OI-Public-LIbrary/国家集训队论文1999-2019/2005/胡伟栋/附件/polygon-1.2.rtf

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\hich\af0\dbch\af36\loch\f0 Version 1.}{\b\deleted\fs24\dbch\af36\revauthdel1\revdttmdel109667828\insrsid4791013\delrsid4220756 \hich\af0\dbch\af36\loch\f0 1}{\b\fs24\cf0\revised\revauth1\revdttm109667828\dbch\af36\insrsid4220756 
\hich\af0\dbch\af36\loch\f0 2}{\b\fs24\dbch\af36\insrsid11802560 
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\hich\af0\dbch\af13\loch\f0 X}{\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 and }{
\i\deleted\f0\fs24\lang1033\langfe1045\revauthdel1\revdttmdel109667815\langnp1033\insrsid7428569\delrsid5511802 \hich\af0\dbch\af13\loch\f0 Q}{\deleted\f0\fs24\lang1033\langfe1045\revauthdel1\revdttmdel109667815\langnp1033\insrsid14754587\delrsid5511802 
\hich\af0\dbch\af13\loch\f0  }{\i\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 \hich\af0\dbch\af13\loch\f0 Y}{\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 
\hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14754587 \hich\af0\dbch\af13\loch\f0 of the polygon, }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 the }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14754587 \hich\af0\dbch\af13\loch\f0 line }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 segmen\hich\af0\dbch\af13\loch\f0 t connecting }{
\i\deleted\f0\fs24\lang1033\langfe1045\revauthdel1\revdttmdel109667815\langnp1033\insrsid14175626\delrsid5511802 \hich\af0\dbch\af13\loch\f0 P}{\deleted\f0\fs24\lang1033\langfe1045\revauthdel1\revdttmdel109667815\langnp1033\insrsid7428569\delrsid5511802 
\hich\af0\dbch\af13\loch\f0  }{\i\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 \hich\af0\dbch\af13\loch\f0 X}{\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 
\hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 and }{\i\deleted\f0\fs24\lang1033\langfe1045\revauthdel1\revdttmdel109667815\langnp1033\insrsid14175626\delrsid5511802 
\hich\af0\dbch\af13\loch\f0 Q}{\deleted\f0\fs24\lang1033\langfe1045\revauthdel1\revdttmdel109667815\langnp1033\insrsid7428569\delrsid5511802 \hich\af0\dbch\af13\loch\f0  }{
\i\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 \hich\af0\dbch\af13\loch\f0 Y}{\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 \hich\af0\dbch\af13\loch\f0  }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 is }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6168808 \hich\af0\dbch\af13\loch\f0 inside }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 the polygon. }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0 All polygons in this task are convex polygons }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6449303 
\hich\af0\dbch\af13\loch\f0 with at least two }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 vertices}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6449303 \hich\af0\dbch\af13\loch\f0 , and all vertices }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16543568 \hich\af0\dbch\af13\loch\f0 in a polygon }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6449303 \hich\af0\dbch\af13\loch\f0 are different}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14754587 
\hich\af0\dbch\af13\loch\f0  and }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6449303 \hich\af0\dbch\af13\loch\f0 have }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14754587 \hich\af0\dbch\af13\loch\f0 integer coordinates}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6449303 \hich\af0\dbch\af13\loch\f0 .}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2429262 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid5787905 \hich\af0\dbch\af13\loch\f0 
No three vertices of the polygon are collinear.}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2429262 \hich\af0\dbch\af13\loch\f0  The }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6688771 \hich\af0\dbch\af13\loch\f0 \hich\f0 word \'93
\hich\af0\dbch\af13\loch\f0 \hich\f0 polygon\'94\loch\f0  below always refers to such polygons.}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid7428569 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid14175626 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 Given two polygons }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626\charrsid14175626 \hich\af0\dbch\af13\loch\f0 ,}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0 the}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  Minkowski sum of }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  consists of all the points of the form (}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 x}{
\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 1}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 +}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 x}{\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 2}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 , }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 y}{\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 1}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 +}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 y}{\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 2}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 ) where (}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 x}{\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 1}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 , }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 y}{
