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<title>2024 AMC 10B Problems/Problem 15</title>
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<updated>2024-11-15T01:28:20Z</updated>
<summary type="html"><p><span dir="auto"><span class="autocomment">Solution 1 (AMC 10)</span></span></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 01:28, 15 November 2024</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 1 (AMC 10)==</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 1 (AMC 10)==</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 sum of the six existing numbers is <math>24.8</math>. For the mean to be an integer, the sum of the remaining three numbers <math>x,y,z</math> and the remaining six must be divisible by <math>9</math>; thus <math>x+y+z</math> must equal either <math>2.2,11<del class="diffchange diffchange-inline">,</del>2,20.2</math>. However, one of <math>x,y,z</math> must be the median since the middle four numbers in the original set are all non-integral. As a result, we can eliminate <math>2.2</math> since the sum would be too small to allow for one of <math>x,y,z</math> to be the median (notice that negative numbers are not permitted since the range would be greater than <math>7</math>). This leaves two cases.</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 sum of the six existing numbers is <math>24.8</math>. For the mean to be an integer, the sum of the remaining three numbers <math>x,y,z</math> and the remaining six must be divisible by <math>9</math>; thus <math>x+y+z</math> must equal either <math>2.2,11<ins class="diffchange diffchange-inline">.</ins>2,20.2</math>. However, one of <math>x,y,z</math> must be the median since the middle four numbers in the original set are all non-integral. As a result, we can eliminate <math>2.2</math> since the sum would be too small to allow for one of <math>x,y,z</math> to be the median (notice that negative numbers are not permitted since the range would be greater than <math>7</math>). This leaves two cases.</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>For the <math>x+y+z=20.2</math> case, notice that all three must be less than <math>8</math> due to the range restriction. However, also notice that <math>y</math> and <math>z</math> must be on the higher end of the range, meaning that the new median would have to fall between the new middle numbers <math>5.2</math> and <math>6.2</math>. Thus <math>x=6</math>, and we can manipulate the numbers to make the range <math>7</math> by making <math>z=8</math> and thus <math>y=6.2</math>, providing one case in <math>(6,6.2,8)</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>For the <math>x+y+z=20.2</math> case, notice that all three must be less than <math>8</math> due to the range restriction. However, also notice that <math>y</math> and <math>z</math> must be on the higher end of the range, meaning that the new median would have to fall between the new middle numbers <math>5.2</math> and <math>6.2</math>. Thus <math>x=6</math>, and we can manipulate the numbers to make the range <math>7</math> by making <math>z=8</math> and thus <math>y=6.2</math>, providing one case in <math>(6,6.2,8)</math>.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14" >Line 14:</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>~eevee9406</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>~eevee9406</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;"></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 2==</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 2==</div></td></tr>
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