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2007 Alabama ARML TST Problems/Problem 7 - Revision history
2024-03-29T00:38:09Z
Revision history for this page on the wiki
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Misof at 21:35, 28 January 2009
2009-01-28T21:35:32Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:35, 28 January 2009</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>Find the number of distinct integers in the list</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>Find the number of distinct integers in the list</div></td></tr>
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Misof
https://artofproblemsolving.com/wiki/index.php?title=2007_Alabama_ARML_TST_Problems/Problem_7&diff=27944&oldid=prev
1=2: /* See also */
2008-09-15T14:17:22Z
<p><span dir="auto"><span class="autocomment">See also</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:17, 15 September 2008</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>==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>
<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 style="font-weight: bold; text-decoration: none;">{{ARML box|year=2007|state=Alabama|num-b=6|num-a=8}}</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 style="font-weight: bold; text-decoration: none;"></ins></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;"><div>[[Category:Intermediate Number Theory Problems]]</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>[[Category:Intermediate Number Theory Problems]]</div></td></tr>
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1=2
https://artofproblemsolving.com/wiki/index.php?title=2007_Alabama_ARML_TST_Problems/Problem_7&diff=27943&oldid=prev
1=2 at 14:16, 15 September 2008
2008-09-15T14:16:29Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:16, 15 September 2008</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="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>The first time that the difference of two consecutive squares is greater than or equal to 2007 is <math>1004^2-1003^2=2007</math>. Below <math>\lfloor \frac{1003^2}{2007}\rfloor =501</math>, every non-negative integer can be reached. Then above that, each number is distinct. So there are <math>502+(2007-1004+1)=\boxed{1506}</math> distinct integers in the given list.</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>The first time that the difference of two consecutive squares is greater than or equal to 2007 is <math>1004^2-1003^2=2007</math>. Below <math><ins class="diffchange diffchange-inline">\left</ins>\lfloor \frac{1003^2}{2007}<ins class="diffchange diffchange-inline">\right</ins>\rfloor =501</math>, every non-negative integer can be reached. Then above that, each number is distinct. So there are <math>502+(2007-1004+1)=\boxed{1506}</math> distinct integers in the given list.</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>
<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>[[Category:Intermediate Number Theory Problems]]</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>[[Category:Intermediate Number Theory Problems]]</div></td></tr>
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1=2
https://artofproblemsolving.com/wiki/index.php?title=2007_Alabama_ARML_TST_Problems/Problem_7&diff=27942&oldid=prev
1=2: New page: =Problem== Find the number of distinct integers in the list <cmath>\left\lfloor \dfrac{1^2}{2007}\right\rfloor , \left\lfloor \dfrac{2^2}{2007}\right\rfloor , \left\lfloor \dfrac{3^2}{200...
2008-09-15T14:16:03Z
<p>New page: =Problem== Find the number of distinct integers in the list <cmath>\left\lfloor \dfrac{1^2}{2007}\right\rfloor , \left\lfloor \dfrac{2^2}{2007}\right\rfloor , \left\lfloor \dfrac{3^2}{200...</p>
<p><b>New page</b></p><div>=Problem==<br />
Find the number of distinct integers in the list<br />
<br />
<cmath>\left\lfloor \dfrac{1^2}{2007}\right\rfloor , \left\lfloor \dfrac{2^2}{2007}\right\rfloor , \left\lfloor \dfrac{3^2}{2007}\right\rfloor , \left\lfloor \dfrac{4^2}{2007}\right\rfloor , \cdots , \left\lfloor \dfrac{2007^2}{2007}\right\rfloor ,</cmath><br />
<br />
where <math>\lfloor x \rfloor</math> represents the greatest integer less than or equal to <math>x</math>.<br />
<br />
==Solution==<br />
The first time that the difference of two consecutive squares is greater than or equal to 2007 is <math>1004^2-1003^2=2007</math>. Below <math>\lfloor \frac{1003^2}{2007}\rfloor =501</math>, every non-negative integer can be reached. Then above that, each number is distinct. So there are <math>502+(2007-1004+1)=\boxed{1506}</math> distinct integers in the given list.<br />
<br />
==See also==<br />
[[Category:Intermediate Number Theory Problems]]</div>
1=2