\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 1}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 ) is a point in }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and (}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 x}{
\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 2}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 , }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 y}{\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 2}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 ) is a point in }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 .}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 
\hich\af0\dbch\af13\loch\f0   It turns out}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  that the Minkowski}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7807050 \hich\af0\dbch\af13\loch\f0  sum of polygons }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 is also a}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7807050 \hich\af0\dbch\af13\loch\f0  polygon}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 .  The figure below shows an example: two triangles and their Minkowski sum.}{\f0\lang1033\langfe1045\langnp1033\insrsid7428569 
\par }\pard \s23\qc \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid7428569 {\f0\cf0\revised\lang1033\langfe1045\revauth2\revauth1\revdttm646540433\revdttm109667828\langnp1033\insrsid15034933 
{\pict{\*\picprop\shplid1028{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn fLine}{\sv 0}}{\sp{\sn fPreferRelativeResize}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 1}}}
\picscalex151\picscaley51\piccropl0\piccropr0\piccropt0\piccropb0\picw7620\pich7620\picwgoal4320\pichgoal4320\wmetafile8\bliptag-1704515076\blipupi290{\*\blipuid 9a6729fcb1116ac6d4439947abc834a3}
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\f0\lang1033\langfe1045\langnp1033\insrsid7428569 
\par }\pard \s23\ql \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid7428569 {\f0\lang1033\langfe1045\langnp1033\insrsid7428569 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid7428569 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 We study a reverse ope}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2363506 \hich\af0\dbch\af13\loch\f0 ration to the Minkowski sum}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 .  For a given polygon }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 P}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0 , we are looking for }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 two polygons }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid5787905 
\hich\af0\dbch\af13\loch\f0 such }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 that:
\par {\listtext\pard\plain\s23 \lang1033\langfe1045\dbch\af0\langfenp1045\insrsid7428569 \hich\af0\dbch\af0\loch\f0 -\tab}}\pard \s23\ql \fi-360\li720\ri0\widctlpar\jclisttab\tx720\aspalpha\aspnum\faauto\ls29
\jclisttab\tx720\aspalpha\aspnum\faauto\ls29\pnrauth1\pnrdate109667815\pnrstart0\pnrxst1\pnrxst0\pnrxst45\pnrxst0\pnrstop4\pnrstart1\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrstop9
\pnrstart2\pnrnfc23\pnrnfc23\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrstop18\pnrstart3\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0
\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0
\pnrpnbr0\pnrpnbr0\pnrstop36\adjustright\rin0\lin720\itap0\pararsid7428569 {\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 P}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 
 is the Minkowski sum of }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 ,  
\par {\listtext\pard\plain\s23 \lang1033\langfe1045\dbch\af0\langfenp1045\insrsid7428569 \hich\af0\dbch\af0\loch\f0 -\tab}}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  has }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6054904 \hich\af0\dbch\af13\loch\f0 from 2 to 4 different vertices, }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 i.e. it is a segment}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6054904 \hich\af0\dbch\af13\loch\f0  (2 vertices)}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 , a triangle }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6054904 \hich\af0\dbch\af13\loch\f0 (3 vertices) }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0 or a quadrilateral}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6054904 \hich\af0\dbch\af13\loch\f0  (4 vertices}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 ), 
\par {\listtext\pard\plain\s23 \lang1033\langfe1045\dbch\af0\langfenp1045\insrsid7428569 \hich\af0\dbch\af0\loch\f0 -\tab}}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  should have as many vertices, as possible, i.e.:
\par {\listtext\pard\plain\s23 \f10\lang1033\langfe1045\langfenp1045\insrsid7428569 \loch\af10\dbch\af13\hich\f10 \'a7\tab}}\pard \s23\ql \fi-360\li1440\ri0\widctlpar\jclisttab\tx1440\aspalpha\aspnum\faauto\ls30\ilvl1
\jclisttab\tx1440\aspalpha\aspnum\faauto\ls30\ilvl1\pnrauth1\pnrdate109667815\pnrstart0\pnrxst1\pnrxst0\pnrxst167\pnrxst240\pnrstop4\pnrstart1\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrrgb0\pnrstop9
\pnrstart2\pnrnfc23\pnrnfc23\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc0\pnrnfc1\pnrnfc0\pnrnfc0\pnrstop18\pnrstart3\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr1\pnrpnbr0\pnrpnbr0
\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0\pnrpnbr0
\pnrpnbr0\pnrpnbr0\pnrstop36\adjustright\rin0\lin1440\itap0\pararsid7428569 {\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 
 should be a quadrilateral, if possible, 
