83 lines
5.7 KiB
HTML
83 lines
5.7 KiB
HTML
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<h2>Uphill paths</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 411</h3><span style="width:300px;color:#666;">Published on Saturday, 19th January 2013, 10:00 pm; Solved by 270</span></div>
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<p>
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Let <var>n</var> be a positive integer. Suppose there are stations at the coordinates (<var>x</var>, <var>y</var>) = (2<sup><var>i</var></sup> mod <var>n</var>, 3<sup><var>i</var></sup> mod <var>n</var>) for 0 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>i</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 2<var>n</var>. We will consider stations with the same coordinates as the same station.
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</p><p>
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We wish to form a path from (0, 0) to (<var>n</var>, <var>n</var>) such that the x and y coordinates never decrease.<br />
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Let S(<var>n</var>) be the maximum number of stations such a path can pass through.
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For example, if <var>n</var> = 22, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, S(22) = 5.
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The case is illustrated below, with an example of an optimal path:
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<p>
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It can also be verified that S(123) = 14 and S(10000) = 48.
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</p><p>
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Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /> S(<var>k</var><sup>5</sup>) for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>k</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 30.
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