Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2009 AIME II Problems/Problem 7"

m (→‎Solution)
Line 31: Line 31:
  
 
And thus we have <math>a=2\cdot 2009 - p(c) = 4010</math>, and the answer is <math>\dfrac{ab}{10} = \dfrac{4010\cdot 1}{10} = \boxed{401}</math>.
 
And thus we have <math>a=2\cdot 2009 - p(c) = 4010</math>, and the answer is <math>\dfrac{ab}{10} = \dfrac{4010\cdot 1}{10} = \boxed{401}</math>.
 +
 +
----
 +
Additionally, once you count the number of factors of 2 in the summation, one can consider the fact that, since <math>b</math> must be odd, it has to take on a value of <math>1,3,5,7, or 9</math> (Because the number of 2s in the summation is clearly greater than <math>1000</math>, dividing by <math>10</math> will yield a number greater than 100, and multiplying this number by any odd number greater than <math>9</math> will yield an answer <math>>999</math>, which cannot happen on the AIME.) Once you calculate the value of <math>4010</math>, and divide by <math>10</math>, <math>b</math> must be equal to 1, as any other value of b will result in an answer >999. This gives <math>\boxed{401}</math> as the answer.
  
 
== See Also ==
 
== See Also ==

Revision as of 21:43, 17 July 2016

Problem

Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$.

Solution

First, note that $(2n)!! = 2^n \cdot n!$, and that $(2n)!! \cdot (2n-1)!! = (2n)!$.

We can now take the fraction $\dfrac{(2i-1)!!}{(2i)!!}$ and multiply both the numerator and the denominator by $(2i)!!$. We get that this fraction is equal to $\dfrac{(2i)!}{(2i)!!^2} = \dfrac{(2i)!}{2^{2i}(i!)^2}$.

Now we can recognize that $\dfrac{(2i)!}{(i!)^2}$ is simply ${2i \choose i}$, hence this fraction is $\dfrac{{2i\choose i}}{2^{2i}}$, and our sum turns into $S=\sum_{i=1}^{2009} \dfrac{{2i\choose i}}{2^{2i}}$.

Let $c = \sum_{i=1}^{2009} {2i\choose i} \cdot 2^{2\cdot 2009 - 2i}$. Obviously $c$ is an integer, and $S$ can be written as $\dfrac{c}{2^{2\cdot 2009}}$. Hence if $S$ is expressed as a fraction in lowest terms, its denominator will be of the form $2^a$ for some $a\leq 2\cdot 2009$.

In other words, we just showed that $b=1$. To determine $a$, we need to determine the largest power of $2$ that divides $c$.

Let $p(i)$ be the largest $x$ such that $2^x$ that divides $i$.

We can now return to the observation that $(2i)! = (2i)!! \cdot (2i-1)!! = 2^i \cdot i! \cdot (2i-1)!!$. Together with the obvious fact that $(2i-1)!!$ is odd, we get that $p((2i)!)=p(i!)+i$.

It immediately follows that $p\left( {2i\choose i} \right) = p((2i)!) - 2p(i!) = i - p(i!)$, and hence $p\left( {2i\choose i} \cdot 2^{2\cdot 2009 - 2i} \right) = 2\cdot 2009 - i - p(i!)$.

Obviously, for $i\in\{1,2,\dots,2009\}$ the function $f(i)=2\cdot 2009 - i - p(i!)$ is is a strictly decreasing function. Therefore $p(c) = p\left( {2\cdot 2009\choose 2009} \right) = 2009 - p(2009!)$.

We can now compute $p(2009!) = \sum_{k=1}^{\infty} \left\lfloor \dfrac{2009}{2^k} \right\rfloor = 1004 + 502 + \cdots + 3 + 1 = 2001$. Hence $p(c)=2009-2001=8$.

And thus we have $a=2\cdot 2009 - p(c) = 4010$, and the answer is $\dfrac{ab}{10} = \dfrac{4010\cdot 1}{10} = \boxed{401}$.

Additionally, once you count the number of factors of 2 in the summation, one can consider the fact that, since $b$ must be odd, it has to take on a value of $1,3,5,7, or 9$ (Because the number of 2s in the summation is clearly greater than $1000$, dividing by $10$ will yield a number greater than 100, and multiplying this number by any odd number greater than $9$ will yield an answer $>999$, which cannot happen on the AIME.) Once you calculate the value of $4010$, and divide by $10$, $b$ must be equal to 1, as any other value of b will result in an answer >999. This gives $\boxed{401}$ as the answer.

See Also

2009 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2009_AIME_II_Problems/Problem_7&oldid=79383"