Difference between revisions of "2010 AIME I Problems/Problem 14"

(credit to IDEALLY.PERFECT)
 
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'''Note:''' <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.
 
'''Note:''' <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.
  
== Solution ==
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== Solution 1 ==
 
Observe that <math>f</math> is strictly increasing in <math>n</math>. We realize that we need <math>100</math> terms to add up to around <math>300</math>, so we need some sequence of <math>2</math>s, <math>3</math>s, and then <math>4</math>s.
 
Observe that <math>f</math> is strictly increasing in <math>n</math>. We realize that we need <math>100</math> terms to add up to around <math>300</math>, so we need some sequence of <math>2</math>s, <math>3</math>s, and then <math>4</math>s.
  
 
It follows that <math>n \approx 100</math>. Manually checking shows that <math>f(109) = 300</math> and <math>f(110) > 300</math>. Thus, our answer is <math>\boxed{109}</math>.
 
It follows that <math>n \approx 100</math>. Manually checking shows that <math>f(109) = 300</math> and <math>f(110) > 300</math>. Thus, our answer is <math>\boxed{109}</math>.
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 +
== Solution 2 ==
 +
Because we want the value for which <math>f(n)=300</math>, the average value of the 100 terms of the sequence should be around <math>3</math>. For the value of <math>\lfloor log_{10} (kn) \rfloor</math> to be <math>3</math>, <math>1000 \le kn < 10000</math>. We want kn to be around the middle of that range, and for k to be in the middle of 0 and 100, let <math>k=50</math>, so <math>50n=\frac{10000+1000}{2}=\frac{11000}{2}=5500</math>, and <math>n = 110</math>. <math>f(110) = 301</math>, so we want to lower <math>n</math>. Testing <math>109</math> yields <math>300</math>, so our answer is still <math>\boxed{109}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 16:16, 20 March 2010

Problem

For each positive integer n, let $f(n) = \sum_{k = 1}^{100} \lfloor log_{10} (kn) \rfloor$. Find the largest value of n for which $f(n) \le 300$.

Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

Solution 1

Observe that $f$ is strictly increasing in $n$. We realize that we need $100$ terms to add up to around $300$, so we need some sequence of $2$s, $3$s, and then $4$s.

It follows that $n \approx 100$. Manually checking shows that $f(109) = 300$ and $f(110) > 300$. Thus, our answer is $\boxed{109}$.

Solution 2

Because we want the value for which $f(n)=300$, the average value of the 100 terms of the sequence should be around $3$. For the value of $\lfloor log_{10} (kn) \rfloor$ to be $3$, $1000 \le kn < 10000$. We want kn to be around the middle of that range, and for k to be in the middle of 0 and 100, let $k=50$, so $50n=\frac{10000+1000}{2}=\frac{11000}{2}=5500$, and $n = 110$. $f(110) = 301$, so we want to lower $n$. Testing $109$ yields $300$, so our answer is still $\boxed{109}$.

See also

2010 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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