document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202567>Question 1202567</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of 2000 numbers is 2007. Find maximum product of these numbers. (There are more than one set of such numbers)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837517 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=837517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "By a general principle, given a set of numbers with a particular sum, the maximum product of the numbers is when the numbers are all equal.<br><BR>\n" );
document.write( "2000 numbers, all equal, with a sum of 2007, means each number is 2007/2000.<br><BR>\n" );
document.write( "The maximum product is<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282007%2F2000%29%5E2000 ALIGN=MIDDLE  ALT=\"%282007%2F2000%29%5E2000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Note the problem is not defined precisely enough.  If some of the numbers are allowed to be negative, then there is no maximum product.  Given a positive number x, the set consisting of -x, -1, 1997 1's, and x+11 has a sum of 2007, and the product of the numbers is x(x+11), which clearly has no maximum.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );