https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Mbtint&feedformat=atom AoPS Wiki - User contributions [en] 2022-05-17T18:54:44Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2010_AIME_II_Problems/Problem_7&diff=79785 2010 AIME II Problems/Problem 7 2016-08-01T01:31:13Z <p>Mbtint: /* Problem 7 */</p> <hr /> <div>== Problem 7 ==<br /> &lt;!-- don't remove the following tag, for PoTW on the Wiki front page--&gt;&lt;onlyinclude&gt;Let &lt;math&gt;P(z)=z^3+az^2+bz+c&lt;/math&gt;, where a, b, and c are real. There exists a complex number &lt;math&gt;w&lt;/math&gt; such that the three roots of &lt;math&gt;P(z)&lt;/math&gt; are &lt;math&gt;w+3i&lt;/math&gt;, &lt;math&gt;w+9i&lt;/math&gt;, and &lt;math&gt;2w-4&lt;/math&gt;, where &lt;math&gt;i^2=-1&lt;/math&gt;. Find &lt;math&gt;|a+b+c|&lt;/math&gt;.&lt;!-- don't remove the following tag, for PoTW on the Wiki front page--&gt;&lt;/onlyinclude&gt;<br /> <br /> == Solution ==<br /> Set &lt;math&gt;w=x+yi&lt;/math&gt;, so &lt;math&gt;x_1 = x+(y+3)i&lt;/math&gt;, &lt;math&gt;x_2 = x+(y+9)i&lt;/math&gt;, &lt;math&gt;x_3 = 2x-4+2yi&lt;/math&gt;.<br /> <br /> Since &lt;math&gt;a,b,c\in{R}&lt;/math&gt;, the imaginary part of a,b,c must be 0.<br /> <br /> Start with a, since it's the easiest one to do: &lt;math&gt;y+3+y+9+2y=0, y=-3&lt;/math&gt;,<br /> <br /> and therefore: &lt;math&gt;x_1 = x&lt;/math&gt;, &lt;math&gt;x_2 = x+6i&lt;/math&gt;, &lt;math&gt;x_3 = 2x-4-6i&lt;/math&gt;.<br /> <br /> Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do: <br /> &lt;math&gt;x(x+6i)(2x-4-6i)&lt;/math&gt;. The imaginary part is: &lt;math&gt;6x^2-24x&lt;/math&gt;, which is 0, and therefore x=4, since x=0 doesn't work.<br /> <br /> So now, &lt;math&gt;x_1 = 4, x_2 = 4+6i, x_3 = 4-6i&lt;/math&gt;,<br /> <br /> and therefore: &lt;math&gt;a=-12, b=84, c=-208&lt;/math&gt;. Finally, we have &lt;math&gt;|a+b+c|=|-12+84-208|=\boxed{136}&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{AIME box|year=2010|num-b=6|num-a=8|n=II}}<br /> {{MAA Notice}}</div> Mbtint