2008 iTest Problems

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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
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37 Problem 37
38 Problem 38
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48 Problem 48
49 Problem 49
50 Problem 50
51 Problem 51
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80 Problem 80
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82 Problem 82
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84 Problem 84
85 Problem 85
86 Problem 86
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88 Problem 88
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94 Problem 94
95 Problem 95
96 Problem 96
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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 $$$$ 33 for adults (Jerry and Hannah), $$$$ 22 for children (Michael is still young enough to get the children’s rate), and family memberships (which allow the whole family in at once) cost $$$$ 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