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Difference between revisions of "2008 iTest Problems"

(→‎Problem 25)
(Added 99, 67, 63, 62, 54, 52, 49, 47, and 27.)
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[[2008 iTest Problems/Problem 26]]
 
[[2008 iTest Problems/Problem 26]]
 
==Problem 27==
 
==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?
  
 
[[2008 iTest Problems/Problem 27]]
 
[[2008 iTest Problems/Problem 27]]
Line 181: Line 182:
 
[[2008 iTest Problems/Problem 46]]
 
[[2008 iTest Problems/Problem 46]]
 
==Problem 47==
 
==Problem 47==
 +
Find <math>a + b + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the hundreds, tens, and units digits of the six-digit integer
 +
<math>123abc</math>, which is a multiple of 990.
  
 
[[2008 iTest Problems/Problem 47]]
 
[[2008 iTest Problems/Problem 47]]
Line 187: Line 190:
 
[[2008 iTest Problems/Problem 48]]
 
[[2008 iTest Problems/Problem 48]]
 
==Problem 49==
 
==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 <math>m/n</math> be the probability that the yarns intersect, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>.
  
 
[[2008 iTest Problems/Problem 49]]
 
[[2008 iTest Problems/Problem 49]]
Line 196: Line 200:
 
[[2008 iTest Problems/Problem 51]]
 
[[2008 iTest Problems/Problem 51]]
 
==Problem 52==
 
==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 <math>c</math> and <math>d</math> are relatively prime positive integers such that <math>c/d</math> is the length of a side of the square, find the value of <math>c + d</math>.
  
 
[[2008 iTest Problems/Problem 52]]
 
[[2008 iTest Problems/Problem 52]]
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[[2008 iTest Problems/Problem 53]]
 
[[2008 iTest Problems/Problem 53]]
 
==Problem 54==
 
==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?
  
 
[[2008 iTest Problems/Problem 54]]
 
[[2008 iTest Problems/Problem 54]]
Line 226: Line 232:
 
[[2008 iTest Problems/Problem 61]]
 
[[2008 iTest Problems/Problem 61]]
 
==Problem 62==
 
==Problem 62==
 +
Find the number of values of <math>x</math> such that the number of square units in the area of the isosceles triangle with sides <math>x</math>, 65, and 65 is a positive integer.
  
 
[[2008 iTest Problems/Problem 62]]
 
[[2008 iTest Problems/Problem 62]]
 
==Problem 63==
 
==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?
  
 
[[2008 iTest Problems/Problem 63]]
 
[[2008 iTest Problems/Problem 63]]
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[[2008 iTest Problems/Problem 66]]
 
[[2008 iTest Problems/Problem 66]]
 
==Problem 67==
 
==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.)
  
 
[[2008 iTest Problems/Problem 67]]
 
[[2008 iTest Problems/Problem 67]]
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[[2008 iTest Problems/Problem 98]]
 
[[2008 iTest Problems/Problem 98]]
 
==Problem 99==
 
==Problem 99==
 +
Given a convex, <math>n</math>-sided polygon <math>P</math>, form a <math>2n</math>-sided polygon <math>\text{clip}(P)</math> by cutting off each corner of <math>P</math> at the edges’ trisection points. In other words, <math>\text{clip}(P)</math> is the polygon whose vertices are the <math>2n</math> edge trisection points of <math>P</math>, connected in order around the boundary of <math>P</math>. Let <math>P_1</math> be an isosceles trapezoid with side lengths <math>13, 13, 13</math>, and <math>3</math>, and for each <math>i > 2</math>, let <math>P_i = \text{clip}(P_{i-1})</math>. This iterative clipping process approaches a limiting shape <math>P_1 = \lim_{i \rightarrow \infty} P_i</math>. If the difference of the areas of <math>P_{10}</math> and <math>P_{1}</math> is written as a fraction <math>\frac{x}{y}</math> in lowest terms, calculate the number of positive integer factors of <math>x \cdot y</math>.
  
 
[[2008 iTest Problems/Problem 99]]
 
[[2008 iTest Problems/Problem 99]]

Revision as of 15:35, 6 May 2009

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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 $$33 for adults (Jerry and Hannah), $$22 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. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family?

2008 iTest Problems/Problem 1

Problem 2

2008 iTest Problems/Problem 2

Problem 3

2008 iTest Problems/Problem 3

Problem 4

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

2008 iTest Problems/Problem 4

Problem 5

2008 iTest Problems/Problem 5

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 $L$.

2008 iTest Problems/Problem 6

Problem 7

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

2008 iTest Problems/Problem 7

Problem 8

(story eliminated)

Given the system of equations

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

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

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

find $x+y+z$.

2008 iTest Problems/Problem 8

Problem 9

(story eliminated)

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

2008 iTest Problems/Problem 9

Problem 10

2008 iTest Problems/Problem 10

Problem 11

2008 iTest Problems/Problem 11

Problem 12

2008 iTest Problems/Problem 12

Problem 13

2008 iTest Problems/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.

