115 lines
6.7 KiB
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115 lines
6.7 KiB
HTML
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<h2>Convex Holes</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 252</h3><span style="width:300px;color:#666;">Published on Saturday, 27th June 2009, 01:00 pm; Solved by 555</span></div>
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Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).
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</p>
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<p>
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As an example, the image below shows a set of twenty points and a few such convex holes.
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The convex hole shown as a red heptagon has an area equal to 1049694.5 square units, which is the highest possible area for a convex hole on the given set of points.
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<div style="text-align:center;">
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<img src="project/images/p_252_convexhole.gif" alt="" /></div>
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<style type="text/css">
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table.p252 td {
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padding: 0px 3px 0px 3px;
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}
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<p>For our example, we used the first 20 points (<var>T</var><sub>2<var>k</var><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1</sub>, <var>T</var><sub>2<var>k</var></sub>), for <var>k</var> = 1,2,…,20, produced with the pseudo-random number generator:</p>
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<center><table class="p252">
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<tr>
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<td style="text-align:right"><var>S</var><sub>0</sub></td>
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<td>=<sub> </sub></td>
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<td>290797<sub> </sub></td>
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</tr><tr>
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<td><var>S</var><sub><var>n</var>+1</sub></td>
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<td>=<sub> </sub></td>
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<td><var>S</var><sub><var>n</var></sub><sup>2</sup> mod 50515093</td>
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</tr><tr>
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<td style="text-align:right"><var>T</var><sub><var>n</var></sub></td>
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<td>=<sub> </sub></td>
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<td>( <var>S</var><sub><var>n</var></sub> mod 2000 ) <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 1000<sup> </sup></td>
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</tr>
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</table></center>
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<p>
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<i>i.e.</i> (527, 144), (<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />488, 732), (<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />454, <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />947), …
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</p>
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<p>
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What is the maximum area for a convex hole on the set containing the first 500 points in the pseudo-random sequence?<br /> Specify your answer including one digit after the decimal point.
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