Difference between revisions of "2015 AMC 10A Problems/Problem 23"

(Solution)
Line 5: Line 5:
  
  
==Solution==
+
==Solution 1==
 
By Vieta's Formula, <math>a</math> is the sum of the integral zeros of the function, and so <math>a</math> is integral.
 
By Vieta's Formula, <math>a</math> is the sum of the integral zeros of the function, and so <math>a</math> is integral.
  
Line 13: Line 13:
 
<cmath>((a-4) - k)((a-4) + k) = 16.</cmath>
 
<cmath>((a-4) - k)((a-4) + k) = 16.</cmath>
 
Let <math>(a-4) - k = u</math> and <math>(a-4) + k = v</math>; then, <math>a-4 = \dfrac{u+v}{2}</math> and so <math>a = \dfrac{u+v}{2} + 4</math>. Listing all possible <math>(u, v)</math> pairs (not counting transpositions because this does not affect <math>u + v</math>), <math>(2, 8), (4, 4), (-2, -8), (-4, -4)</math>, yields <math>a = 9, 8, -1, 0</math>. These <math>a</math> sum to 16, so our answer is <math>\textbf{(C)}</math>.
 
Let <math>(a-4) - k = u</math> and <math>(a-4) + k = v</math>; then, <math>a-4 = \dfrac{u+v}{2}</math> and so <math>a = \dfrac{u+v}{2} + 4</math>. Listing all possible <math>(u, v)</math> pairs (not counting transpositions because this does not affect <math>u + v</math>), <math>(2, 8), (4, 4), (-2, -8), (-4, -4)</math>, yields <math>a = 9, 8, -1, 0</math>. These <math>a</math> sum to 16, so our answer is <math>\textbf{(C)}</math>.
 +
 +
==Solution 2==
 +
 +
Let <math>r_1</math> and <math>r_2</math> be the integer zeroes of the quadratic.
 +
 +
Since the coefficent of the <math>x^2</math> term is <math>1</math>, the quadratic can be written as <math>(x - r_1)(x - r_2)</math> or <math>x^2 - (r_1 + r_2)x + r_1r_2)</math>.
 +
 +
By comparing this with <math>x^2 - ax + a^2</math>, <math>r_1 + r_2 = a</math> and <math>r_1r_2 = 2a</math>.
 +
 +
Plugging the first equation in the second, <math>r_1r_2 = 2(r_1 + r_2). Rearranging gives </math>r_1r_2 - 2r_1 - 2r_2 = 0<math>.
 +
 +
This can be factored as </math>(r_1 - 2)(r_2 - 2) = 4<math>.
 +
 +
These factors can be: </math>(1, 4), (-1, -4), (4, 1), (-4, -1), (2, 2), (-2, -2)<math>.
 +
 +
We want the number of distinct </math>a = r_1 + r_2<math>, and these factors gives </math>a = {-1, 8, 9}.
 +
 +
So the answer is <math>-1 + 8 + 9 = \boxed{\textbf{(C) }16}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2015|ab=A|before=First Problem|num-a=2}}
 
{{AMC10 box|year=2015|ab=A|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:48, 4 February 2015

Problem

The zeroes of the function $f(x)=x^2-ax+2a$ are integers .What is the sum of the possible values of a?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}}\ 17\qquad\textbf{(E)}\ 18$ (Error compiling LaTeX. ! Extra }, or forgotten $.)


Solution 1

By Vieta's Formula, $a$ is the sum of the integral zeros of the function, and so $a$ is integral.

Because the zeros are integral, the discriminant of the function, $a^2 - 8a$, is a perfect square, say $k^2$. Then adding 16 to both sides and completing the square yields \[(a - 4)^2 = k^2 + 16.\] Hence $(a-4)^2 - k^2 = 16$ and \[((a-4) - k)((a-4) + k) = 16.\] Let $(a-4) - k = u$ and $(a-4) + k = v$; then, $a-4 = \dfrac{u+v}{2}$ and so $a = \dfrac{u+v}{2} + 4$. Listing all possible $(u, v)$ pairs (not counting transpositions because this does not affect $u + v$), $(2, 8), (4, 4), (-2, -8), (-4, -4)$, yields $a = 9, 8, -1, 0$. These $a$ sum to 16, so our answer is $\textbf{(C)}$.

Solution 2

Let $r_1$ and $r_2$ be the integer zeroes of the quadratic.

Since the coefficent of the $x^2$ term is $1$, the quadratic can be written as $(x - r_1)(x - r_2)$ or $x^2 - (r_1 + r_2)x + r_1r_2)$.

By comparing this with $x^2 - ax + a^2$, $r_1 + r_2 = a$ and $r_1r_2 = 2a$.

Plugging the first equation in the second, $r_1r_2 = 2(r_1 + r_2). Rearranging gives$r_1r_2 - 2r_1 - 2r_2 = 0$.

This can be factored as$ (Error compiling LaTeX. ! Missing $ inserted.)(r_1 - 2)(r_2 - 2) = 4$.

These factors can be:$ (Error compiling LaTeX. ! Missing $ inserted.)(1, 4), (-1, -4), (4, 1), (-4, -1), (2, 2), (-2, -2)$.

We want the number of distinct$ (Error compiling LaTeX. ! Missing $ inserted.)a = r_1 + r_2$, and these factors gives$a = {-1, 8, 9}.

So the answer is $-1 + 8 + 9 = \boxed{\textbf{(C) }16}$.

See Also

2015 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
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

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

Invalid username
Login to AoPS