Difference between revisions of "2002 AMC 12B Problems/Problem 12"
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+ | {{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #12]] and [[2002 AMC 10B Problems|2002 AMC 10B #16]]}} | ||
== Problem == | == Problem == | ||
For how many integers <math>n</math> is <math>\dfrac n{20-n}</math> the [[perfect square|square]] of an integer? | For how many integers <math>n</math> is <math>\dfrac n{20-n}</math> the [[perfect square|square]] of an integer? | ||
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=== Solution 1 === | === Solution 1 === | ||
− | Let <math>x^2 = \frac{n}{20-n} </math>, with <math>x \ge 0</math> (note that the solutions <math>x < 0</math> do not give any additional solutions for <math>n</math>). Then rewriting, <math>n = \frac{20x^2}{x^2 + 1}</math>. Since <math>\text{gcd}(x^2, x^2 + 1) = 1</math>, it follows that <math>x^2 + 1</math> divides <math>20</math>. Listing the factors of <math>20</math>, we find that <math>x = 0, 1, 2 , 3</math> are the only <math> | + | Let <math>x^2 = \frac{n}{20-n} </math>, with <math>x \ge 0</math> (note that the solutions <math>x < 0</math> do not give any additional solutions for <math>n</math>). Then rewriting, <math>n = \frac{20x^2}{x^2 + 1}</math>. Since <math>\text{gcd}(x^2, x^2 + 1) = 1</math>, it follows that <math>x^2 + 1</math> divides <math>20</math>. Listing the factors of <math>20</math>, we find that <math>x = 0, 1, 2 , 3</math> are the only <math>\boxed{\mathrm{(D)}\ 4}</math> solutions (respectively yielding <math>n = 0, 10, 16, 18</math>). |
=== Solution 2 === | === Solution 2 === | ||
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For prime <math>n</math> the fraction will not be an integer, as the denominator will not contain the prime in the numerator. | For prime <math>n</math> the fraction will not be an integer, as the denominator will not contain the prime in the numerator. | ||
− | This leaves <math>n\in\{12,14,15,16,18\}</math>, and a quick substitution shows that out of these only <math>n=16</math> and <math>n=18</math> yield a square. | + | This leaves <math>n\in\{12,14,15,16,18\}</math>, and a quick substitution shows that out of these only <math>n=16</math> and <math>n=18</math> yield a square. Therefore, there are only <math>\boxed{\mathrm{(D)}\ 4}</math> solutions (respectively yielding <math>n = 0, 10, 16, 18</math>). |
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+ | === Solution 3 === | ||
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+ | If <math>\frac{n}{20-n} = k^2 \ge 0</math>, then <math>n \ge 0</math> and <math>20-n > 0</math>, otherwise <math>\frac{n}{20-n}</math> will be negative. Thus <math>0 \le n \le 19</math> and <cmath>0 = \frac{0}{20-(0)} \le \frac{n}{20-n} \le \frac{19}{20-(19)} = 19</cmath> Checking all <math>k</math> for which <math>0 \le k^2 \le 19</math>, we have <math>0</math>, <math>1</math>, <math>2</math>, <math>3</math> as the possibilities. <math>\boxed{(D)}</math> | ||
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+ | ~ Nafer | ||
+ | |||
+ | === Solution 4 === | ||
+ | For all integers x, <math>x^2</math> is always a positive integer. So solve for <math>\frac{n}{20-n} = 0</math>, getting <math>n=0</math> and <math>\frac{n}{20-n} = 1</math>, getting <math>n=10</math>. For all values n less than 0 and greater than 20, the value <math>\frac{n}{20-n}</math> is negative, so now try values of n between 10 and 20. Quick substitution finds <math>0</math>, <math>10</math>, <math>16</math>, and <math>18</math> which yields <math>x=0</math>, <math>x=1</math>, <math>x=2</math>, and <math>x=3</math> respectively. 4 values, or <math>\boxed{(D)}</math> | ||
== See also == | == See also == | ||
+ | {{AMC10 box|year=2002|ab=B|num-b=15|num-a=17}} | ||
{{AMC12 box|year=2002|ab=B|num-b=11|num-a=13}} | {{AMC12 box|year=2002|ab=B|num-b=11|num-a=13}} | ||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 03:53, 3 September 2021
- The following problem is from both the 2002 AMC 12B #12 and 2002 AMC 10B #16, so both problems redirect to this page.
Contents
Problem
For how many integers is the square of an integer?
Solution
Solution 1
Let , with (note that the solutions do not give any additional solutions for ). Then rewriting, . Since , it follows that divides . Listing the factors of , we find that are the only solutions (respectively yielding ).
Solution 2
For and the fraction is negative, for it is not defined, and for it is between 0 and 1.
Thus we only need to examine and .
For and we obviously get the squares and respectively.
For prime the fraction will not be an integer, as the denominator will not contain the prime in the numerator.
This leaves , and a quick substitution shows that out of these only and yield a square. Therefore, there are only solutions (respectively yielding ).
Solution 3
If , then and , otherwise will be negative. Thus and Checking all for which , we have , , , as the possibilities.
~ Nafer
Solution 4
For all integers x, is always a positive integer. So solve for , getting and , getting . For all values n less than 0 and greater than 20, the value is negative, so now try values of n between 10 and 20. Quick substitution finds , , , and which yields , , , and respectively. 4 values, or
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.