87 lines
7.3 KiB
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87 lines
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HTML
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<h2>Cutting Rectangular Grid Paper</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 338</h3><span style="width:300px;color:#666;">Published on Sunday, 15th May 2011, 01:00 am; Solved by 229</span></div>
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<p>A rectangular sheet of grid paper with integer dimensions <var>w</var> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <var>h</var> is given. Its grid spacing is 1.<br />
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When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.</p>
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<p>For example, from a sheet with dimensions 9 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 4 , we can make rectangles with dimensions 18 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 2, 12 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 3 and 6 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 6 by cutting and rearranging as below:</p>
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<p>Similarly, from a sheet with dimensions 9 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 8 , we can make rectangles with dimensions 18 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 4 and 12 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 6 .</p>
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<p>For a pair <var>w</var> and <var>h</var>, let F(<var>w</var>,<var>h</var>) be the number of distinct rectangles that can be made from a sheet with dimensions <var>w</var> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <var>h</var> .<br />
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For example, F(2,1) = 0, F(2,2) = 1, F(9,4) = 3 and F(9,8) = 2. <br />
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Note that rectangles congruent to the initial one are not counted in F(<var>w</var>,<var>h</var>).<br />
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Note also that rectangles with dimensions <var>w</var> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <var>h</var> and dimensions <var>h</var> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <var>w</var> are not considered distinct.</p>
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<p>For an integer <var>N</var>, let G(<var>N</var>) be the sum of F(<var>w</var>,<var>h</var>) for all pairs <var>w</var> and <var>h</var> which satisfy 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>h</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>w</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>N</var>.<br />
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We can verify that G(10) = 55, G(10<sup>3</sup>) = 971745 and G(10<sup>5</sup>) = 9992617687.</p>
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<p>Find G(10<sup>12</sup>). Give your answer modulo 10<sup>8</sup>.</p>
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