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2002 IMO Shortlist Problems/N1 - Revision history
2024-03-28T17:59:54Z
Revision history for this page on the wiki
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Scorpius119 at 19:53, 24 March 2007
2007-03-24T19:53:38Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:53, 24 March 2007</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{<del class="diffchange diffchange-inline">solution</del>}}</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Observe that <math>2002^</ins>{<ins class="diffchange diffchange-inline">2002}\equiv 4^{2002}\equiv 64^{667}\cdot 4\equiv 4\pmod{9}</math>. On the other hand, each cube is congruent to 0, 1, or -1 modulo 9. So a sum of at most three cubes modulo 9 must among <math>0,\pm 1,\pm 2,\pm 3</math> none of which are congruent to 4. Therefore <math>t\geq 4</math>.</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">To show that '''4''' is the minimum value of <math>t</math>, note that</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><math>(10\cdot 2002^{667})^3+(10\cdot 2002^{667})^3+(2002^</ins>{<ins class="diffchange diffchange-inline">667</ins>}<ins class="diffchange diffchange-inline">)^3+(2002^{667})^3=2002^{2002</ins>}<ins class="diffchange diffchange-inline"></math></ins></div></td></tr>
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Scorpius119
https://artofproblemsolving.com/wiki/index.php?title=2002_IMO_Shortlist_Problems/N1&diff=12034&oldid=prev
Boy Soprano II: category
2006-12-24T21:05:29Z
<p>category</p>
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Boy Soprano II
https://artofproblemsolving.com/wiki/index.php?title=2002_IMO_Shortlist_Problems/N1&diff=12031&oldid=prev
Chess64 at 20:50, 24 December 2006
2006-12-24T20:50:51Z
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Chess64
https://artofproblemsolving.com/wiki/index.php?title=2002_IMO_Shortlist_Problems/N1&diff=12030&oldid=prev
Chess64 at 20:50, 24 December 2006
2006-12-24T20:50:14Z
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<p><b>New page</b></p><div>==Problem==<br />
What is the smallest positive integer <math>t</math> such that there exist integers <math>x_1,x_2,\ldots,x_t</math> with <p><center><math>x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}</math>?</center></p><br />
==Solution==<br />
<br />
{{solution}}<br />
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*[[2002 IMO Shortlist]]</div>
Chess64