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Difference between revisions of "2007 iTest Problems"

(problem 10 link...)
(Added the headings for all the problems, as well as the sections. You can find the problems here: http://www.mistacademy.com/Contests/iTest/2007iTest.pdf)
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==Problem 1==
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==Multiple Choice Section==
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===Problem 1===
 
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>.  What is the arithmetic mean of the two primes in the smallest twin prime pair?  
 
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>.  What is the arithmetic mean of the two primes in the smallest twin prime pair?  
  
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[[2007 iTest Problems/Problem 1|Solution]]
 
[[2007 iTest Problems/Problem 1|Solution]]
  
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==Problem 2==
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===Problem 2===
 
Find  <math>a + b</math> if <math>a</math> and <math>b</math> satisfy
 
Find  <math>a + b</math> if <math>a</math> and <math>b</math> satisfy
 
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.
 
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.
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[[2007 iTest Problems/Problem 2|Solution]]
 
[[2007 iTest Problems/Problem 2|Solution]]
  
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==Problem 3==
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===Problem 3===
 
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?
 
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?
  
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[[2007 iTest Problems/Problem 3|Solution]]
 
[[2007 iTest Problems/Problem 3|Solution]]
  
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==Problem 4==
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===Problem 4===
 
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.
 
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.
  
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[[2007 iTest Problems/Problem 4|Solution]]
 
[[2007 iTest Problems/Problem 4|Solution]]
  
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==Problem 5==
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===Problem 5===
 
Compute the sum of all twenty-one terms of the geometric series
 
Compute the sum of all twenty-one terms of the geometric series
 
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.
 
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.
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[[2007 iTest Problems/Problem 5|Solution]]
 
[[2007 iTest Problems/Problem 5|Solution]]
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==Problem 6==
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===Problem 6===
 
Find the units digit of the sum
 
Find the units digit of the sum
  
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[[2007 iTest Problems/Problem 6|Solution]]
 
[[2007 iTest Problems/Problem 6|Solution]]
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==Problem 7==
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===Problem 7===
 
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.
 
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.
 
Find <math>s</math>.
 
Find <math>s</math>.
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[[2007 iTest Problems/Problem 7|Solution]]
 
[[2007 iTest Problems/Problem 7|Solution]]
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==Problem 8==
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===Problem 8===
 
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?
 
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?
  
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[[2007 iTest Problems/Problem 8|Solution]]
 
[[2007 iTest Problems/Problem 8|Solution]]
  
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==Problem 9==
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===Problem 9===
 
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?
 
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?
  
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[[2007 iTest Problems/Problem 9|Solution]]
 
[[2007 iTest Problems/Problem 9|Solution]]
  
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==Problem 10==
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===Problem 10===
 
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?
 
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?
  
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[[2007 iTest Problems/Problem 10|Solution]]
 
[[2007 iTest Problems/Problem 10|Solution]]
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===Problem 11===
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===Problem 12===
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===Problem 13===
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===Problem 14===
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===Problem 15===
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===Problem 16===
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===Problem 17===
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===Problem 18===
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===Problem 19===
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===Problem 20===
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===Problem 21===
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===Problem 22===
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===Problem 23===
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===Problem 24===
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===Problem 25===
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==Short Answer Section==
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===Problem 26===
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===Problem 27===
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===Problem 28===
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===Problem 29===
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===Problem 30===
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===Problem 31===
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===Problem 32===
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===Problem 33===
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===Problem 34===
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===Problem 35===
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===Problem 36===
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===Problem 37===
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===Problem 38===
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===Problem 39===
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===Problem 40===
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===Problem 41===
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===Problem 42===
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===Problem 43===
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===Problem 44===
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===Problem 45===
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===Problem 46===
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===Problem 47===
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===Problem 48===
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===Problem 49===
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===Problem 50===
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==Ultimate Question==
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===Problem 51===
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===Problem 52===
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===Problem 53===
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===Problem 54===
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===Problem 55===
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===Problem 56===
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===Problem 57===
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===Problem 58===
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===Problem 59===
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===Problem 60===
  
 
{{incomplete|problem page}}
 
{{incomplete|problem page}}

Revision as of 11:00, 13 May 2009

Multiple Choice Section

Problem 1

A twin prime pair is a set of two primes $(p, q)$ such that $q$ is $2$ greater than $p$. What is the arithmetic mean of the two primes in the smallest twin prime pair?

$\mathrm{(A)}\, 4$

Solution

Problem 2

Find $a + b$ if $a$ and $b$ satisfy $3a + 7b = 1977$ and $5a + b = 2007$.

$\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498$

Solution

Problem 3

An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?

$\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56$

Solution

Problem 4

Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.

$\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}$

Solution

Problem 5

Compute the sum of all twenty-one terms of the geometric series \[1 + 2 + 4 + 8 + \ldots + 1048576\].

$\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161$

Solution

Problem 6

Find the units digit of the sum

\[\sum_{i=1}^{100}(i!)^{2}\]

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9$

Solution

Problem 7

An equilateral triangle with side length $1$ has the same area as a square with side length $s$. Find $s$.

$\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}$

Solution

Problem 8

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?

$\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789$

Solution

Problem 9

Suppose that $m$ and $n$ are positive integers such that $m < n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m, n)$ satisfy these conditions?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007$

Solution

Problem 10

My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007$

Solution

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Short Answer Section

Problem 26

Problem 27

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

Problem 48

Problem 49

Problem 50

Ultimate Question

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

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