https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&feed=atom&action=history
1995 USAMO Problems - Revision history
2024-03-28T19:27:48Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&diff=54723&oldid=prev
Nathan wailes at 17:31, 4 July 2013
2013-07-04T17:31:55Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:31, 4 July 2013</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</div></td></tr>
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Nathan wailes
https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&diff=48475&oldid=prev
1=2: /* Resources */
2012-09-17T18:54:50Z
<p><span dir="auto"><span class="autocomment">Resources</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:54, 17 September 2012</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>[[1995 USAMO Problems/Problem 5|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>[[1995 USAMO Problems/Problem 5|Solution]]</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">Resources </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">See Also </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>{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}</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>{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</div></td></tr>
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1=2
https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&diff=48473&oldid=prev
1=2: /* Resources */
2012-09-17T18:51:26Z
<p><span dir="auto"><span class="autocomment">Resources</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:51, 17 September 2012</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>== Resources ==</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>== Resources ==</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;"></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;"><div>{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}</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>{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}</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;"></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;">* [http://www.artofproblemsolving.com/resources.php?c=182&cid=27&year=1995 1995 USAMO Problems on the resources page]</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;"><div>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</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>* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</div></td></tr>
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1=2
https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&diff=27398&oldid=prev
1=2: delta is not a triangle.
2008-08-12T16:49:18Z
<p>delta is not a triangle.</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>==Problem 3==</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 3==</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>Given a nonisosceles, nonright triangle <math>\, ABC, \,</math> let <math>\, O \,</math> denote the center of its circumscribed circle, and let <math>\, A_1, \, B_1, \,</math> and <math>\, C_1 \,</math> be the midpoints of sides <math>\, BC, \, CA, \,</math> and <math>\, AB, \,</math> respectively.  Point <math>\, A_2 \,</math> is located on the ray <math>\, OA_1 \,</math> so that <math>\, \<del class="diffchange diffchange-inline">Delta </del>OAA_1 \,</math> is similar to <math>\, \<del class="diffchange diffchange-inline">Delta </del>OA_2A</math>.  Points <math>\, B_2 \,</math> and <math>\, C_2 \,</math> on rays <math>\, OB_1 \,</math> and <math>\, OC_1, \,</math> respectively, are defined similarly.  Prove that lines <math>\, AA_2, \, BB_2, \,</math> and <math>\, CC_2 \,</math> are concurrent, i.e. these three lines intersect at a point.</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>Given a nonisosceles, nonright triangle <math>\, ABC, \,</math> let <math>\, O \,</math> denote the center of its circumscribed circle, and let <math>\, A_1, \, B_1, \,</math> and <math>\, C_1 \,</math> be the midpoints of sides <math>\, BC, \, CA, \,</math> and <math>\, AB, \,</math> respectively.  Point <math>\, A_2 \,</math> is located on the ray <math>\, OA_1 \,</math> so that <math>\, \<ins class="diffchange diffchange-inline">triangle </ins>OAA_1 \,</math> is similar to <math>\, \<ins class="diffchange diffchange-inline">triangle </ins>OA_2A</math>.  Points <math>\, B_2 \,</math> and <math>\, C_2 \,</math> on rays <math>\, OB_1 \,</math> and <math>\, OC_1, \,</math> respectively, are defined similarly.  Prove that lines <math>\, AA_2, \, BB_2, \,</math> and <math>\, CC_2 \,</math> are concurrent, i.e. these three lines intersect at a point.</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>[[1995 USAMO Problems/Problem 3|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>[[1995 USAMO Problems/Problem 3|Solution]]</div></td></tr>
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1=2
https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&diff=23137&oldid=prev
Boy Soprano II: revert to original wording
2008-02-10T17:22:50Z
