https://artofproblemsolving.com/wiki/index.php?title=1989_AHSME_Problems/Problem_11&feed=atom&action=history
1989 AHSME Problems/Problem 11 - Revision history
2024-03-28T22:28:21Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=1989_AHSME_Problems/Problem_11&diff=65418&oldid=prev
Timneh at 11:49, 22 October 2014
2014-10-22T11:49:51Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:49, 22 October 2014</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>Note that the statement <math>a<2b<6c<24d<2400</math> is true, but does not specify the distances between each pair of values.</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>Note that the statement <math>a<2b<6c<24d<2400</math> is true, but does not specify the distances between each pair of values.</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 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;">== See also ==</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;">{{AHSME box|year=1989|num-b=10|num-a=12}}  </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 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;">[[Category: Introductory Algebra Problems]]</ins></div></td></tr>
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Timneh
https://artofproblemsolving.com/wiki/index.php?title=1989_AHSME_Problems/Problem_11&diff=56378&oldid=prev
Nathan wailes at 17:48, 5 July 2013
2013-07-05T17:48:07Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:48, 5 July 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l15" >Line 15:</td>
<td colspan="2" class="diff-lineno">Line 15:</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>Note that the statement <math>a<2b<6c<24d<2400</math> is true, but does not specify the distances between each pair of values.</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>Note that the statement <math>a<2b<6c<24d<2400</math> is true, but does not specify the distances between each pair of values.</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;">{{MAA Notice}}</ins></div></td></tr>
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Nathan wailes
https://artofproblemsolving.com/wiki/index.php?title=1989_AHSME_Problems/Problem_11&diff=48702&oldid=prev
Zimbalono: Created page with "== Problem == Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be positive integers with <math> a < 2b</math>, <math>b < 3c </math>, and <math>c<4d</math>...."
2012-09-24T16:54:27Z
<p>Created page with "== Problem == Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be positive integers with <math> a < 2b</math>, <math>b < 3c </math>, and <math>c<4d</math>...."</p>
<p><b>New page</b></p><div>== Problem ==<br />
<br />
Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be positive integers with <math> a < 2b</math>, <math>b < 3c </math>, and <math>c<4d</math>. If <math>d<100</math>, the largest possible value for <math>a</math> is<br />
<br />
<math> \textrm{(A)}\ 2367\qquad\textrm{(B)}\ 2375\qquad\textrm{(C)}\ 2391\qquad\textrm{(D)}\ 2399\qquad\textrm{(E)}\ 2400 </math><br />
<br />
== Solution ==<br />
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Each of these integers is bounded above by the next one.<br />
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*<math>d<100</math>, so the maximum <math>d</math> is <math>99</math>.<br />
*<math>c<4d\le396</math>, so the maximum <math>c</math> is <math>395</math>.<br />
*<math>b<3c\le1185</math>, so the maximum <math>b</math> is <math>1184</math>.<br />
*<math>a<2b\le2368</math>, so the maximum <math>a</math> is <math>\boxed{2367}</math>.<br />
<br />
Note that the statement <math>a<2b<6c<24d<2400</math> is true, but does not specify the distances between each pair of values.</div>
Zimbalono