2013 AMC 12B Problems/Problem 17

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Let $a,b,$ and $c$ be real numbers such that

\[a+b+c=2, \text{ and}\] \[a^2+b^2+c^2=12\]

What is the difference between the maximum and minimum possible values of $c$?

$\text{(A) }2\qquad \text{ (B) }\frac{10}{3}\qquad \text{ (C) }4 \qquad \text{ (D) }\frac{16}{3}\qquad \text{ (E) }\frac{20}{3}$

Solution 1

$a+b= 2-c$. Now, by Cauchy-Schwarz, we have that $(a^2+b^2) \ge \frac{(2-c)^2}{2}$. Therefore, we have that $\frac{(2-c)^2}{2}+c^2 \le 12$. We then find the roots of $c$ that satisfy equality and find the difference of the roots. This gives the answer, $\boxed{\textbf{(D)} \ \frac{16}{3}}$.

Solution 2

This is similar to the first solution but is far more intuitive. From the given, we have \[a + b = 2 - c\] \[a^2 + b^2 = 12 - c^2\] This immediately suggests use of the Cauchy-Schwarz inequality. By Cauchy, we have \[2\,(a^2 + b^2) \geq (a + b)^2\] Substitution of the above results and some algebra yields \[3c^2 - 4c - 20 \leq 0\] This quadratic inequality is easily solved, and it is seen that equality holds for $c = -2$ and $c = \frac{10}{3}$.

The difference between these two values is $\boxed{\textbf{(D)} \ \frac{16}{3}}$.

Solution 3

(no Cauchy-Schwarz)

From the first equation, we know that $c=2-a-b$. We substitute this into the second equation to find that \[a^2+b^2+(2-a-b)^2=12.\] This simplifies to $2a^2+2b^2-4a-4b+2ab=8$, which we can write as the quadratic $a^2+(b-2)a+(b^2-2b-4)=0$. We wish to find real values for $a$ and $b$ that satisfy this equation. Therefore, the discriminant is nonnegative. Hence, \[(b-2)^2-4(b^2-2b-4)\ge0,\] or $-3b^2+4b+20\ge 0$. This factors as $-(3b-10)(b+2)\ge 0$. Therefore, $-2\le b\le \frac{10}{3}$, and by symmetry this must be true for $a$ and $c$ as well.

Now $a=b=2$ and $c=-2$ satisfy both equations, so we see that $c=-2$ must be the minimum possible value of $c$. Also, $c=\frac{10}{3}$ and $a=b=-\frac{2}{3}$ satisfy both equations, so we see that $c=\frac{10}{3}$ is the maximum possible value of $c$. The difference between these is $\frac{10}{3}-(-2)=\frac{16}{3}$, or $\boxed{\textbf{(D)}}$.

See also

2013 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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

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