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2006 AIME I Problems/Problem 11 - Revision history
2024-03-29T14:39:13Z
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Pi is 3.14: /* Solution */
2022-11-04T09:40:48Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 09:40, 4 November 2022</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 we cannot say, "there is one tower using the block <math>1</math>, so there are <math>3^7</math> towers using the blocks <math>1, 2, \ldots, 8</math>."  The reason this fails is that our recursion only worked when <math>m \geq 2</math>: when <math>m = 1</math>, there are only 2 places to insert a block of size <math>m + 1 = 2</math>, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking the number of towers where we change the second rule to read, "The cube immediately on top of a cube with edge-length <math>k</math> must have edge-length at most <math>k + n</math>," where <math>n</math> can be any fixed integer.)</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 we cannot say, "there is one tower using the block <math>1</math>, so there are <math>3^7</math> towers using the blocks <math>1, 2, \ldots, 8</math>."  The reason this fails is that our recursion only worked when <math>m \geq 2</math>: when <math>m = 1</math>, there are only 2 places to insert a block of size <math>m + 1 = 2</math>, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking the number of towers where we change the second rule to read, "The cube immediately on top of a cube with edge-length <math>k</math> must have edge-length at most <math>k + n</math>," where <math>n</math> can be any fixed integer.)</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;">== Video Solution by OmegaLearn ==</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;">https://youtu.be/WpSpnx8PPnc?t=732</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;">~ pi_is_3.14</ins></div></td></tr>
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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>
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Pi is 3.14
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=76431&oldid=prev
Smurty: /* See also */
2016-02-18T08:00:29Z
<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 08:00, 18 February 2016</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;">* [https://www.artofproblemsolving.com/wiki/index.php?title=2015_AIME_II_Problems/Problem_10 2015 AIME II Problem 10]</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>{{AIME box|year=2006|n=I|num-b=10|num-a=12}}</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>{{AIME box|year=2006|n=I|num-b=10|num-a=12}}</div></td></tr>
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Smurty
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=68996&oldid=prev
NormanWho at 14:59, 16 March 2015
2015-03-16T14:59:33Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:59, 16 March 2015</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>We proceed [[recursion | recursively]].  Suppose we can build <math>T_m</math> towers using blocks of size <math>1, 2, \ldots, m</math>.  How many towers can we build using blocks of size <math>1, 2, \ldots, m, m + 1</math>?  If we remove the block of size <math>m + 1</math> from such a tower (keeping all other blocks in order), we get a valid tower using blocks <math>1, 2, \ldots, m</math>.  Given a tower using blocks <math>1, 2, \ldots, m</math> (with <math>m \geq 2</math>), we can insert the block of size <math>m + 1</math> in exactly 3 places: at the beginning, immediately following the block of size <math>m - 1</math> or immediately following the block of size <math>m</math>.  Thus, there are 3 times as many towers using blocks of size <math>1, 2, \ldots, m, m + 1</math> as there are towers using only <math>1, 2, \ldots, m</math>.  There are 2 towers which use blocks <math>1, 2</math>, so there are <math>2\cdot 3^6 = 1458</math> towers using blocks <math>1, 2, \ldots, 8</math>, so the answer is <math>458</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>We proceed [[recursion | recursively]].  Suppose we can build <math>T_m</math> towers using blocks of size <math>1, 2, \ldots, m</math>.  How many towers can we build using blocks of size <math>1, 2, \ldots, m, m + 1</math>?  If we remove the block of size <math>m + 1</math> from such a tower (keeping all other blocks in order), we get a valid tower using blocks <math>1, 2, \ldots, m</math>.  Given a tower using blocks <math>1, 2, \ldots, m</math> (with <math>m \geq 2</math>), we can insert the block of size <math>m + 1</math> in exactly 3 places: at the beginning, immediately following the block of size <math>m - 1</math> or immediately following the block of size <math>m</math>.  