Difference between revisions of "2014 AMC 12A Problems/Problem 19"

Latest revision as of 19:33, 4 April 2024

Problem

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$ . What is $N$ ?

$\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$

Solution 1

Factor the quadratic into $\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0$ where $-n$ is our integer solution. Then, $k = \frac{12}{n} + 5n,$ which takes rational values between $-200$ and $200$ when $|n| \leq 39$ , excluding $n = 0$ . This leads to an answer of $2 \cdot 39 = \boxed{\textbf{(E) } 78}$ .

Solution 2

Solve for $k$ so $k=-\frac{12}{x}-5x.$ Note that $x$ can be any integer in the range $[-39,0)\cup(0,39]$ so $k$ is rational with $\lvert k\rvert<200$ . Hence, there are $39+39=\boxed{\textbf{(E) } 78}.$

Solution 3

Plug in $k=200$ to find the upper limit. You will find the limit to be a number from $0<x<-1$ and one that is just below $-39.$ All the integer values from $-1$ to $-39$ can be attainable through some value of $k$ . Since the question asks for the absolute value of $k$ , we see that the answer is $39\cdot2 = \boxed{\textbf{(E) }78.}$

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Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=BoPnuYKBq30

2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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All AMC 12 Problems and Solutions

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

@@ Line 19: / Line 19: @@
 ==Solution 3==
-Plug in <math>k=200</math> to find the upper limit. You will find the limit to be a number from <math>0<x<-1</math> and one that is just below <math>-39.</math> All the integer values from <math>-1</math> to <math>-39</math> can be attainable through some value of <math>k</math>. Since the question asks for the absolute value of <math>k</math>, we see that the answer is <math>39\cdot2 = \boxed{\textbf{(C)  }78.}</math>
+Plug in <math>k=200</math> to find the upper limit. You will find the limit to be a number from <math>0<x<-1</math> and one that is just below <math>-39.</math> All the integer values from <math>-1</math> to <math>-39</math> can be attainable through some value of <math>k</math>. Since the question asks for the absolute value of <math>k</math>, we see that the answer is <math>39\cdot2 = \boxed{\textbf{(E)  }78.}</math>
 iron

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