\par {\listtext\pard\plain\s23 \f10\lang1033\langfe1045\langfenp1045\insrsid7428569 \loch\af10\dbch\af13\hich\f10 \'a7\tab}\hich\af0\dbch\af13\loch\f0 if }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  cannot be a quadrilateral, it should be a triangle, if possible, 
\par {\listtext\pard\plain\s23 \f10\lang1033\langfe1045\langfenp1045\insrsid7428569 \loch\af10\dbch\af13\hich\f10 \'a7\tab}\hich\af0\dbch\af13\loch\f0 otherwise it should be a segment. 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid7428569 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 
\par }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 Clearly, neither }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 
\hich\af0\dbch\af13\loch\f0  nor }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  can }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid5787905 \hich\af0\dbch\af13\loch\f0 be }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 equal }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid5787905 
\hich\af0\dbch\af13\loch\f0 to }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 P}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 
 because then the other summand would have to be a poi\hich\af0\dbch\af13\loch\f0 nt, which is not a valid polygon.
\par }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid14175626 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 
You are given a set of input files, each containing a description of a polygon }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 P}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 
.  For e}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0 ach input file you should find}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0 the}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 
\hich\af0\dbch\af13\loch\f0 polygons }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569\charrsid14175626 \hich\af0\dbch\af13\loch\f0 ,}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626\charrsid14175626 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0 as required above, }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 and create an output file containing descriptions of }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0  and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2363506 
\hich\af0\dbch\af13\loch\f0 .\hich\af0\dbch\af13\loch\f0   }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 For the given input files such}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 
\hich\af0\dbch\af13\loch\f0  polygons }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626\charrsid5787905 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0  and }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626\charrsid5787905 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid14175626 \hich\af0\dbch\af13\loch\f0  can always be found}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2363506 \hich\af0\dbch\af13\loch\f0 .}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0   If there are many correct results, you should }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid15007950 \hich\af0\dbch\af13\loch\f0 find and output }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid7428569 \hich\af0\dbch\af13\loch\f0 
one of them.  You should not submit any programs, just the output files. }{\f0\lang1033\langfe1045\langnp1033\insrsid7428569 
\par }\pard\plain \ql \li0\ri0\nowidctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang1033\langfe2052\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\fs24\dbch\af36\insrsid11802560 
\par }\pard\plain \s1\ql \li0\ri0\keepn\nowidctlpar\faauto\outlinelevel0\adjustright\rin0\lin0\itap0 \b\fs24\lang1033\langfe2052\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\dbch\af36\insrsid11802560 \hich\af0\dbch\af36\loch\f0 INPUT
\par }\pard\plain \ql \li0\ri0\nowidctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang1033\langfe2052\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\fs24\insrsid11802560 
\par }\pard\plain \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid11670116 \fs20\lang1045\langfe1045\loch\af2\hich\af2\dbch\af13\cgrid\langnp1045\langfenp1045 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 
\hich\af0\dbch\af13\loch\f0 You are given 10 problem instances in the text files named }{\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af2\dbch\af13\loch\f2 polygon1.in }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 
\hich\af0\dbch\af13\loch\f0 to}{\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af2\dbch\af13\loch\f2  polygon10.}{\fs24\lang1033\langfe1045\langnp1033\insrsid11670116\charrsid11670116 \hich\af2\dbch\af13\loch\f2 in}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 , where the number after }{\fs24\lang1033\langfe1045\langnp1033\insrsid11670116\charrsid11670116 \hich\af2\dbch\af13\loch\f2 polygon}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0  is the input number. Each input file is organized as follows. }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 