Thus, there are 3 times as many towers using blocks of size <math>1, 2, \ldots, m, m + 1</math> as there are towers using only <math>1, 2, \ldots, m</math>.  There are 2 towers which use blocks <math>1, 2</math>, so there are <math>2\cdot 3^6 = 1458</math> towers using blocks <math>1, 2, \ldots, 8</math>, so the answer is <math><ins class="diffchange diffchange-inline">\boxed{</ins>458<ins class="diffchange diffchange-inline">}</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>(Note that we cannot say, "there is one tower using the block <math>1</math>, so there are <math>3^7</math> towers using the blocks <math>1, 2, \ldots, 8</math>."  The reason this fails is that our recursion only worked when <math>m \geq 2</math>: when <math>m = 1</math>, there are only 2 places to insert a block of size <math>m + 1 = 2</math>, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking the number of towers where we change the second rule to read, "The cube immediately on top of a cube with edge-length <math>k</math> must have edge-length at most <math>k + n</math>," where <math>n</math> can be any fixed integer.)</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 we cannot say, "there is one tower using the block <math>1</math>, so there are <math>3^7</math> towers using the blocks <math>1, 2, \ldots, 8</math>."  The reason this fails is that our recursion only worked when <math>m \geq 2</math>: when <math>m = 1</math>, there are only 2 places to insert a block of size <math>m + 1 = 2</math>, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking the number of towers where we change the second rule to read, "The cube immediately on top of a cube with edge-length <math>k</math> must have edge-length at most <math>k + n</math>," where <math>n</math> can be any fixed integer.)</div></td></tr>
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NormanWho
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=55212&oldid=prev
Nathan wailes at 00:05, 5 July 2013
2013-07-05T00:05:59Z
<p></p>
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Nathan wailes
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=16999&oldid=prev
Azjps: rv
2007-09-25T23:33:38Z
<p>rv</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 23:33, 25 September 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;"><div>== Problem ==</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>== Problem ==</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;">A collection of 8 [[cube (geometry) | cube]]s consists of one cube with [[edge]]-[[length]] <math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: </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;"></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 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;">* Any cube may be the bottom cube in the tower.</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;">* The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math> k+2. </math> </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;"></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">A sequence is defined as follows </del><math> <del class="diffchange diffchange-inline">a_1=a_2=a_3=1, </del></math> <del class="diffchange diffchange-inline">and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n</del>. <del class="diffchange diffchange-inline"></math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find </del>the remainder when <math><del class="diffchange diffchange-inline">\sum^{28}_{k=1} a_k </del></math> is divided by 1000<del class="diffchange diffchange-inline">.</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">Let </ins><math> <ins class="diffchange diffchange-inline">T </ins></math> <ins class="diffchange diffchange-inline">be the number of different towers than can be constructed</ins>. <ins class="diffchange diffchange-inline">What is </ins>the <ins class="diffchange diffchange-inline">[[</ins>remainder<ins class="diffchange diffchange-inline">]] </ins>when <math> <ins class="diffchange diffchange-inline">T </ins></math> is divided by 1000<ins class="diffchange diffchange-inline">?</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;"></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>== 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><del class="diffchange diffchange-inline">Define the sum as </del><math><del class="diffchange diffchange-inline">x</del></math>. <del class="diffchange diffchange-inline">Notice that </del><math><del class="diffchange diffchange-inline">a_n</del>\ <del class="diffchange diffchange-inline">= a_{n + 3} - a_{n + 2} - a_{n </del>+ 1<del class="diffchange diffchange-inline">} </del></math><del class="diffchange diffchange-inline">, so </del>the <del class="diffchange diffchange-inline">sum will be:</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">We proceed [[recursion | recursively]].  