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2008 iTest Problems

Revision as of 14:01, 22 March 2012 by 1=2 (talk | contribs) (added the problem)

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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?

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

The difference between two prime numbers is 11. Find their sum.

Solution

Problem 5

Solution

Problem 6

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than $L$.

Solution

Problem 7

Find the number of integers $n$ for which $n^2 + 10n < 2008$.

Solution

Problem 8

(story eliminated)

Given the system of equations

$2x + 3y + 3z = 8$,

$3x + 2y + 3z = 808$,

$3x + 3y + 2z = 80808$,

find $x+y+z$.

Solution

Problem 9

(story eliminated)

What is the units digit of $2008^{2008}$?

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

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.

Solution

Problem 15

How many four-digit multiples of 8 are greater than 2008?

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Find the number of lattice points that the line $19x+20y = 1909$ passes through in Quadrant I.

Solution

Problem 19

Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A \cap B$.

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities

$4n-18 < 2008$,
$7n + 17 > 2008$.

Solution

Problem 24

Solution

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 $n$ cm, where $n$ is an integer, find the number of possible values of $n$.

Solution

Problem 26

Solution

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?

Solution

Problem 28

Solution

Problem 29

Solution

Problem 30

Solution

Problem 31

Solution

Problem 32

Solution

Problem 33

Solution

Problem 34

Solution

Problem 35

Solution

Problem 36

Solution

Problem 37

Solution

Problem 38

Solution

Problem 39

Solution

Problem 40

Solution

Problem 41

Solution

Problem 42

Solution

Problem 43

Solution

Problem 44

Solution

Problem 45

Solution

Problem 46

Solution

Problem 47

Find $a + b + c$, where $a$, $b$, and $c$ are the hundreds, tens, and units digits of the six-digit integer $123abc$, which is a multiple of 990.

Solution

Problem 48

A repunit is a natural number whose digits are all $1$. For instance,

\[1,11,111,1111, \ldots\]

are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97$?

Solution

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 $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Solution

Problem 50

Solution

Problem 51

Solution

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 $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c + d$.

Solution

Problem 53

Solution

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?

Solution

Problem 55

Solution

Problem 56

Solution

Problem 57

Solution

Problem 58

Solution

Problem 59

Solution

Problem 60

Solution

Problem 61

Solution

Problem 62

Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, 65, and 65 is a positive integer.

Solution

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?

Solution

Problem 64

Solution

Problem 65

Solution

Problem 66

Solution

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.)

Solution

Problem 68

Solution

Problem 69

Solution

Problem 70

Solution

Problem 71

Solution

Problem 72

Solution

Problem 73

Solution

Problem 74

Solution

Problem 75

Solution

Problem 76

Solution

Problem 77

Solution

Problem 78

Solution

Problem 79

Solution

Problem 80

Let

$p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$

and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.

Solution

Problem 81

Solution

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 .

Solution

Problem 83

Solution

Problem 84

Solution

Problem 85

Solution

Problem 86

Solution

Problem 87

Solution

Problem 88

Solution

Problem 89

Solution

Problem 90

Solution

Problem 91

Solution

Problem 92

Solution

Problem 93

Solution

Problem 94

Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence

\[\left\lfloor \frac{2008}{1} \right\rfloor, \left\lfloor \frac{2008^2}{2} \right\rfloor, \left\lfloor \frac{2008^3}{3}\right\rfloor, \left\lfloor \frac{2008^4}{4} \right\rfloor, \cdots\]

Solution

Problem 95

Solution

Problem 96

Solution

Problem 97

(storyline deleted) Let $k$ be the number of students in a circle. Then let $m$ be the number of ways they can rearrange ourselves so that each of them is in the same spot or within one spot of where they started, and no two people are ever on the same spot. If $m$ leaves a remainder of $1$ when divided by $5$, how many possible values are there of $k$, where $k$ is at least $3$ and at most $2008$?

Solution

Problem 98

Solution

Problem 99

Given a convex, $n$-sided polygon $P$, form a $2n$-sided polygon $\text{clip}(P)$ by cutting off each corner of $P$ at the edges’ trisection points. In other words, $\text{clip}(P)$ is the polygon whose vertices are the $2n$ edge trisection points of $P$, connected in order around the boundary of $P$. Let $P_1$ be an isosceles trapezoid with side lengths $13, 13, 13$, and $3$, and for each $i > 2$, let $P_i = \text{clip}(P_{i-1})$. This iterative clipping process approaches a limiting shape $P_1 = \lim_{i \rightarrow \infty} P_i$. If the difference of the areas of $P_{10}$ and $P_{1}$ is written as a fraction $\frac{x}{y}$ in lowest terms, calculate the number of positive integer factors of $x \cdot y$.

Solution

Problem 100

Solution

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