The first line contains one integer }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 N}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 :}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  the number of vertices of the polygo\hich\af0\dbch\af13\loch\f0 n }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 P}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 .  The following }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 N}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0  lines describe the vertices in a counter-clockwise order, one vertex per line.  Line }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 I}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 +1 (for }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 I}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0  = 1, 2, ..., }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 N}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 ) contains two integers }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 X}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 I}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0  and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 Y}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 I}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 , separated by a space:}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  coordinates of the }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 \hich\af0\dbch\af13\loch\f0 I}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 th vertex of the polygon.  All }{
\f0\fs24\cf0\revised\lang1033\langfe1045\revauth1\revdttm109667815\langnp1033\insrsid5511802 \hich\af0\dbch\af13\loch\f0 input }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 coordin\hich\af0\dbch\af13\loch\f0 
ates are non-negative integers.
\par }\pard\plain \s2\ql \li0\ri0\keepn\nowidctlpar\faauto\outlinelevel1\adjustright\rin0\lin0\itap0 \b\fs20\lang1033\langfe2052\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\fs24\insrsid11802560 
\par \hich\af0\dbch\af13\loch\f0 OUTPUT
\par }\pard\plain \ql \li0\ri0\nowidctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang1033\langfe2052\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\b\fs24\insrsid11802560 
\par }\pard\plain \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid16256136 \fs20\lang1045\langfe1045\loch\af2\hich\af2\dbch\af13\cgrid\langnp1045\langfenp1045 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 
\hich\af0\dbch\af13\loch\f0 You are to submit 10 output files corresponding to the given input files}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6822233 \hich\af0\dbch\af13\loch\f0  which describe}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0  the required polygons }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0  and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 
\hich\af0\dbch\af13\loch\f0 .}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0  The first line is to contain }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6822233 \hich\af0\dbch\af13\loch\f0 the }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 text}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid6822233 \hich\af0\dbch\af13\loch\f0 :}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid11670116 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid12403209 {\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af2\dbch\af13\loch\f2 #FILE polygon I}{
\fs24\lang1033\langfe1045\langnp1033\insrsid16256136\charrsid16256136 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid16256136 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 where integer }{
\fs24\lang1033\langfe1045\langnp1033\insrsid16256136\charrsid16256136 \hich\af2\dbch\af13\loch\f2 I}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid15007950 
\hich\af0\dbch\af13\loch\f0 (1}{\fs24\dbch\af36\insrsid15007950 \loch\af2\dbch\af36\hich\f2 \u8804\'3d}{\fs24\lang1033\langfe1045\langnp1033\insrsid15007950\charrsid15007950 \hich\af2\dbch\af13\loch\f2 I}{\fs24\dbch\af36\insrsid15007950 
\loch\af2\dbch\af36\hich\f2 \u8804\'3d}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid15007950 \hich\af0\dbch\af13\loch\f0 10)}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 is the numbe}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid4548687 \hich\af0\dbch\af13\loch\f0 r of the respective input file.}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 
\par 
\par }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 The}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3806791 \hich\af0\dbch\af13\loch\f0  output format is similar to the input format}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 . The second }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2980745 \hich\af0\dbch\af13\loch\f0 line }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 
\hich\af0\dbch\af13\loch\f0 is to contain one integer }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 : the number of vertices in }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136\charrsid15007950 \hich\af0\dbch\af13\loch\f0 A}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 (}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0 2}{\fs24\dbch\af36\insrsid16256136 \loch\af2\dbch\af36\hich\f2 \u8804\'3d}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 N}{
\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 A}{\fs24\dbch\af36\insrsid16256136 \loch\af2\dbch\af36\hich\f2 \u8804\'3d}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0 4}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 ).}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  The following }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0  lines describe the vertices of }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0  in the}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  counter-clockwise order, one vertex per line.  Line }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 I}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid1144037 \hich\af0\dbch\af13\loch\f0 +2}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  (for }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 