Suppose we can build <math>T_m</math> towers using blocks of size </ins><math><ins class="diffchange diffchange-inline">1, 2, \ldots, m</ins></math>. <ins class="diffchange diffchange-inline"> How many towers can we build using blocks of size </ins><math><ins class="diffchange diffchange-inline">1, 2, </ins>\<ins class="diffchange diffchange-inline">ldots, m, m </ins>+ 1</math><ins class="diffchange diffchange-inline">?  If we remove </ins>the <ins class="diffchange diffchange-inline">block of size </ins><math><ins class="diffchange diffchange-inline">m </ins>+ <ins class="diffchange diffchange-inline">1</math> from such a tower </ins>(<ins class="diffchange diffchange-inline">keeping all other blocks in order</ins>)<ins class="diffchange diffchange-inline">, we get a valid tower using blocks <math>1, 2, \ldots, m</math>.  Given a tower using blocks <math>1, 2, </ins>\ldots<ins class="diffchange diffchange-inline">, m</math> </ins>(<ins class="diffchange diffchange-inline">with <math>m \geq 2</math></ins>)<ins class="diffchange diffchange-inline">, we can insert the block of size <math>m </ins>+ <ins class="diffchange diffchange-inline">1</ins></math> <ins class="diffchange diffchange-inline">in exactly 3 places</ins>: <ins class="diffchange diffchange-inline">at the beginning, immediately following the block of size <math>m - 1</math> or immediately following the block of size <math>m</math>.  Thus, there are 3 times as many towers using blocks of size </ins><math><ins class="diffchange diffchange-inline">1, 2, </ins>\ldots<ins class="diffchange diffchange-inline">, m, m </ins>+ <ins class="diffchange diffchange-inline">1</math> as there are towers using only <math>1, 2, </ins>\ldots<ins class="diffchange diffchange-inline">, m</math>.  There are 2 towers which use blocks <math>1, 2</math>, so there are <math>2\cdot 3^6 = 1458</math> towers using blocks <math>1, 2, </ins>\ldots<ins class="diffchange diffchange-inline">, 8</math>, so the answer is <math>458</ins></math><ins class="diffchange diffchange-inline">.</ins></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">:</del><math><del class="diffchange diffchange-inline">x = (a_4 - a_3 - a_2) </del>+ (<del class="diffchange diffchange-inline">a_5 - a_4 - a_3</del>) <del class="diffchange diffchange-inline">+ </del>\ldots (<del class="diffchange diffchange-inline">a_{30} - a_{29} - a_{28}</del>) + <del class="diffchange diffchange-inline">a_{28}</del></math></div></td><td colspan="2"> </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>:<math><del class="diffchange diffchange-inline">x = (a_4+ a_5 </del>\ldots <del class="diffchange diffchange-inline">a_{30}) - (a_3 + a_4 </del>+ \ldots <del class="diffchange diffchange-inline">a_{29}) - (a_2 + a_3 + </del>\ldots <del class="diffchange diffchange-inline">a_{28}) + a_{28} + (a_1 - a_1)</del></math></div></td><td colspan="2"> </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><del class="diffchange diffchange-inline">The first two groupings almost completely cancel. The third resembles </del><math><del class="diffchange diffchange-inline">x</del></math><del class="diffchange diffchange-inline">.</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">(Note that we cannot say, "there is one tower using the block </ins><math><ins class="diffchange diffchange-inline">1</ins></math><ins class="diffchange diffchange-inline">, so there are </ins><math><ins class="diffchange diffchange-inline">3^7</ins></math> <ins class="diffchange diffchange-inline">towers using the blocks </ins><math><ins class="diffchange diffchange-inline">1, 2, </ins>\<ins class="diffchange diffchange-inline">ldots, 8</ins></math><ins class="diffchange diffchange-inline">."  The reason this fails is that our recursion only worked when </ins><math><ins class="diffchange diffchange-inline">m </ins>\<ins class="diffchange diffchange-inline">geq </ins>2</math><ins class="diffchange diffchange-inline">: when </ins><math><ins class="diffchange diffchange-inline">m = 1</ins></math><ins class="diffchange diffchange-inline">, there are only 2 places to insert a block of size </ins><math><ins class="diffchange diffchange-inline">m + 1 = 2</ins></math><ins class="diffchange diffchange-inline">, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking </ins>the <ins class="diffchange diffchange-inline">number </ins>of <ins class="diffchange diffchange-inline">towers where we change </ins>the <ins class="diffchange diffchange-inline">second rule to read, "The cube immediately on top of a cube with edge-length </ins><math><ins class="diffchange diffchange-inline">k</ins></math> <ins class="diffchange diffchange-inline">must have edge-length at most </ins><math><ins class="diffchange diffchange-inline">k + n</ins></math>,<ins class="diffchange diffchange-inline">" where </ins><math><ins class="diffchange diffchange-inline">n</ins></math> <ins class="diffchange diffchange-inline">can be any fixed integer</ins>.