<p>revert to original wording</p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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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 style="font-weight: bold; text-decoration: none;">Problems of the [[1995 USAMO | 1995]] [[USAMO]].</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 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 1==</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 1==</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">The sequence </del><math><del class="diffchange diffchange-inline">a_i</del></math> <del class="diffchange diffchange-inline">of </del>[[<del class="diffchange diffchange-inline">nonnegative</del>]] <del class="diffchange diffchange-inline">[[integer]]s </del>is defined as follows: <del class="diffchange diffchange-inline">The first </del><math><del class="diffchange diffchange-inline">p-1</del></math> <del class="diffchange diffchange-inline">terms are </del><math><del class="diffchange diffchange-inline">0</del>, <del class="diffchange diffchange-inline">1</del>, 2, <del class="diffchange diffchange-inline">3</del>, <del class="diffchange diffchange-inline">... </del>, p-<del class="diffchange diffchange-inline">2</del></math><del class="diffchange diffchange-inline">. Then </del><math>a_n</math> is the least positive integer <del class="diffchange diffchange-inline">so </del>that <del class="diffchange diffchange-inline">there is no </del>arithmetic <del class="diffchange diffchange-inline">progression </del>of length <math>p</math> <del class="diffchange diffchange-inline">in </del>the <del class="diffchange diffchange-inline">first n+1 </del>terms. <del class="diffchange diffchange-inline">If </del><math><del class="diffchange diffchange-inline">p</del></math> <del class="diffchange diffchange-inline">is an odd prime</del>, <del class="diffchange diffchange-inline">show that an </del>is the number obtained by writing <math>n</math> in base <math>p-1</math><del class="diffchange diffchange-inline">, then treating </del>the result <del class="diffchange diffchange-inline">as a number </del>in base <math>p</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> </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">Let </ins><math><ins class="diffchange diffchange-inline">\, p \,</ins></math> <ins class="diffchange diffchange-inline">be an odd </ins>[[<ins class="diffchange diffchange-inline">prime</ins>]]<ins class="diffchange diffchange-inline">.  The sequence <math>(a_n)_{n \geq 0}</math> </ins>is defined as follows: <math><ins class="diffchange diffchange-inline">\, a_0 = 0, </ins></math> <math><ins class="diffchange diffchange-inline">a_1 = 1, \, \ldots</ins>, <ins class="diffchange diffchange-inline">\</ins>, <ins class="diffchange diffchange-inline">a_{p-2} = p-</ins>2 <ins class="diffchange diffchange-inline">\</ins>,<ins class="diffchange diffchange-inline"></math> and</ins>, <ins class="diffchange diffchange-inline">for all <math>\</ins>, <ins class="diffchange diffchange-inline">n \geq </ins>p-<ins class="diffchange diffchange-inline">1, \,</ins></math> <math><ins class="diffchange diffchange-inline">\, </ins>a_n <ins class="diffchange diffchange-inline">\,</ins></math> is the least positive integer that <ins class="diffchange diffchange-inline">does not form an </ins>arithmetic <ins class="diffchange diffchange-inline">sequence </ins>of length <math><ins class="diffchange diffchange-inline">\, </ins>p <ins class="diffchange diffchange-inline">\,</ins></math> <ins class="diffchange diffchange-inline">with any of </ins>the <ins class="diffchange diffchange-inline">preceding </ins>terms. <ins class="diffchange diffchange-inline">Prove that, for all </ins><math><ins class="diffchange diffchange-inline">\, n, \,</ins></math> <ins class="diffchange diffchange-inline"><math>\, a_n \</ins>,<ins class="diffchange diffchange-inline"></math> </ins>is the number obtained by writing <math><ins class="diffchange diffchange-inline">\, </ins>n <ins class="diffchange diffchange-inline">\,</ins></math> in base <math><ins class="diffchange diffchange-inline">\, </ins>p-1 <ins class="diffchange diffchange-inline">\,</ins></math> <ins class="diffchange diffchange-inline">and reading </ins>the result in base <math><ins class="diffchange diffchange-inline">\, </ins>p</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>[[1995 USAMO Problems/Problem 1|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>[[1995 USAMO Problems/Problem 1|Solution]]</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 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 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>==Problem 2==</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 <del class="diffchange diffchange-inline">trigonometric map </del>is <del class="diffchange diffchange-inline">any one of </del><math>\sin, \cos, \tan, \<del class="diffchange diffchange-inline">arcsin</del>, \<del class="diffchange diffchange-inline">arccos</del></math> and <math>\<del class="diffchange diffchange-inline">arctan</del></math>. <del class="diffchange diffchange-inline">Show that given </del>any <del class="diffchange diffchange-inline">[[</del>positive<del class="diffchange diffchange-inline">]] [[</del>rational<del class="diffchange