\hich\af0\dbch\af13\loch\f0 I}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  = 1, 2, ..., }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 N}{
\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 ) contains two integers }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 X}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 
\hich\af0\dbch\af13\loch\f0 Y}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 , separated by a space: }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 coordinates of the }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 I}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 th vertex of the polygon }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 . }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 
\par 
\par }\pard \s23\qj \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid3632309 {\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 Line }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 
\hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid16256136\charrsid16256136 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 +}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid1144037 \hich\af0\dbch\af13\loch\f0 3}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  should contain one integer }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 
\hich\af0\dbch\af13\loch\f0 : the number }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid2363506 \hich\af0\dbch\af13\loch\f0 of }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid16256136 \hich\af0\dbch\af13\loch\f0 vertices in}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0 , }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 (2}{\fs24\dbch\af36\insrsid3632309 \loch\af2\dbch\af36\hich\f2 \u8804\'3d}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 
\hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid3632309\charrsid3632309 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 ). }{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 The following }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid3632309 
\hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  lines describe the vertices of }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 B}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0  in the}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  counter-clockwise order, one vertex per line.  Line }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 N}{\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 A}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0 +}{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 J}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 +}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid1144037 \hich\af0\dbch\af13\loch\f0 3}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  (for }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 
\hich\af0\dbch\af13\loch\f0 J}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  = 1, 2, ..., }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 N}{
\i\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 B}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 ) contains two integers }{
\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 X}{\f0\fs24\lang1033\langfe1045\sub\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0  }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 
\hich\af0\dbch\af13\loch\f0 and }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 Y}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 , separated by a space}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 \hich\af0\dbch\af13\loch\f0 : }{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 coordinates of the }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid3632309 
\hich\af0\dbch\af13\loch\f0 J}{\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 th vertex of the polygon }{\i\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 B}{
\f0\fs24\lang1033\langfe1045\langnp1033\insrsid12403209 \hich\af0\dbch\af13\loch\f0 . 
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\par }\pard \s2\ql \li0\ri0\keepn\widctlpar\faauto\outlinelevel1\adjustright\rin0\lin0\itap0 {\cf0\revised\lang1024\langfe1024\revauth2\revdttm646540433\noproof\insrsid13592033 
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\hich\af0\dbch\af13\loch\f0 For the above input, either of the }{\fs24\insrsid2363506 \hich\af0\dbch\af13\loch\f0 below }{\fs24\insrsid3632309\charrsid3632309 \hich\af0\dbch\af13\loch\f0 output files (see also }{\fs24\insrsid2899922 
\hich\af0\dbch\af13\loch\f0 the }{\fs24\insrsid3632309\charrsid3632309 \hich\af0\dbch\af13\loch\f0 figure}{\fs24\insrsid2899922 \hich\af0\dbch\af13\loch\f0 s}{\fs24\insrsid3632309\charrsid3632309 \hich\af0\dbch\af13\loch\f0 
) is correct, since in both cases }{\i\fs24\insrsid3632309\charrsid3632309 \hich\af0\dbch\af13\loch\f0 A}{\fs24\insrsid3632309\charrsid3632309 \hich\af0\dbch\af13\loch\f0  is a triangle and it cannot be a quadrilateral. 
\par }{\cf0\revised\lang1024\langfe1024\revauth2\revdttm646540433\noproof\insrsid13592033 {\shp{\*\shpinst\shpleft3624\shptop2642\shpright7120\shpbottom3983\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr2\shpwrk0\shpfblwtxt0\shpz3\shplid1031
{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn pib}{\sv {\pict\picscalex33\picscaley33\piccropl0\piccropr0\piccropt0\piccropb0
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