<ins class="diffchange diffchange-inline">)</ins></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> </div></td><td colspan="2"> </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">:</del><math><del class="diffchange diffchange-inline">x\ = a_1 - a_3 + a_{28} + a_{30} - x</del></math></div></td><td colspan="2"> </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">:</del><math><del class="diffchange diffchange-inline">2x</del>\ <del class="diffchange diffchange-inline">= a_{28} + a_{30}</del></math></div></td><td colspan="2"> </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">:</del><math><del class="diffchange diffchange-inline">x\ = </del>\<del class="diffchange diffchange-inline">frac{a_{28} + a_{30}}{</del>2<del class="diffchange diffchange-inline">}</del></math>  </div></td><td colspan="2"> </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> </div></td><td colspan="2"> </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><math><del class="diffchange diffchange-inline">a_{28}</del></math> <del class="diffchange diffchange-inline">and </del><math><del class="diffchange diffchange-inline">a_{30}</del></math> <del class="diffchange diffchange-inline">are both given; </del>the <del class="diffchange diffchange-inline">last four digits </del>of the <del class="diffchange diffchange-inline">sum is </del><math><del class="diffchange diffchange-inline">3668</del></math><del class="diffchange diffchange-inline">, and half of that is </del><math><del class="diffchange diffchange-inline">1834</del></math><del class="diffchange diffchange-inline">. Therefore</del>, <del class="diffchange diffchange-inline">the answer is </del><math><del class="diffchange diffchange-inline">834</del></math>.</div></td><td colspan="2"> </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>
</table>
Azjps
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=16855&oldid=prev
1=2: /* Solution */
2007-09-25T19:43:53Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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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 19:43, 25 September 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5" >Line 5:</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="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><del class="diffchange diffchange-inline">We proceed [[recursion | recursively]].  Suppose we can build </del><math><del class="diffchange diffchange-inline">T_m</del></math> <del class="diffchange diffchange-inline">towers using blocks of size </del><math><del class="diffchange diffchange-inline">1, 2, </del>\<del class="diffchange diffchange-inline">ldots, m</math>.  How many towers can we build using blocks of size <math>1, </del>2<del class="diffchange diffchange-inline">, \ldots, m, m </del>+ 1</math><del class="diffchange diffchange-inline">?  If we remove </del>the <del class="diffchange diffchange-inline">block of size </del><math><del class="diffchange diffchange-inline">m </del>+ <del class="diffchange diffchange-inline">1</math> from such a tower </del>(<del class="diffchange diffchange-inline">keeping all other blocks in order</del>)<del class="diffchange diffchange-inline">, we get a valid tower using blocks <math>1, 2, \ldots, m</math>.  Given a tower using blocks <math>1, 2, </del>\ldots<del class="diffchange diffchange-inline">, m</math> </del>(<del class="diffchange diffchange-inline">with <math>m \geq 2</math></del>)<del class="diffchange diffchange-inline">, we can insert the block of size <math>m </del>+ <del class="diffchange diffchange-inline">1</del></math> <del class="diffchange diffchange-inline">in exactly 3 places</del>: <del class="diffchange diffchange-inline">at the beginning, immediately following the block of size </del><math><del class="diffchange diffchange-inline">m - 1</math> or immediately following the block of size <math>m</math>.  Thus, there are 3 times as many towers using blocks of size <math>1, 2, </del>\ldots<del class="diffchange diffchange-inline">, m, m </del>+ <del class="diffchange diffchange-inline">1</math> as there are towers using only <math>1, 2, </del>\ldots<del class="diffchange diffchange-inline">, m</math>.  There are 2 towers which use blocks <math>1, 2</math>, so there are <math>2\cdot 3^6 = 1458</math> towers using blocks <math>1, 2, </del>\ldots<del class="diffchange diffchange-inline">, 8</math>, so the answer is <math>458</del></math><del class="diffchange diffchange-inline">.