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Difference between revisions of "1986 AHSME Problems/Problem 30"

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Consider the cases <math>x>0</math> and <math>x<0</math>, and also note that by AM-GM, for any positive number <math>a</math>, we have <math>a+\frac{17}{a} \geq 2\sqrt{17}</math>, with equality only if <math>a = \sqrt{17}</math>. Thus, if <math>x>0</math>, considering each equation in turn, we get that <math>y \geq \sqrt{17}, z \geq \sqrt{17}, w \geq \sqrt{17}</math>, and finally <math>x \geq \sqrt{17}</math>.  
 
Consider the cases <math>x>0</math> and <math>x<0</math>, and also note that by AM-GM, for any positive number <math>a</math>, we have <math>a+\frac{17}{a} \geq 2\sqrt{17}</math>, with equality only if <math>a = \sqrt{17}</math>. Thus, if <math>x>0</math>, considering each equation in turn, we get that <math>y \geq \sqrt{17}, z \geq \sqrt{17}, w \geq \sqrt{17}</math>, and finally <math>x \geq \sqrt{17}</math>.  
  
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Now suppose <math>x > \sqrt{17}</math>. Then <math>y - \sqrt{17} = \frac{x^{2}+17}{2x} - \sqrt{17} = (\frac{x-\sqrt{17}}{2x})(x-\sqrt{17}) < \frac{1}{2}(x-\sqrt{17})</math>, so that <math>x > y</math>. Similarly, we can get <math>y > z</math>, <math>z > w</math>, and <math>w > x</math>, and combining these gives <math>x > x</math>, an obvious contradiction.
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Now suppose <math>x > \sqrt{17}</math>.  
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Then <math>y - \sqrt{17} = \frac{x^{2}+17}{2x} - \sqrt{17} = (\frac{x-\sqrt{17}}{2x})(x-\sqrt{17}) < \frac{1}{2}(x-\sqrt{17})</math>, so that <math>x > y</math>. Similarly, we can get <math>y > z</math>, <math>z > w</math>, and <math>w > x</math>, and combining these gives <math>x > x</math>, an obvious contradiction.
  
 
Thus we must have <math>x \geq \sqrt{17}</math>, but <math>x \ngtr \sqrt{17}</math>, so if <math>x > 0</math>, the only possibility is <math>x = \sqrt{17}</math>, and analogously from the other equations we get <math>x = y = z = w = \sqrt{17}</math>; indeed, by substituting, we verify that this works.
 
Thus we must have <math>x \geq \sqrt{17}</math>, but <math>x \ngtr \sqrt{17}</math>, so if <math>x > 0</math>, the only possibility is <math>x = \sqrt{17}</math>, and analogously from the other equations we get <math>x = y = z = w = \sqrt{17}</math>; indeed, by substituting, we verify that this works.
  
 
As for the other case, <math>x < 0</math>, notice that <math>(x,y,z,w)</math> is a solution if and only if <math>(-x,-y,-z,-w)</math> is a solution, since this just negates both sides of each equation and so they are equivalent. Thus the only other solution is <math>x = y = z = w = -\sqrt{17}</math>, so that we have <math>2</math> solutions in total, and therefore the answer is <math>\boxed{B}</math>.
 
As for the other case, <math>x < 0</math>, notice that <math>(x,y,z,w)</math> is a solution if and only if <math>(-x,-y,-z,-w)</math> is a solution, since this just negates both sides of each equation and so they are equivalent. Thus the only other solution is <math>x = y = z = w = -\sqrt{17}</math>, so that we have <math>2</math> solutions in total, and therefore the answer is <math>\boxed{B}</math>.
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== See also ==
 
== See also ==

Revision as of 13:04, 21 June 2019

Problem

The number of real solutions $(x,y,z,w)$ of the simultaneous equations $2y = x + \frac{17}{x}, 2z = y + \frac{17}{y}, 2w = z + \frac{17}{z}, 2x = w + \frac{17}{w}$ is

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

Solution

Consider the cases $x>0$ and $x<0$, and also note that by AM-GM, for any positive number $a$, we have $a+\frac{17}{a} \geq 2\sqrt{17}$, with equality only if $a = \sqrt{17}$. Thus, if $x>0$, considering each equation in turn, we get that $y \geq \sqrt{17}, z \geq \sqrt{17}, w \geq \sqrt{17}$, and finally $x \geq \sqrt{17}$.

Now suppose $x > \sqrt{17}$. Then $y - \sqrt{17} = \frac{x^{2}+17}{2x} - \sqrt{17} = (\frac{x-\sqrt{17}}{2x})(x-\sqrt{17}) < \frac{1}{2}(x-\sqrt{17})$, so that $x > y$. Similarly, we can get $y > z$, $z > w$, and $w > x$, and combining these gives $x > x$, an obvious contradiction.

Thus we must have $x \geq \sqrt{17}$, but $x \ngtr \sqrt{17}$, so if $x > 0$, the only possibility is $x = \sqrt{17}$, and analogously from the other equations we get $x = y = z = w = \sqrt{17}$; indeed, by substituting, we verify that this works.

As for the other case, $x < 0$, notice that $(x,y,z,w)$ is a solution if and only if $(-x,-y,-z,-w)$ is a solution, since this just negates both sides of each equation and so they are equivalent. Thus the only other solution is $x = y = z = w = -\sqrt{17}$, so that we have $2$ solutions in total, and therefore the answer is $\boxed{B}$.

See also

1986 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 29 		Followed by
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