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 "1995 AIME Problems/Problem 13"

(→‎Solution)
Line 11: Line 11:
  
 
But this only accounts for <math>\sum_{k = 1}^{6} (4k^3 + k) = 4\left(\frac{6(6+1)}{2}\right)^2 + \frac{6(6+1)}{2} = 1764 + 21 = 1785</math> terms, so we still have <math>1995 - 1785 = 210</math> terms with <math>f(n) = 7</math>. This adds <math>210 \cdot \frac {1}{7} = 30</math> to our summation, giving <math>{400}</math>.
 
But this only accounts for <math>\sum_{k = 1}^{6} (4k^3 + k) = 4\left(\frac{6(6+1)}{2}\right)^2 + \frac{6(6+1)}{2} = 1764 + 21 = 1785</math> terms, so we still have <math>1995 - 1785 = 210</math> terms with <math>f(n) = 7</math>. This adds <math>210 \cdot \frac {1}{7} = 30</math> to our summation, giving <math>{400}</math>.
 +
 +
==Solution 2==
 +
This is a pretty easy problem just to bash. Since the max number we can get is <math>7</math>, we just need to test n values for <math>1.5,2.5,3.5,4.5,5.5</math> and <math>6.5</math>. Then just do how many numbers there are times <math>\frac{1}{\lfloor n \rfloor}</math>, which should be <math>5+17+37+65+101+145+30 = \boxed{400}</math>
  
 
== See also ==
 
== See also ==

Revision as of 13:36, 24 July 2019

Problem

Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

Solution

When $\left(k - \frac {1}{2}\right)^4 \leq n < \left(k + \frac {1}{2}\right)^4$, then $f(n) = k$. Thus there are $\left \lfloor \left(k + \frac {1}{2}\right)^4 - \left(k - \frac {1}{2}\right)^4 \right\rfloor$ values of $n$ for which $f(n) = k$. Expanding using the binomial theorem,

\begin{align*} \left(k + \frac {1}{2}\right)^4 - \left(k - \frac {1}{2}\right)^4 &= \left(k^4 + 2k^3 + \frac 32k^2 + \frac 12k + \frac 1{16}\right) - \left(k^4 - 2k^3 + \frac 32k^2 - \frac 12k + \frac 1{16}\right)\\ &= 4k^3 + k. \end{align*}

Thus, $\frac{1}{k}$ appears in the summation $4k^3 + k$ times, and the sum for each $k$ is then $(4k^3 + k) \cdot \frac{1}{k} = 4k^2 + 1$. From $k = 1$ to $k = 6$, we get $\sum_{k=1}^{6} 4k^2 + 1 = 364 + 6 = 370$ (either adding or using the sum of consecutive squares formula).

But this only accounts for $\sum_{k = 1}^{6} (4k^3 + k) = 4\left(\frac{6(6+1)}{2}\right)^2 + \frac{6(6+1)}{2} = 1764 + 21 = 1785$ terms, so we still have $1995 - 1785 = 210$ terms with $f(n) = 7$. This adds $210 \cdot \frac {1}{7} = 30$ to our summation, giving ${400}$.

Solution 2

This is a pretty easy problem just to bash. Since the max number we can get is $7$, we just need to test n values for $1.5,2.5,3.5,4.5,5.5$ and $6.5$. Then just do how many numbers there are times $\frac{1}{\lfloor n \rfloor}$, which should be $5+17+37+65+101+145+30 = \boxed{400}$

See also

1995 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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=1995_AIME_Problems/Problem_13&oldid=107871"