</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">Define the sum as </ins><math><ins class="diffchange diffchange-inline">x</ins></math><ins class="diffchange diffchange-inline">. Notice that </ins><math><ins class="diffchange diffchange-inline">a_n</ins>\ <ins class="diffchange diffchange-inline">= a_{n + 3} - a_{n + </ins>2<ins class="diffchange diffchange-inline">} - a_{n </ins>+ 1<ins class="diffchange diffchange-inline">} </ins></math><ins class="diffchange diffchange-inline">, so </ins>the <ins class="diffchange diffchange-inline">sum will be:</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">:</ins><math><ins class="diffchange diffchange-inline">x = (a_4 - a_3 - a_2) </ins>+ (<ins class="diffchange diffchange-inline">a_5 - a_4 - a_3</ins>) <ins class="diffchange diffchange-inline">+ </ins>\ldots (<ins class="diffchange diffchange-inline">a_{30} - a_{29} - a_{28}</ins>) + <ins class="diffchange diffchange-inline">a_{28}</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>:<math><ins class="diffchange diffchange-inline">x = (a_4+ a_5 </ins>\ldots <ins class="diffchange diffchange-inline">a_{30}) - (a_3 + a_4 </ins>+ \ldots <ins class="diffchange diffchange-inline">a_{29}) - (a_2 + a_3 + </ins>\ldots <ins class="diffchange diffchange-inline">a_{28}) + a_{28} + (a_1 - a_1)</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="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">(Note that we cannot say, "there is one tower using the block </del><math><del class="diffchange diffchange-inline">1</del></math><del class="diffchange diffchange-inline">, so there are </del><math><del class="diffchange diffchange-inline">3^7</del></math> <del class="diffchange diffchange-inline">towers using the blocks </del><math><del class="diffchange diffchange-inline">1, 2, </del>\<del class="diffchange diffchange-inline">ldots, 8</del></math><del class="diffchange diffchange-inline">."  The reason this fails is that our recursion only worked when </del><math><del class="diffchange diffchange-inline">m </del>\<del class="diffchange diffchange-inline">geq </del>2</math><del class="diffchange diffchange-inline">: when </del><math><del class="diffchange diffchange-inline">m = 1</del></math><del class="diffchange diffchange-inline">, there are only 2 places to insert a block of size </del><math><del class="diffchange diffchange-inline">m + 1 = 2</del></math><del class="diffchange diffchange-inline">, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking </del>the <del class="diffchange diffchange-inline">number </del>of <del class="diffchange diffchange-inline">towers where we change </del>the <del class="diffchange diffchange-inline">second rule to read, "The cube immediately on top of a cube with edge-length </del><math><del class="diffchange diffchange-inline">k</del></math> <del class="diffchange diffchange-inline">must have edge-length at most </del><math><del class="diffchange diffchange-inline">k + n</del></math>,<del class="diffchange diffchange-inline">" where </del><math><del class="diffchange diffchange-inline">n</del></math> <del class="diffchange diffchange-inline">can be any fixed integer</del>.<del class="diffchange diffchange-inline">)</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">The first two groupings almost completely cancel. The third resembles </ins><math><ins class="diffchange diffchange-inline">x</ins></math><ins class="diffchange diffchange-inline">.</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">:</ins><math><ins class="diffchange diffchange-inline">x\ = a_1 - a_3 + a_{28} + a_{30} - x</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><ins class="diffchange diffchange-inline">:</ins><math><ins class="diffchange diffchange-inline">2x</ins>\ <ins class="diffchange diffchange-inline">= a_{28} + a_{30}</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><ins class="diffchange diffchange-inline">:</ins><math><ins class="diffchange diffchange-inline">x\ = </ins>\<ins class="diffchange diffchange-inline">frac{a_{28} + a_{30}}{</ins>2<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><math><ins class="diffchange diffchange-inline">a_{28}</ins></math> <ins class="diffchange diffchange-inline">and </ins><math><ins class="diffchange diffchange-inline">a_{30}</ins></math> <ins class="diffchange diffchange-inline">are both given; </ins>the <ins class="diffchange diffchange-inline">last four digits </ins>of the <ins class="diffchange diffchange-inline">sum is </ins><math><ins class="diffchange diffchange-inline">3668</ins></math><ins class="diffchange diffchange-inline">, and half of that is </ins><math><ins class="diffchange diffchange-inline">1834</ins></math><ins class="diffchange diffchange-inline">. Therefore</ins>, <ins class="diffchange diffchange-inline">the answer is </ins><math><ins class="diffchange diffchange-inline">834</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>== 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>