diffchange-inline">]] </del>number <math><del class="diffchange diffchange-inline">x</del></math><del class="diffchange diffchange-inline">, one can find a </del>finite sequence of <del class="diffchange diffchange-inline">trigonometric maps which take </del><math><del class="diffchange diffchange-inline">0</math> to <math>x</del></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> </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>A <ins class="diffchange diffchange-inline">calculator </ins>is <ins class="diffchange diffchange-inline">broken so that the only keys that still work are the </ins><math><ins class="diffchange diffchange-inline">\, </ins>\sin, <ins class="diffchange diffchange-inline">\; </ins>\cos, <ins class="diffchange diffchange-inline"></math>  <math></ins>\tan, \<ins class="diffchange diffchange-inline">; \sin^{-1}</ins>, \<ins class="diffchange diffchange-inline">; \cos^{-1}, \,</ins></math> and <math>\<ins class="diffchange diffchange-inline">, \tan^{-1} \,</ins></math> <ins class="diffchange diffchange-inline">buttons</ins>. <ins class="diffchange diffchange-inline"> The display initially shows 0. Given </ins>any positive rational number <math><ins class="diffchange diffchange-inline">\, q, \,</ins></math> <ins class="diffchange diffchange-inline">show that pressing some </ins>finite sequence of <ins class="diffchange diffchange-inline">buttons will yield </ins><math><ins class="diffchange diffchange-inline">\, q</ins></math><ins class="diffchange diffchange-inline">.  Assume that the calculator does real number calculations with infinite precision.  All functions are in terms of radians</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>[[1995 USAMO Problems/Problem 2|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>[[1995 USAMO Problems/Problem 2|Solution]]</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 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 3==</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 3==</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">The circumcenter </del><math>O</math> of <del class="diffchange diffchange-inline">the triangle </del><math>\<del class="diffchange diffchange-inline">triangle ABC</del></math> <del class="diffchange diffchange-inline">does not lie on any side or [[median]]. Let </del>the midpoints of <math>BC, CA, <del class="diffchange diffchange-inline">AB</del></math> <del class="diffchange diffchange-inline">be </del><math><del class="diffchange diffchange-inline">L</del>, <del class="diffchange diffchange-inline">M</del>, <del class="diffchange diffchange-inline">N</del></math> respectively. <del class="diffchange diffchange-inline">Construct </del><math><del class="diffchange diffchange-inline">P</del>, <del class="diffchange diffchange-inline">Q</del>, <del class="diffchange diffchange-inline">R</del></math> on the <del class="diffchange diffchange-inline">rays </del><math><del class="diffchange diffchange-inline">OL</del>, <del class="diffchange diffchange-inline">OM</del>, <del class="diffchange diffchange-inline">ON</del></math> <del class="diffchange diffchange-inline">respectively </del>so that <math>\<del class="diffchange diffchange-inline">angle OPA = </del>\<del class="diffchange diffchange-inline">angle OAL</del>, \<del class="diffchange diffchange-inline">angle OQB = </del>\<del class="diffchange diffchange-inline">angle OBM </del>and \<del class="diffchange diffchange-inline">angle ORC = </del>\<del class="diffchange diffchange-inline">angle OCN</del></math>. <del class="diffchange diffchange-inline">Show </del>that <math><del class="diffchange diffchange-inline">AP</del>, <del class="diffchange diffchange-inline">BQ</del></math> and <math><del class="diffchange diffchange-inline">CR</del></math> are <del class="diffchange diffchange-inline">[[</del>concurrent<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> </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">Given a nonisosceles, nonright triangle <math>\, ABC, \,</ins><<ins class="diffchange diffchange-inline">/</ins>math> <ins class="diffchange diffchange-inline">let <math>\, </ins>O <ins class="diffchange diffchange-inline">\,</ins></math> <ins class="diffchange diffchange-inline">denote the center </ins>of <ins class="diffchange diffchange-inline">its circumscribed circle, and let <math>\, A_1, \, B_1, \,</math> and </ins><math>\<ins class="diffchange diffchange-inline">, C_1 \,</ins></math> <ins class="diffchange diffchange-inline">be </ins>the midpoints of <ins class="diffchange diffchange-inline">sides </ins><math><ins class="diffchange diffchange-inline">\, </ins>BC<ins class="diffchange diffchange-inline">, \</ins>, CA, <ins class="diffchange diffchange-inline">\,</ins></math> <ins class="diffchange diffchange-inline">and </ins><math><ins class="diffchange diffchange-inline">\</ins>, <ins class="diffchange diffchange-inline">AB, \</ins>,</math> respectively. <ins class="diffchange diffchange-inline"> Point </ins><math><ins class="diffchange diffchange-inline">\</ins>, <ins class="diffchange diffchange-inline">A_2 \</ins>,</math> <ins class="diffchange diffchange-inline">is located </ins>on the <ins class="diffchange diffchange-inline">ray </ins><math><ins class="diffchange diffchange-inline">\</ins>, <ins class="diffchange diffchange-inline">OA_1 \</ins>,</math> so that <math>\<ins class="diffchange diffchange-inline">, \Delta OAA_1 </ins>\,<ins class="diffchange diffchange-inline"></math> is similar to <math>\, \Delta OA_2A</math>.  Points <math>\, B_2 \,</math> and <math>\, C_2 \,</math> on rays <math></ins>\<ins class="diffchange diffchange-inline">, OB_1 </ins>\<ins class="diffchange diffchange-inline">,</math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>\<ins class="diffchange diffchange-inline">, OC_1, </ins>\<ins class="diffchange diffchange-inline">,</ins></math> <ins class="diffchange diffchange-inline">respectively, are defined similarly</ins>. <ins class="diffchange diffchange-inline"> Prove </ins>that <ins class="diffchange diffchange-inline">lines </ins><math><ins class="diffchange diffchange-inline">\, AA_2, \, BB_2, \</ins>,</math> and <math><ins class="diffchange diffchange-inline">\, CC_2 \,</ins></math> are concurrent<ins class="diffchange diffchange-inline">, i.e. these three lines intersect at a point</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>[[1995 USAMO Problems/Problem 3|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>[[1995 USAMO Problems/Problem 3|Solution]]</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>==Problem 4==</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 4==</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><math><del class="diffchange diffchange-inline">a_1</del></math> is an <del class="diffchange diffchange-inline">[[</del>infinite<del class="diffchange diffchange-inline">]] </del>sequence of <del class="diffchange diffchange-inline">[[</del>integers<del class="diffchange diffchange-inline">]] such that </del><math><del class="diffchange diffchange-inline">a_n </del>- <del class="diffchange diffchange-inline">a_m</del></math> <del class="diffchange diffchange-inline">is divisible by </del><math><del class="diffchange diffchange-inline">n </del>- <del class="diffchange diffchange-inline">m</del></math> for <del class="diffchange diffchange-inline">all </del><math>n</math> <del class="diffchange diffchange-inline">and </del><math><del class="diffchange diffchange-inline">m</del></math> such that <math><del class="diffchange diffchange-inline">n</del>\<del class="diffchange diffchange-inline">ne m</math>. For some polynomial </del><<del class="diffchange diffchange-inline">math>p(x)</math> we have <math>p</del>(n) <del class="diffchange diffchange-inline">> |a_n|</del></math> for all <math>n</math>. <del class="diffchange diffchange-inline">Show </del>that there is a polynomial <math><del class="diffchange diffchange-inline">q(x)</del></math> such that <math><del class="diffchange diffchange-inline">q</del>(n) <del class="diffchange diffchange-inline">= a_n</del></math> for all <math>n</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> </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">Suppose </ins><math><ins class="diffchange diffchange-inline">\, q_0, \, q_1, \,  q_2, \ldots \; \,</ins></math> is an infinite sequence of integers <ins class="diffchange diffchange-inline">satisfying the following two conditions:<br></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">(i)  </ins><math><ins class="diffchange diffchange-inline">\, m</ins>-<ins class="diffchange diffchange-inline">n \,</ins></math> <ins class="diffchange diffchange-inline">divides </ins><math><ins class="diffchange diffchange-inline">\, q_m </ins>- <ins class="diffchange diffchange-inline">q_n \,</ins></math> for <math<ins class="diffchange diffchange-inline">>\, m </ins>> n <ins class="diffchange diffchange-inline">\geq 0,</ins></math> <ins class="diffchange diffchange-inline"><br></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">(ii) there is a polynomial </ins><math><ins class="diffchange diffchange-inline">\, P \,</ins></math> such that <math>\<ins class="diffchange diffchange-inline">, |q_n| </ins>< <ins class="diffchange diffchange-inline">P</ins>(n) <ins class="diffchange diffchange-inline">\,</ins></math> for all <math><ins class="diffchange diffchange-inline">\, </ins>n</math>. <ins class="diffchange diffchange-inline"><br></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 </ins>that there is a polynomial <math><ins class="diffchange diffchange-inline">\, Q \,</ins></math> such that <math><ins class="diffchange diffchange-inline">\, q_n = Q</ins>(n) <ins class="diffchange diffchange-inline">\,</ins></math> for all <math><ins class="diffchange diffchange-inline">\, </ins>n</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>[[1995 USAMO Problems/Problem 4|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>[[1995 USAMO Problems/Problem 4|Solution]]</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>==Problem 5==</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 5==</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">A [[graph (graph theory)|graph]] with </del><math>n</math> <del class="diffchange diffchange-inline">[[vertex|vertices]] </del>and <math><del class="diffchange diffchange-inline">k</del></math> <del class="diffchange diffchange-inline">edges has no faces </del>of <del class="diffchange diffchange-inline">degree </del>three. <del class="diffchange diffchange-inline">Show that it has a vertice <math>P</math> such </del>that there <del class="diffchange diffchange-inline">are </del>at <del class="diffchange diffchange-inline">most </del><math><del class="diffchange diffchange-inline">k</del>(1 - <del class="diffchange diffchange-inline">\frac{4k}{</del>n^2<del class="diffchange diffchange-inline">}</del>)</math> <del class="diffchange diffchange-inline">edges between points not joined to <math>P</math></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> </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">Suppose that in a certain society, each pair of persons can be classified as either ''amicable'' or ''hostile''. We shall say that each member of an amicable pair is a ''friend'' of the other, and each member of a hostile pair is a ''foe'' of the other.  Suppose that the society has </ins><math><ins class="diffchange diffchange-inline">\, </ins>n <ins class="diffchange diffchange-inline">\,</ins></math> <ins class="diffchange diffchange-inline">persons </ins>and <math><ins class="diffchange diffchange-inline">\, q \,</ins></math> <ins class="diffchange diffchange-inline">amicable pairs, and that for every set </ins>of three <ins class="diffchange diffchange-inline">persons, at least one pair is hostile</ins>. <ins class="diffchange diffchange-inline"> Prove </ins>that there <ins class="diffchange diffchange-inline">is </ins>at <ins class="diffchange diffchange-inline">least one member of the society whose foes include </ins><math><ins class="diffchange diffchange-inline">\, q</ins>(1 - <ins class="diffchange diffchange-inline">4q/</ins>n^2) <ins class="diffchange diffchange-inline">\,</ins></math> <ins class="diffchange diffchange-inline">or fewer amicable pairs</ins>. <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>[[1995 USAMO Problems/Problem 5|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>[[1995 USAMO Problems/Problem 5|Solution]]</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;">== Resources ==</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;">{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}</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;">* [http://www.artofproblemsolving.com/resources.php?c=182&cid=27&year=1995 1995 USAMO Problems on the resources page]</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;">* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]</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;">* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]</ins></div></td></tr>
</table>
Boy Soprano II
https://artofproblemsolving.com/wiki/index.php?title=1995_USAMO_Problems&diff=23108&oldid=prev
Temperal: problems page
2008-02-10T03:41:12Z
<p>problems page</p>
<p><b>New page</b></p><div>==Problem 1==<br />
The sequence <math>a_i</math> of [[nonnegative]] [[integer]]s is defined as follows: The first <math>p-1</math> terms are <math>0, 1, 2, 3, ... , p-2</math>. Then <math>a_n</math> is the least positive integer so that there is no arithmetic progression of length <math>p</math> in the first n+1 terms. If <math>p</math> is an odd prime, show that an is the number obtained by writing <math>n</math> in base <math>p-1</math>, then treating the result as a number in base <math>p</math>.<br />
<br />
[[1995 USAMO Problems/Problem 1|Solution]]<br />
==Problem 2==<br />
A trigonometric map is any one of <math>\sin, \cos, \tan, \arcsin, \arccos</math> and <math>\arctan</math>. Show that given any [[positive]] [[rational]] number <math>x</math>, one can find a finite sequence of trigonometric maps which take <math>0</math> to <math>x</math>.<br />
<br />
[[1995 USAMO Problems/Problem 2|Solution]]<br />
==Problem 3==<br />
The circumcenter <math>O</math> of the triangle <math>\triangle ABC</math> does not lie on any side or [[median]]. Let the midpoints of <math>BC, CA, AB</math> be <math>L, M, N</math> respectively. Construct <math>P, Q, R</math> on the rays <math>OL, OM, ON</math> respectively so that <math>\angle OPA = \angle OAL, \angle OQB = \angle OBM and \angle ORC = \angle OCN</math>. Show that <math>AP, BQ</math> and <math>CR</math> are [[concurrent]].<br />
<br />
[[1995 USAMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<math>a_1</math> is an [[infinite]] sequence of [[integers]] such that <math>a_n - a_m</math> is divisible by <math>n - m</math> for all <math>n</math> and <math>m</math> such that <math>n\ne m</math>. For some polynomial <math>p(x)</math> we have <math>p(n) > |a_n|</math> for all <math>n</math>. Show that there is a polynomial <math>q(x)</math> such that <math>q(n) = a_n</math> for all <math>n</math>. <br />
<br />
[[1995 USAMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
A [[graph (graph theory)|graph]] with <math>n</math> [[vertex|vertices]] and <math>k</math> edges has no faces of degree three. Show that it has a vertice <math>P</math> such that there are at most <math>k(1 - \frac{4k}{n^2})</math> edges between points not joined to <math>P</math>.<br />
<br />
[[1995 USAMO Problems/Problem 5|Solution]]</div>
Temperal