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2001 JBMO Problems/Problem 1 - Revision history
2024-03-29T10:06:50Z
Revision history for this page on the wiki
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Rockmanex3 at 01:48, 12 August 2018
2018-08-12T01:48:16Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 01:48, 12 August 2018</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;"><div><br></div></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;"><div><br></div></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>Now <math>b^3 + c^3 = 1001.</math>  Since <math>b^3 \ge c^3,</math> we find that <math>2b^3 \ge 1001.</math>  <del class="diffchange diffchange-inline">Thus, </del><math>b = 10</math> and <math>c = 1.</math></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>Now <math>b^3 + c^3 = 1001.</math>  Since <math>b^3 \ge c^3,</math> we find that <math>2b^3 \ge 1001.</math>  <ins class="diffchange diffchange-inline">That means </ins><math>b = 10</math> and <math>c = 1.</math></div></td></tr>
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Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=2001_JBMO_Problems/Problem_1&diff=97187&oldid=prev
Rockmanex3: Completely FIXED the solution
2018-08-12T01:47:23Z
<p>Completely FIXED the solution</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 01:47, 12 August 2018</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;"><div>==Solution==</div></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;"><div>==Solution==</div></td></tr>
<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>Note that for all positive integers <math>n,</math> the value <math>n^3</math> is congruent to <math>-1,0,1</math> [[modulo]] <math>9.</math>  Since <math>2001 \equiv 3 \pmod{9},</math> we find that <math>a,b,c \equiv 1 \pmod{9}.</math></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>Note that for all positive integers <math>n,</math> the value <math>n^3</math> is congruent to <math>-1,0,1</math> [[modulo]] <math>9.</math>  Since <math>2001 \equiv 3 \pmod{9},</math> we find that <math>a<ins class="diffchange diffchange-inline">^3</ins>,b<ins class="diffchange diffchange-inline">^3</ins>,c<ins class="diffchange diffchange-inline">^3 </ins>\equiv 1 \pmod{9}<ins class="diffchange diffchange-inline">.</math> Thus, <math>a,b,c \equiv 1 \pmod{3},</math> and the only numbers congruent to <math>1</math> modulo <math>3</math> are <math>1,4,7,10</ins>.</math></div></td></tr>
<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="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;"><div><br></div></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;"><div><br></div></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">The only numbers congruent to </del><math><del class="diffchange diffchange-inline">1</del></math> <del class="diffchange diffchange-inline">modulo </del><math><del class="diffchange diffchange-inline">9</del></math> <del class="diffchange diffchange-inline">that are also less than </del><math>2001</math> <del class="diffchange diffchange-inline">are </del><math>10</math> <del class="diffchange diffchange-inline">and </del><math><del class="diffchange diffchange-inline">1</del>.</math>  <del class="diffchange diffchange-inline">With some quick checking</del>, we find that the only solutions are <math>\boxed{(10,10,1),(10,1,10),(1,10,10)}.</math></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">[[WLOG]], let </ins><math><ins class="diffchange diffchange-inline">a \ge b \ge c.</ins></math> <ins class="diffchange diffchange-inline"> That means </ins><math><ins class="diffchange diffchange-inline">a^3 \ge b^3, c^3</ins></math> <ins class="diffchange diffchange-inline">and </ins><math><ins class="diffchange diffchange-inline">3a^3 \ge </ins>2001<ins class="diffchange diffchange-inline">.</ins></math> <ins class="diffchange diffchange-inline"> Thus, <math>a^3 \ge 667,</ins><<ins class="diffchange diffchange-inline">/</ins>math> <ins class="diffchange diffchange-inline">so <math>a = </ins>10<ins class="diffchange diffchange-inline">.</ins></math></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> </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"><br></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">Now </ins><math><ins class="diffchange diffchange-inline">b^3 + c^3 = 1001</ins>.</math>  <ins class="diffchange diffchange-inline">Since <math>b^3 \ge c^3</ins>,<ins class="diffchange diffchange-inline"></math> </ins>we find that <ins class="diffchange diffchange-inline"><math>2b^3 \ge 1001.</math>  Thus, <math>b = 10</math> and <math>c = 1.</math></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> </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"><br></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">In summary, </ins>the only solutions are <math>\boxed{(10,10,1),(10,1,10),(1,10,10)}.</math></div></td></tr>
<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="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;"><div>==See Also==</div></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;"><div>==See Also==</div></td></tr>
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Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=2001_JBMO_Problems/Problem_1&diff=97153&oldid=prev
Rockmanex3: Solution to Problem 1 — mods are a game changer
2018-08-11T14:01:13Z
<p>Solution to Problem 1 — mods are a game changer</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
Solve the equation <math>a^3 + b^3 + c^3 = 2001</math> in positive integers.<br />
<br />
==Solution==<br />
<br />
Note that for all positive integers <math>n,</math> the value <math>n^3</math> is congruent to <math>-1,0,1</math> [[modulo]] <math>9.</math> Since <math>2001 \equiv 3 \pmod{9},</math> we find that <math>a,b,c \equiv 1 \pmod{9}.</math><br />
<br />
<br><br />
The only numbers congruent to <math>1</math> modulo <math>9</math> that are also less than <math>2001</math> are <math>10</math> and <math>1.</math> With some quick checking, we find that the only solutions are <math>\boxed{(10,10,1),(10,1,10),(1,10,10)}.</math><br />
<br />
==See Also==<br />
{{JBMO box|year=2001|before=First Problem|num-a=2|five=}}<br />
<br />
[[Category:Intermediate Number Theory Problems]]</div>
Rockmanex3