76 lines
6.1 KiB
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76 lines
6.1 KiB
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<h2>Best Approximations</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 192</h3><span style="width:300px;color:#666;">Published on Saturday, 3rd May 2008, 05:00 am; Solved by 825</span></div>
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<p>Let <var>x</var> be a real number.<br />
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A <i>best approximation</i> to <var>x</var> for the <i>denominator bound</i> <var>d</var> is a rational number <var>r</var>/<var>s</var> <i>in reduced form</i>, with <var>s</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>d</var>, such that any rational number which is closer to <var>x</var> than <var>r</var>/<var>s</var> has a denominator larger than <var>d</var>:</p>
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<div style="text-align:center;">|<var>p</var>/<var>q</var>-<var>x</var>| <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> |<var>r</var>/<var>s</var>-<var>x</var>| <img src='images/symbol_implies.gif' width='15' height='11' alt='⇒' border='0' style='vertical-align:middle;' /> <var>q</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> <var>d</var></div>
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<p>For example, the best approximation to <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />13 for the denominator bound 20 is 18/5 and the best approximation to <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />13 for the denominator bound 30 is 101/28.</p>
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<p>Find the sum of all denominators of the best approximations to <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' /><var>n</var> for the denominator bound 10<sup>12</sup>, where <var>n</var> is not a perfect square and 1 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 100000.</p>
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