document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=744549>Question 744549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are n>=3 people positioned on a field (euclidean plane)so that each has a unique neighbor. Each person has a cream pie.At a signal,everybody hurles his or her pie at the nearest neighbor.assuming that n is odd and that nobody can miss his or her target,true or false:there always remains at least one person not hit by a pie?Explain the method so that i can implement a program on it...<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852667 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53107)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852667\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "There are n>=3 people positioned on a field (Euclidean plane)so that each has a unique <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28closest%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28closest%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > neighbor. <BR>\n" );
document.write( "Each person has a cream pie. At a signal, everybody hurls his or her pie at the nearest neighbor.<BR>\n" );
document.write( "assuming that n is odd and that nobody can miss his or her target, true or false: there always <BR>\n" );
document.write( "remains at least one person not hit by a pie? <BR>\n" );
document.write( "Explain the method so that i can implement a program on it...<BR>\n" );
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document.write( "According to the problem, we have a finite number of points (persons) at the plane, and each point has a unique <BR>\n" );
document.write( "closest point.\r<BR>\n" );
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document.write( "So, let's take a point (a person) 'A', for which this distance to the closest neighbor is maximal (greatest) among<BR>\n" );
document.write( "al other points (persons).\r<BR>\n" );
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document.write( "Since the set of points is finite, we can do it.\r<BR>\n" );
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document.write( "According to the problem, point 'A' has a unique closest point 'B', but by the choice of 'A', <BR>\n" );
document.write( "for point B its closest point is positioned closer to 'B' than 'A'.\r<BR>\n" );
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document.write( "It means that 'B' will not hurl his/her pie at 'A':  'B' will hurl his/her pie to his/her closest neighbor,<BR>\n" );
document.write( "which is different from 'A'.\r<BR>\n" );
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document.write( "So and thus, NOBODY will hurl his/her pie to 'A', so 'A' will remain not hit by a pie.\r<BR>\n" );
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document.write( "Thus, I proved that under given conditions, the answer is \"TRUE\", <BR>\n" );
document.write( "and I constructed/pointed such exclusive person/point explicitly:\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;it is the unique person/point A, whose distance <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;from other people/points is maximal.\r<BR>\n" );
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