</table>
1=2
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=16852&oldid=prev
1=2: /* Problem */
2007-09-25T19:43:21Z
<p><span dir="auto"><span class="autocomment">Problem</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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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 19:43, 25 September 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</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>== Problem ==</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>== Problem ==</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 style="font-weight: bold; text-decoration: none;">A collection of 8 [[cube (geometry) | cube]]s consists of one cube with [[edge]]-[[length]] <math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: </del></div></td><td colspan="2"> </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><del style="font-weight: bold; text-decoration: none;">* Any cube may be the bottom cube in the tower.</del></div></td><td colspan="2"> </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 style="font-weight: bold; text-decoration: none;">* The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math> k+2. </math> </del></div></td><td colspan="2"> </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><del class="diffchange diffchange-inline">Let </del><math> <del class="diffchange diffchange-inline">T </del></math> <del class="diffchange diffchange-inline">be the number of different towers than can be constructed</del>. <del class="diffchange diffchange-inline">What is </del>the <del class="diffchange diffchange-inline">[[</del>remainder<del class="diffchange diffchange-inline">]] </del>when <math> <del class="diffchange diffchange-inline">T </del></math> is divided by 1000<del class="diffchange diffchange-inline">?</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">A sequence is defined as follows </ins><math> <ins class="diffchange diffchange-inline">a_1=a_2=a_3=1, </ins></math> <ins class="diffchange diffchange-inline">and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n</ins>. <ins class="diffchange diffchange-inline"></math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find </ins>the remainder when <math><ins class="diffchange diffchange-inline">\sum^{28}_{k=1} a_k </ins></math> is divided by 1000<ins class="diffchange diffchange-inline">.</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;"></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>== 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>
</table>
1=2
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=13943&oldid=prev
K1c2h2o: /* See also */
2007-03-12T02:21:02Z
<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 02:21, 12 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;"><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="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>{{AIME box|year=2006|n=I|num-b=<del class="diffchange diffchange-inline">4</del>|num-a=<del class="diffchange diffchange-inline">6</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>{{AIME box|year=2006|n=I|num-b=<ins class="diffchange diffchange-inline">10</ins>|num-a=<ins class="diffchange diffchange-inline">12</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;"></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>[[Category:Intermediate Combinatorics 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 Combinatorics Problems]]</div></td></tr>
</table>
K1c2h2o
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=13934&oldid=prev
K1c2h2o: /* See also */
2007-03-12T02:16:20Z
<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 02:16, 12 March 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13" >Line 13:</td>
<td colspan="2" class="diff-lineno">Line 13:</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="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">* [[2006 </del>AIME <del class="diffchange diffchange-inline">I Problems/Problem 10 </del>| <del class="diffchange diffchange-inline">Previous problem]]</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">{{</ins>AIME <ins class="diffchange diffchange-inline">box</ins>|<ins class="diffchange diffchange-inline">year=</ins>2006<ins class="diffchange diffchange-inline">|n=</ins>I|<ins class="diffchange diffchange-inline">num-b=4|num-a=6}}</ins></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">* [[</del>2006 <del class="diffchange diffchange-inline">AIME </del>I <del class="diffchange diffchange-inline">Problems/Problem 12 </del>| <del class="diffchange diffchange-inline">Next problem]]</del></div></td><td colspan="2"> </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">* [[2006 AIME I Problems]]</del></div></td><td colspan="2"> </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>[[Category:Intermediate Combinatorics 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 Combinatorics Problems]]</div></td></tr>
