document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209935>Question 1209935</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a, b, and c be positive real numbers.  If a + b + c = 1, then find the minimum value of<BR>\n" );
document.write( "\frac{1}{a} + \frac{1}{b} + \frac{1}{c*a^2} + \frac{2}{ab^2} + \frac{8}{c^3}. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851082 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{2}{ab^2} + \frac{8}{c^3}$.<BR>\n" );
document.write( "We are given that $a, b, c > 0$ and $a + b + c = 1$.\r<BR>\n" );
document.write( "\n" );
document.write( "We will use the AM-GM inequality.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's rewrite the terms:\r<BR>\n" );
document.write( "\n" );
document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{2}{ab^2} + \frac{8}{c^3}$$\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find $a, b, c$ such that the terms are equal.\r<BR>\n" );
document.write( "\n" );
document.write( "Let $\frac{1}{a} = \frac{1}{b} = \frac{1}{ca^2} = \frac{2}{ab^2} = \frac{8}{c^3} = k$.\r<BR>\n" );
document.write( "\n" );
document.write( "Then $a = \frac{1}{k}$, $b = \frac{1}{k}$, $c = \frac{1}{ka^2} = \frac{k}{k} = 1/k$.\r<BR>\n" );
document.write( "\n" );
document.write( "From $\frac{2}{ab^2} = k$, we have $2 = kab^2 = k \cdot \frac{1}{k} \cdot \frac{1}{k^2} = \frac{1}{k^2}$. So $k^2 = \frac{1}{2}$ and $k = \frac{1}{\sqrt{2}}$.\r<BR>\n" );
document.write( "\n" );
document.write( "From $\frac{8}{c^3} = k$, we have $8 = kc^3 = k \cdot \frac{1}{k^3} = \frac{1}{k^2}$. So $k^2 = \frac{1}{8}$ and $k = \frac{1}{2\sqrt{2}}$.\r<BR>\n" );
document.write( "\n" );
document.write( "This leads to a contradiction, so the terms cannot be equal.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's try to apply AM-GM with some adjustments.\r<BR>\n" );
document.write( "\n" );
document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{2}{ab^2} + \frac{8}{c^3}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Let's consider the AM-GM inequality with $a+b+c=1$.<BR>\n" );
document.write( "We can rewrite the expression as:\r<BR>\n" );
document.write( "\n" );
document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{1}{ab^2} + \frac{1}{ab^2} + \frac{8}{c^3}$$\r<BR>\n" );
document.write( "\n" );
document.write( "We have 6 terms.<BR>\n" );
document.write( "By AM-GM,<BR>\n" );
document.write( "$$S \ge 6 \sqrt[6]{\frac{1}{a} \cdot \frac{1}{b} \cdot \frac{1}{ca^2} \cdot \frac{1}{ab^2} \cdot \frac{1}{ab^2} \cdot \frac{8}{c^3}} = 6 \sqrt[6]{\frac{8}{a^4b^5c^4}}$$\r<BR>\n" );
document.write( "\n" );
document.write( "This doesn't seem to lead to a solution.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's try to manipulate the expression to have terms of the form $a, b, c$.<BR>\n" );
document.write( "We have $a+b+c=1$.\r<BR>\n" );
document.write( "\n" );
document.write( "We want to minimize $S$.<BR>\n" );
document.write( "Let $a=1/4, b=1/2, c=1/4$.<BR>\n" );
document.write( "Then $S = 4 + 2 + \frac{1}{(1/4)(1/16)} + \frac{2}{(1/4)(1/4)} + \frac{8}{(1/64)} = 6 + 64 + 32 + 512 = 614$.\r<BR>\n" );
document.write( "\n" );
document.write( "Let $a=1/5, b=2/5, c=2/5$.<BR>\n" );
document.write( "Then $S = 5 + 5/2 + \frac{1}{(2/5)(1/25)} + \frac{2}{(1/5)(4/25)} + \frac{8}{(8/125)} = 5 + 2.5 + 62.5 + 62.5 + 125 = 257.5$.\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{81}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );