https://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&feed=atom&action=history2021 USAMO Problems/Problem 2 - Revision history2024-03-28T15:03:48ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=190257&oldid=prevIntegralarefun: /* Problem */ ; diagram issues2023-02-26T16:30:43Z<p><span dir="auto"><span class="autocomment">Problem: </span> ; diagram issues</span></p>
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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>The Planar National Park is an undirected 3-regular planar graph (i.e. all vertices have degree 3). That is: every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.)</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 Planar National Park is an undirected 3-regular planar graph (i.e. all vertices have degree 3). That is: every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.)</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><center></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><center></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><img<del class="diffchange diffchange-inline">></del>https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9mLzc1YmNjN2YxMWZhZTNhMTVkZTQ4NWE1ZDIyMDNhN2I5NzY0NTBlLnBuZw==&rn=Z3JhcGguUE5H<del class="diffchange diffchange-inline"><</del>/<del class="diffchange diffchange-inline">img</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><img <ins class="diffchange diffchange-inline">src = "</ins>https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9mLzc1YmNjN2YxMWZhZTNhMTVkZTQ4NWE1ZDIyMDNhN2I5NzY0NTBlLnBuZw==&rn=Z3JhcGguUE5H<ins class="diffchange diffchange-inline">" alt = "Diagram"</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>A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?</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>A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?</div></td></tr>
</table>Integralarefunhttps://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=169724&oldid=prevSrit123: /* Problem */2022-01-12T15:37:56Z<p><span dir="auto"><span class="autocomment">Problem</span></span></p>
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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></center></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></center></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 visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park<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>A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park<ins class="diffchange diffchange-inline">?</ins></div></td></tr>
</table>Srit123https://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=169723&oldid=prevSrit123: /* Problem */2022-01-12T15:37:40Z<p><span dir="auto"><span class="autocomment">Problem</span></span></p>
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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></center></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></center></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 visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park<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>A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
</table>Srit123https://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=162297&oldid=prevDneary: This was Q2 (not Q4)2021-09-15T18:20:00Z<p>This was Q2 (not Q4)</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:20, 15 September 2021</td>
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<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 <del class="diffchange diffchange-inline">2021 USAMO </del>has <del class="diffchange diffchange-inline">not been out yet</del>. <del class="diffchange diffchange-inline">Please do not edit this page</del>. <del class="diffchange diffchange-inline">If you </del>are a <del class="diffchange diffchange-inline">site admin</del>, <del class="diffchange diffchange-inline">please delete </del>this <del class="diffchange diffchange-inline">page</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">==Problem==</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>The <ins class="diffchange diffchange-inline">Planar National Park is an undirected 3-regular planar graph (i.e. all vertices have degree 3). That is: every trail </ins>has <ins class="diffchange diffchange-inline">its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints)</ins>. <ins class="diffchange diffchange-inline">Finally, no trails begin and end at the same two junctions</ins>. <ins class="diffchange diffchange-inline">(An example of one possible layout of the park is shown to the left below, in which there </ins>are <ins class="diffchange diffchange-inline">six junctions and nine trails.)</ins></div></td></tr>
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<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"><img>https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9mLzc1YmNjN2YxMWZhZTNhMTVkZTQ4NWE1ZDIyMDNhN2I5NzY0NTBlLnBuZw==&rn=Z3JhcGguUE5H</img></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"></center></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">A visitor walks through the park as follows: she begins at </ins>a <ins class="diffchange diffchange-inline">vertex and starts walking along an edge. When she reaches the other endpoint</ins>, <ins class="diffchange diffchange-inline">she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does </ins>this <ins class="diffchange diffchange-inline">until she gets back to the vertex where she started</ins>. <ins class="diffchange diffchange-inline">What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?</ins></div></td></tr>
</table>Dnearyhttps://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=148400&oldid=prevCryptographer: The 2021 USAMO has not been out yet. Please do not edit this page. If you are a site admin, please delete this page.2021-03-03T20:16:13Z<p>The 2021 USAMO has not been out yet. Please do not edit this page. If you are a site admin, please delete this page.</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:16, 3 March 2021</td>
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<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">Find</del>, <del class="diffchange diffchange-inline">with proof, all even positive integers that cannot be written as the sum of two primes</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 2021 USAMO has not been out yet. Please do not edit this page. If you are a site admin</ins>, <ins class="diffchange diffchange-inline">please delete this page</ins>.</div></td></tr>
</table>Cryptographerhttps://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=143057&oldid=prevAwesome3.14 at 02:05, 24 January 20212021-01-24T02:05:15Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:05, 24 January 2021</td>
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<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">you really expected </del>the <del class="diffchange diffchange-inline">problems to be here?????</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">Find, with proof, all even positive integers that cannot be written as </ins>the <ins class="diffchange diffchange-inline">sum of two primes.</ins></div></td></tr>
</table>Awesome3.14https://artofproblemsolving.com/wiki/index.php?title=2021_USAMO_Problems/Problem_2&diff=138368&oldid=prevMath31415926535: Created page with "you really expected the problems to be here?????"2020-11-24T17:41:33Z<p>Created page with "you really expected the problems to be here?????"</p>
<p><b>New page</b></p><div>you really expected the problems to be here?????</div>Math31415926535