 53 Problem 53
- 54 Problem 54
- 55 Problem 55
- 56 Problem 56
- 57 Problem 57
- 58 Problem 58
- 59 Problem 59
- 60 Problem 60
- 61 Problem 61
- 62 Problem 62
- 63 Problem 63
- 64 Problem 64
- 65 Problem 65
- 66 Problem 66
- 67 Problem 67
- 68 Problem 68
- 69 Problem 69
- 70 Problem 70
- 71 Problem 71
- 72 Problem 72
- 73 Problem 73
- 74 Problem 74
- 75 Problem 75
- 76 Problem 76
- 77 Problem 77
- 78 Problem 78
- 79 Problem 79
- 80 Problem 80
- 81 Problem 81
- 82 Problem 82
- 83 Problem 83
- 84 Problem 84
- 85 Problem 85
- 86 Problem 86
- 87 Problem 87
- 88 Problem 88
- 89 Problem 89
- 90 Problem 90
- 91 Problem 91
- 92 Problem 92
- 93 Problem 93
- 94 Problem 94
- 95 Problem 95
- 96 Problem 96
- 97 Problem 97
- 98 Problem 98
- 99 Problem 99
- 100 Problem 100
Problem 1
(story eliminated)
When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo. Single day passes cost 33 dollars for adults (Jerry and Hannah), 22 dollars for children (Michael is still young enough to get the children’s rate), and family memberships (which allow the whole family in at once) cost 120 dollars. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family?
Problem 2
Problem 3
Problem 4
The difference between two prime numbers is 11. Find their sum.
Problem 5
Problem 6
Let be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than
.
Problem 7
Find the number of integers for which
.
Problem 8
(story eliminated)
Given the system of equations
![$2x + 3y + 3z = 8$](http://latex.artofproblemsolving.com/b/1/a/b1a949a0479b1c302babf706cac5b421735369c9.png)
,
![$3x + 3y + 2z = 80808$](http://latex.artofproblemsolving.com/5/7/e/57e194550059b7d7a487e6beac99396feca762ea.png)
find .
Problem 9
(story eliminated)
What is the units digit of ?
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
The sum of the two perfect cubes that are closest to 500 is 343+512 = 855. Find the sum of the two perfect cubes that are closest to 2008.
Problem 15
How many four-digit multiples of 8 are greater than 2008?
Problem 16
Problem 17
Problem 18
Find the number of lattice points that the line passes through in Quadrant I.
Problem 19
Let be the set of positive integers that are the product of two consecutive integers. Let
be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of
.
Problem 20
Problem 21
Problem 22
Problem 23
Find the number of positive integers that are solutions to the simultaneous system of inequalities
![$4n-18 < 2008$](http://latex.artofproblemsolving.com/1/4/0/14022cc26fd293ac9d876a80e8e4d80fbdede06c.png)
![$7n + 17 > 2008$](http://latex.artofproblemsolving.com/7/2/e/72edd3fde190046a9b28724afdd7cdb505886c51.png)
Problem 24
Problem 25
A cube has edges of length 120 cm. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of even smaller cubes, all of equal size. If the edge length of one of those even smaller cubes is cm, where
is an integer, find the number of possible values of
.
Problem 26
Problem 27
Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children’s Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staff. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children’s Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes?
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Find , where
,
, and
are the hundreds, tens, and units digits of the six-digit integer
, which is a multiple of 990.
Problem 48
A repunit is a natural number whose digits are all . For instance,
are the four smallest repunits. How many digits are there in the smallest repunit that is
divisible by ?
Problem 49
Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy’s friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarns would intersect if possible. If all possible positions of the students around the table are equally likely, let be the probability that the yarns intersect, where
and
are relatively prime positive integers. Compute
.
Problem 50
Problem 51
Problem 52
A triangle has sides of length 48, 55, and 73. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If and
are relatively prime positive integers such that
is the length of a side of the square, find the value of
.