2008 iTest Problems/Problem 14

Problem 15

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

2008 iTest Problems/Problem 15

Problem 16

2008 iTest Problems/Problem 16

Problem 17

2008 iTest Problems/Problem 17

Problem 18

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

2008 iTest Problems/Problem 18

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 $A \cap B$.

2008 iTest Problems/Problem 19

Problem 20

2008 iTest Problems/Problem 20

Problem 21

2008 iTest Problems/Problem 21

Problem 22

2008 iTest Problems/Problem 22

Problem 23

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

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

2008 iTest Problems/Problem 23

Problem 24

2008 iTest Problems/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 $n$ cm, where $n$ is an integer, find the number of possible values of $n$.

2008 iTest Problems/Problem 25

Problem 26

2008 iTest Problems/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?

2008 iTest Problems/Problem 27

Problem 28

2008 iTest Problems/Problem 28

Problem 29

2008 iTest Problems/Problem 29

Problem 30

2008 iTest Problems/Problem 30

Problem 31

2008 iTest Problems/Problem 31

Problem 32

2008 iTest Problems/Problem 32

Problem 33

2008 iTest Problems/Problem 33

Problem 34

2008 iTest Problems/Problem 34

Problem 35

2008 iTest Problems/Problem 35

Problem 36

2008 iTest Problems/Problem 36

Problem 37

2008 iTest Problems/Problem 37

Problem 38

2008 iTest Problems/Problem 38

Problem 39

2008 iTest Problems/Problem 39

Problem 40

2008 iTest Problems/Problem 40

Problem 41

2008 iTest Problems/Problem 41

Problem 42

2008 iTest Problems/Problem 42

Problem 43

2008 iTest Problems/Problem 43

Problem 44

2008 iTest Problems/Problem 44

Problem 45

2008 iTest Problems/Problem 45

Problem 46

2008 iTest Problems/Problem 46

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.

2008 iTest Problems/Problem 47

Problem 48

2008 iTest Problems/Problem 48

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 $m + n$.

2008 iTest Problems/Problem 49

Problem 50

2008 iTest Problems/Problem 50

Problem 51

2008 iTest Problems/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 $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c + d$.

2008 iTest Problems/Problem 52

Problem 53

2008 iTest Problems/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?

2008 iTest Problems/Problem 54

Problem 55

2008 iTest Problems/Problem 55

Problem 56

2008 iTest Problems/Problem 56

Problem 57

2008 iTest Problems/Problem 57

Problem 58

2008 iTest Problems/Problem 58

Problem 59

2008 iTest Problems/Problem 59

Problem 60

2008 iTest Problems/Problem 60

Problem 61

2008 iTest Problems/Problem 61

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.

2008 iTest Problems/Problem 62

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?

2008 iTest Problems/Problem 63

Problem 64

2008 iTest Problems/Problem 64

Problem 65

2008 iTest Problems/Problem 65

Problem 66

2008 iTest Problems/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.)

2008 iTest Problems/Problem 67

Problem 68

2008 iTest Problems/Problem 68

Problem 69

2008 iTest Problems/Problem 69

Problem 70

2008 iTest Problems/Problem 70

Problem 71

2008 iTest Problems/Problem 71

Problem 72

2008 iTest Problems/Problem 72

Problem 73

2008 iTest Problems/Problem 73

Problem 74

2008 iTest Problems/Problem 74

Problem 75

2008 iTest Problems/Problem 75

Problem 76

2008 iTest Problems/Problem 76

Problem 77

2008 iTest Problems/Problem 77

Problem 78

2008 iTest Problems/Problem 78

Problem 79

2008 iTest Problems/Problem 79

Problem 80

2008 iTest Problems/Problem 80

Problem 81

2008 iTest Problems/Problem 81

Problem 82

2008 iTest Problems/Problem 82

Problem 83

2008 iTest Problems/Problem 83

Problem 84

2008 iTest Problems/Problem 84

Problem 85

2008 iTest Problems/Problem 85

Problem 86

2008 iTest Problems/Problem 86

Problem 87

2008 iTest Problems/Problem 87

Problem 88

2008 iTest Problems/Problem 88

Problem 89

2008 iTest Problems/Problem 89

Problem 90

2008 iTest Problems/Problem 90

Problem 91

2008 iTest Problems/Problem 91

Problem 92

2008 iTest Problems/Problem 92

Problem 93

2008 iTest Problems/Problem 93

Problem 94

2008 iTest Problems/Problem 94

Problem 95

2008 iTest Problems/Problem 95

Problem 96

2008 iTest Problems/Problem 96

Problem 97

2008 iTest Problems/Problem 97

Problem 98

2008 iTest Problems/Problem 98

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 $x \cdot y$.

2008 iTest Problems/Problem 99

Problem 100

2008 iTest Problems/Problem 100

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