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Difference between revisions of "2007 iTest Problems"

(creation)
 
(add some more problems)
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<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math>
 
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math>
  
−
[[2007 iTest Problems/Problem 1|Solution]]
+
[[2007 iTest Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
Line 19: Line 19:
 
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math>
 
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math>
  
 +
[[2007 iTest Problems/Problem 3|Solution]]
 +
 +
==Problem 4==
 +
Star flips a quarter four times. Find the probability that the quarter lands heads exactly
 +
twice.
 +
 +
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math>
 +
 +
[[2007 iTest Problems/Problem 3|Solution]]
 +
 +
==Problem 5==
 +
Compute the sum of all twenty-one terms of the geometric series
 +
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.
 +
 +
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math>
 +
 +
[[2007 iTest Problems/Problem 5|Solution]]
 +
==Problem 6==
 +
Find the units digit of the sum
 +
 +
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath>
 +
 +
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math>
 +
 +
[[2007 iTest Problems/Problem 6|Solution]]
 +
==Problem 7==
 +
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>.
 +
 +
<math>\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}</math>
 +
 +
[[2007 iTest Problems/Problem 7|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?
 +
 +
<math>\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</math>
 +
 +
[[2007 iTest Problems/Problem 8|Solution]]
 +
 +
==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?
 +
 +
<math>\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</math>
 +
 +
[[2007 iTest Problems/Problem 9|Solution]]
 +
 +
==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?
 +
 +
<math>\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</math>
 
{{incomplete|problem page}}
 
{{incomplete|problem page}}

Revision as of 12:20, 11 December 2007

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?

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

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