Problem 53
Problem 54
One of Michael’s responsibilities in organizing the family vacation is to call around and find room rates for hotels along the route the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael’s eye. Michael notices that the first four digits of 987-1234 descend (9-8-7-1) and that the last four ascend in order (1-2-3-4). This fact along with the fact that the digits are split into consecutive groups makes that number easier to remember. Looking back at the list of numbers that Michael called already, he notices that several of the phone numbers have the same property: their first four digits are in descending order while the last four are in ascending order. Suddenly, Michael realizes that he can remember all those numbers without looking back at his list of hotel phone numbers. “Wow,” he thinks, “that’s good marketing strategy.” Michael then wonders to himself how many businesses in a single area code could have such phone numbers. How many 7-digit telephone numbers are there such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order?
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Find the number of values of such that the number of square units in the area of the isosceles triangle with sides
, 65, and 65 is a positive integer.
Problem 63
Looking for a little time alone, Michael takes a jog along the beach. The crashing of waves reminds him of the hydroelectric plant his father helped maintain before the family moved to Jupiter Falls. Michael was in elementary school at the time. He thinks for a moment about how much his life has changed in just a few years. Michael looks forward to finishing high school, but isn’t sure what he wants to do next. He thinks about whether he wants to study engineering in college, like both his parents did, or pursue an education in business. His aunt Jessica studied business and appraises budding technology companies for a venture capital firm. Other possibilities also tug a little at Michael for different reasons. Michael stops and watches a group of girls who seem to be around Tony’s age play a game around an ellipse drawn in the sand. There are two softball bats stuck in the sand. Michael recognizes these as the foci of the ellipse. The bats are 24 feet apart. Two children stand on opposite ends of the ellipse where the ellipse intersects the line on which the bats lie. These two children are 40 feet apart. Five other children stand on different points on the ellipse. One of them blows a whistle and all seven children run screaming toward one bat or the other. Each child runs as fast as she can, touching one bat, then the next, and finally returning to the spot on which she started. When the first girl gets back to her place, she declares, “I win this time! I win!” Another of the girls pats her on the back, and the winning girl speaks again, “This time I found the place where I’d have to run the shortest distance.” Michael thinks for a moment, draws some notes in the sand, then compute the shortest possible distance one of the girls could run from her starting point on the ellipse, to one of the bats, to the other bat, then back to her starting point. He smiles for a moment, then keeps jogging. If Michael’s work is correct, what distance did he compute as the shortest possible distance one of the girls could run during the game?
Problem 64
Problem 65
Problem 66
Problem 67
At lunch, the seven members of the Kubik family sits down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if lexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.)
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Let
![$p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$](http://latex.artofproblemsolving.com/2/0/4/2041dade3e62a86d00fe5ef9f77aa0ea17f8a0f8.png)
and let be the polynomial remainder when
is divided by
. Find
the remainder when
is divided by
.
Problem 81
Problem 82
Tony’s favorite “sport” is a spectator event known as the Super Mega Ultra Galactic Thumbwrestling Championship (SMUG TWC). During the 2008 SMUG TWC, 2008 professional thumbwrestlers who have dedicated their lives to earning lithe, powerful thumbs, compete to earn the highest title of Thumbzilla. The SMUG TWC is designed so that, in the end, any set of three participants can share a banana split while telling FOXTM television reporters about a bout between some pair of the three contestants. Given that there are exactly two contestants in each bout, let m be the minimum number of bouts necessary to complete the SMUG TWC (so that the contestants can enjoy their banana splits and chat with reporters). Compute .
Problem 83
Problem 84
Problem 85
Problem 86
Problem 87
Problem 88
Problem 89
Problem 90
Problem 91
Problem 92
Problem 93
Problem 94
Problem 95
Problem 96
Problem 97
Problem 98
Problem 99
Given a convex, -sided polygon
, form a
-sided polygon
by cutting off each corner of
at the edges’ trisection points. In other words,
is the polygon whose vertices are the
edge trisection points of
, connected in order around the boundary of
. Let
be an isosceles trapezoid with side lengths
, and
, and for each
, let
. This iterative clipping process approaches a limiting shape
. If the difference of the areas of
and
is written as a fraction
in lowest terms, calculate the number of positive integer factors of
.