document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209936>Question 1209936</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{4}{c*a^2} + \frac{16}{b^4*a} + \frac{32}{a*b^3} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851081 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=851081\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $S = \frac{1}{a} + \frac{1}{b} + \frac{4}{ca^2} + \frac{16}{b^4a} + \frac{32}{ab^3}$.<BR>\n" );
document.write( "We are given that $a, b, c > 0$ and $a + b + c = 1$.\r<BR>\n" );
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document.write( "We will use the AM-GM inequality.\r<BR>\n" );
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document.write( "We have<BR>\n" );
document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{4}{ca^2} + \frac{16}{b^4a} + \frac{32}{ab^3}$$<BR>\n" );
document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{4}{a^2c} + \frac{16}{ab^4} + \frac{32}{ab^3}$$\r<BR>\n" );
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document.write( "Apply AM-GM inequality to the terms:<BR>\n" );
document.write( "$$\frac{1}{a} + \frac{1}{b} + \frac{4}{a^2c} + \frac{16}{ab^4} + \frac{32}{ab^3} \ge 5 \sqrt[5]{\frac{1}{a} \cdot \frac{1}{b} \cdot \frac{4}{a^2c} \cdot \frac{16}{ab^4} \cdot \frac{32}{ab^3}}$$<BR>\n" );
document.write( "$$S \ge 5 \sqrt[5]{\frac{2048}{a^5b^8c}}$$<BR>\n" );
document.write( "This approach does not seem to lead to a solution.\r<BR>\n" );
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document.write( "Let's try to rewrite the expression in a way that allows us to use AM-GM more effectively.\r<BR>\n" );
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document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{4}{a^2c} + \frac{16}{ab^4} + \frac{32}{ab^3}$$\r<BR>\n" );
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document.write( "Let's try to use AM-GM inequality with $a+b+c=1$.\r<BR>\n" );
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document.write( "Consider the terms:<BR>\n" );
document.write( "$$\frac{1}{a} = \frac{1}{a}$$<BR>\n" );
document.write( "$$\frac{1}{b} = \frac{1}{b}$$<BR>\n" );
document.write( "$$\frac{4}{a^2c} = \frac{4}{a^2c}$$<BR>\n" );
document.write( "$$\frac{16}{ab^4} = \frac{16}{ab^4}$$<BR>\n" );
document.write( "$$\frac{32}{ab^3} = \frac{32}{ab^3}$$\r<BR>\n" );
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document.write( "Let's rewrite the terms:<BR>\n" );
document.write( "$$\frac{1}{a} = \frac{1}{a}$$<BR>\n" );
document.write( "$$\frac{1}{b} = \frac{1}{b}$$<BR>\n" );
document.write( "$$\frac{4}{a^2c} = \frac{2}{a^2c} + \frac{2}{a^2c}$$<BR>\n" );
document.write( "$$\frac{16}{ab^4} = \frac{16}{ab^4}$$<BR>\n" );
document.write( "$$\frac{32}{ab^3} = \frac{32}{ab^3}$$\r<BR>\n" );
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document.write( "We want to find positive real numbers $x_1, x_2, \dots, x_n$ such that $x_1a + x_2b + x_3c = 1$.\r<BR>\n" );
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document.write( "Let's use the inequality:<BR>\n" );
document.write( "$$\sum_{i=1}^n \frac{1}{a_i} \ge \frac{n^2}{\sum a_i}$$<BR>\n" );
document.write( "However, this does not apply directly.\r<BR>\n" );
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document.write( "Consider $S = \frac{1}{a} + \frac{1}{b} + \frac{4}{a^2c} + \frac{16}{ab^4} + \frac{32}{ab^3}$.<BR>\n" );
document.write( "We want to minimize $S$.\r<BR>\n" );
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document.write( "We know that $a + b + c = 1$.<BR>\n" );
document.write( "Let's analyze the terms.\r<BR>\n" );
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document.write( "Let $a=1/6, b=1/3, c=1/2$.<BR>\n" );
document.write( "$S = 6 + 3 + \frac{4}{(1/36)(1/2)} + \frac{16}{(1/6)(1/81)} + \frac{32}{(1/6)(1/27)} = 9 + 288 + 864 \cdot 16 + 1728 \cdot 32 = 9 + 288 + 13824 + 55296 = 69417$\r<BR>\n" );
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document.write( "Let $a=1/6, b=1/2, c=1/3$.<BR>\n" );
document.write( "$S = 6 + 2 + \frac{4}{(1/36)(1/3)} + \frac{16}{(1/6)(1/16)} + \frac{32}{(1/6)(1/8)} = 8 + 432 + 96 + 192 = 728$\r<BR>\n" );
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document.write( "Let's try $a=1/10, b=1/5, c=7/10$.\r<BR>\n" );
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document.write( "By AM-GM,<BR>\n" );
document.write( "$$S = \frac{1}{a} + \frac{1}{b} + \frac{4}{a^2c} + \frac{16}{ab^4} + \frac{32}{ab^3} \ge 5\sqrt[5]{\frac{2048}{a^5b^8c}}$$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{144}$<BR>\n" );
document.write( " </SPAN>\n" );
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