</table>
K1c2h2o
https://artofproblemsolving.com/wiki/index.php?title=2006_AIME_I_Problems/Problem_11&diff=11736&oldid=prev
JBL at 02:40, 15 November 2006
2006-11-15T02:40:59Z
<p></p>
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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 02:40, 15 November 2006</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>== Problem ==</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>== Problem ==</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>A collection of 8 [[cube (geometry) | cube]]s consists of one cube with edge-length <math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules:  </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>A collection of 8 [[cube (geometry) | cube]]s consists of one cube with <ins class="diffchange diffchange-inline">[[</ins>edge<ins class="diffchange diffchange-inline">]]</ins>-<ins class="diffchange diffchange-inline">[[</ins>length<ins class="diffchange diffchange-inline">]] </ins><math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules:  </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>* Any cube may be the bottom cube in the tower.</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>* Any cube may be the bottom cube in the tower.</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>* The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math> k+2. </math>  </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>* The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math> k+2. </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="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>Let <math> T </math> be the number of different towers than can be constructed. What is the remainder when <math> T </math> is divided by 1000?</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>Let <math> T </math> be the number of different towers than can be constructed. What is the <ins class="diffchange diffchange-inline">[[</ins>remainder<ins class="diffchange diffchange-inline">]] </ins>when <math> T </math> is divided by 1000?</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>== 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><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">We proceed [[recursion | recursively]].  Suppose we can build <math>T_m</math> towers using blocks of size <math>1, 2, \ldots, m</math>.  How many towers can we build using blocks of size <math>1, 2, \ldots, m, m + 1</math>?  If we remove the block of size <math>m + 1</math> from such a tower (keeping all other blocks in order), we get a valid tower using blocks <math>1, 2, \ldots, m</math>.  Given a tower using blocks <math>1, 2, \ldots, m</math> (with <math>m \geq 2</math>), we can insert the block of size <math>m + 1</math> in exactly 3 places: at the beginning, immediately following the block of size <math>m - 1</math> or immediately following the block of size <math>m</math>.  Thus, there are 3 times as many towers using blocks of size <math>1, 2, \ldots, m, m + 1</math> as there are towers using only <math>1, 2, \ldots, m</math>.  There are 2 towers which use blocks <math>1, 2</math>, so there are <math>2\cdot 3^6 = 1458</math> towers using blocks <math>1, 2, \ldots, 8</math>, so the answer is <math>458</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">(Note that we cannot say, "there is one tower using the block <math>1</math>, so there are <math>3^7</math> towers using the blocks <math>1, 2, \ldots, 8</math>."  The reason this fails is that our recursion only worked when <math>m \geq 2</math>: when <math>m = 1</math>, there are only 2 places to insert a block of size <math>m + 1 = 2</math>, at the beginning or at the end, rather than the 3 places we have at later stages.  Also, note that this method generalizes directly to seeking the number of towers where we change the second rule to read, "The cube immediately on top of a cube with edge-length <math>k</math> must have edge-length at most <math>k + n</math>," where <math>n</math> can be any fixed integer.)</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 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>* [[2006 AIME I Problems/Problem 10 | Previous problem]]</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>* [[2006 AIME I Problems/Problem 10 | Previous problem]]</div></td></tr>
</table>
JBL