https://artofproblemsolving.com/wiki/index.php?title=1971_IMO_Problems/Problem_2&feed=atom&action=history
1971 IMO Problems/Problem 2 - Revision history
2024-03-28T22:48:15Z
Revision history for this page on the wiki
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Hamstpan38825 at 17:55, 29 January 2021
2021-01-29T17:55:04Z
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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 17:55, 29 January 2021</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>{{solution<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">WLOG let <math>A_1</math> be the origin <math>0</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><ins class="diffchange diffchange-inline">Take any point <math>A_i</math>, then <math>P_i=A_i+P_1</math>, lies in <math>2 P_1</math>, the polyhedron <math>P_1</math> stretched by the factor <math>2</math> on <math>P_1=0</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><ins class="diffchange diffchange-inline">More general: take any <math>p,q</math> in any convex shape <math>S</math>. Then <math>p+q \in 2S</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><ins class="diffchange diffchange-inline">Prove: since <math>S</math> is convex, <math>\frac</ins>{<ins class="diffchange diffchange-inline">p+q}</ins>{<ins class="diffchange diffchange-inline">2} \in S</math>, thus <math>p+q \in 2S</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">Now all these nine polyhedrons lie inside <math>2 P_1</math>. Let <math>V</math> be the volume of <math>P_1</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Then some polyhedrons with total sum of volumes <math>9V</math> lie in a shape of volume <math>8V</math>, thus they must overlap, meaning that they have an interior point in common.</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">The above </ins>solution <ins class="diffchange diffchange-inline">was posted by ZetaX. The original thread for this problem can be found here: [https://aops.com/community/p417224]</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>==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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Hamstpan38825
https://artofproblemsolving.com/wiki/index.php?title=1971_IMO_Problems/Problem_2&diff=91821&oldid=prev
Durianaops: Created page with "==Problem== Consider a convex polyhedron <math>P_1</math> with nine vertices <math>A_1, A_2, \cdots, A_9;</math> let <math>P_i</math> be the polyhedron obtained from <math>P_1..."
2018-02-17T18:59:19Z
<p>Created page with "==Problem== Consider a convex polyhedron <math>P_1</math> with nine vertices <math>A_1, A_2, \cdots, A_9;</math> let <math>P_i</math> be the polyhedron obtained from <math>P_1..."</p>
<p><b>New page</b></p><div>==Problem==<br />
Consider a convex polyhedron <math>P_1</math> with nine vertices <math>A_1, A_2, \cdots, A_9;</math> let <math>P_i</math> be the polyhedron obtained from <math>P_1</math> by a translation that moves vertex <math>A_1</math> to <math>A_i(i=2,3,\cdots, 9).</math> Prove that at least two of the polyhedra <math>P_1, P_2,\cdots, P_9</math> have an interior point in common.<br />
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==Solution==<br />
{{solution}}<br />
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==See Also==<br />
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{{IMO box|year=1971|num-b=1|num-a=3}}<br />
[[Category:Olympiad Geometry Problems]]</div>
Durianaops