Difference between revisions of "2007 iTest Problems"
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==Problem 4== | ==Problem 4== | ||
| − | Star flips a quarter four times. Find the probability that the quarter lands heads exactly | + | Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice. |
| − | 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> | <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> | ||
Revision as of 22:45, 12 December 2007
Contents
Problem 1
A twin prime pair is a set of two primes 
such that 
is 
greater than 
. What is the arithmetic mean of the two primes in the smallest twin prime pair?
Problem 2
Find 
if 
and 
satisfy
and 
.
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?
Problem 4
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.
Problem 5
Compute the sum of all twenty-one terms of the geometric series
![\[1 + 2 + 4 + 8 + \ldots + 1048576\]](http://latex.artofproblemsolving.com/c/c/1/cc1b6f284c8465d2e1aa0af729878c3a9ab4fdbf.png)
.
Problem 6
Find the units digit of the sum
![\[\sum_{i=1}^{100}(i!)^{2}\]](http://latex.artofproblemsolving.com/4/6/d/46d3c9f38511a89b99e238905fb6f3747dfd6176.png)
Problem 7
An equilateral triangle with side length 
has the same area as a square with side length 
.
Find .
![$\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}$](http://latex.artofproblemsolving.com/0/f/d/0fd9825fdc1605c6a9bd15b665c79728f942f87c.png)
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?
Problem 9
Suppose that 
and 
are positive integers such that 
, the geometric mean of and is greater than 
, and the arithmetic mean of and is less than . How many pairs 
satisfy these conditions?
Problem 10
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only 
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?
