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 "2017 AIME II Problems/Problem 6"

(→‎See Also)
(→‎Solution)
Line 2: Line 2:
 
Find the sum of all positive integers <math>n</math> such that <math>\sqrt{n^2+85n+2017}</math> is an integer.
 
Find the sum of all positive integers <math>n</math> such that <math>\sqrt{n^2+85n+2017}</math> is an integer.
  
−
==Solution==
+
==Solution 1==
 
Manipulating the given expression, <math>\sqrt{n^2+85n+2017}=\frac{1}{2}\sqrt{4n^2+340n+8068}=\frac{1}{2}\sqrt{(2n+85)^2+843}</math>. The expression under the radical must be an square number for the entire expression to be an integer, so <math>(2n+85)^2+843=s^2</math>. Rearranging, <math>s^2-(2n+85)^2=843</math>. By difference of squares, <math>(s-(2n+85))(s+(2n+85))=1\times843=3\times281</math>. It is easy to check that those are all the factor pairs of 843. Considering each factor pair separately, <math>2n+85</math> is found to be <math>421</math> and <math>139</math>. The two values of <math>n</math> that satisfy one of the equations are <math>168</math> and <math>27</math>. Summing these together, the answer is <math>168+27=\boxed{195}</math>.
 
Manipulating the given expression, <math>\sqrt{n^2+85n+2017}=\frac{1}{2}\sqrt{4n^2+340n+8068}=\frac{1}{2}\sqrt{(2n+85)^2+843}</math>. The expression under the radical must be an square number for the entire expression to be an integer, so <math>(2n+85)^2+843=s^2</math>. Rearranging, <math>s^2-(2n+85)^2=843</math>. By difference of squares, <math>(s-(2n+85))(s+(2n+85))=1\times843=3\times281</math>. It is easy to check that those are all the factor pairs of 843. Considering each factor pair separately, <math>2n+85</math> is found to be <math>421</math> and <math>139</math>. The two values of <math>n</math> that satisfy one of the equations are <math>168</math> and <math>27</math>. Summing these together, the answer is <math>168+27=\boxed{195}</math>.
 +
 +
==Solution 2==
 +
Clearly, the result when <math>n</math> is plugged into the given expression is larger than <math>n</math> itself. Let <math>x</math> be the positive difference between that result and <math>n</math>, so that <math>\sqrt{n^2+85n+2017}=n+x</math>. Squaring both sides and canceling the <math>n^2</math> terms gives <math>85n+2017=2xn+x^2</math>. Combining like terms, <math>(85-2x)n=x^2-2017</math>, so
 +
 +
<cmath>n=\frac{x^2-2017}{85-2x}.</cmath>
 +
 +
Since <math>n</math> is positive, there are two cases, which are simple (luckily). Remembering that <math>x</math> is a positive integer, then <math>x^2-2017</math> and <math>85-2x</math> are either both positive or both negative. The smallest value for which <math>x^2>2017</math> is 45, which makes the denominator, and the entire expression, negative. Evaluating the other case where numerator and denominator are both negative, then we have that <math>x<45</math> (from the numerator) and <math>85-2x<0</math>, which means <math>x>42</math>. This only gives two solutions, <math>x=43, 44</math>. Plugging these into the expression for <math>n</math>, we find that they result in 27 and 168, which both satisfy the initial question. Therefore, the answer is <math>168+27=\boxed{195}</math>.
  
 
=See Also=
 
=See Also=
 
{{AIME box|year=2017|n=II|num-b=5|num-a=7}}
 
{{AIME box|year=2017|n=II|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 16:00, 23 March 2017

Problem

Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.

Solution 1

Manipulating the given expression, $\sqrt{n^2+85n+2017}=\frac{1}{2}\sqrt{4n^2+340n+8068}=\frac{1}{2}\sqrt{(2n+85)^2+843}$. The expression under the radical must be an square number for the entire expression to be an integer, so $(2n+85)^2+843=s^2$. Rearranging, $s^2-(2n+85)^2=843$. By difference of squares, $(s-(2n+85))(s+(2n+85))=1\times843=3\times281$. It is easy to check that those are all the factor pairs of 843. Considering each factor pair separately, $2n+85$ is found to be $421$ and $139$. The two values of $n$ that satisfy one of the equations are $168$ and $27$. Summing these together, the answer is $168+27=\boxed{195}$.

Solution 2

Clearly, the result when $n$ is plugged into the given expression is larger than $n$ itself. Let $x$ be the positive difference between that result and $n$, so that $\sqrt{n^2+85n+2017}=n+x$. Squaring both sides and canceling the $n^2$ terms gives $85n+2017=2xn+x^2$. Combining like terms, $(85-2x)n=x^2-2017$, so

\[n=\frac{x^2-2017}{85-2x}.\]

Since $n$ is positive, there are two cases, which are simple (luckily). Remembering that $x$ is a positive integer, then $x^2-2017$ and $85-2x$ are either both positive or both negative. The smallest value for which $x^2>2017$ is 45, which makes the denominator, and the entire expression, negative. Evaluating the other case where numerator and denominator are both negative, then we have that $x<45$ (from the numerator) and $85-2x<0$, which means $x>42$. This only gives two solutions, $x=43, 44$. Plugging these into the expression for $n$, we find that they result in 27 and 168, which both satisfy the initial question. Therefore, the answer is $168+27=\boxed{195}$.

See Also

2017 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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=2017_AIME_II_Problems/Problem_6&oldid=84936"