https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&feed=atom&action=history
2016 USAJMO Problems - Revision history
2024-03-29T11:25:47Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&diff=196452&oldid=prev
Etmetalakret at 20:43, 5 August 2023
2023-08-05T20:43:15Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:43, 5 August 2023</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>[[2016 USAJMO Problems/Problem 6|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>[[2016 USAJMO Problems/Problem 6|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 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;">== See Also ==</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;">*[[USAJMO Problems and Solutions]]</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>{{USAJMO box|year=2016|before=[[2015 USAJMO Problems]]|after=[[2017 USAJMO 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>{{USAJMO box|year=2016|before=[[2015 USAJMO Problems]]|after=[[2017 USAJMO Problems]]}}</div></td></tr>
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Etmetalakret
https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&diff=196451&oldid=prev
Etmetalakret at 20:42, 5 August 2023
2023-08-05T20:42:38Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:42, 5 August 2023</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="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>{{USAJMO box|year=2016|before=[[2015 USAJMO Problems<del class="diffchange diffchange-inline">|2015 USAJMO</del>]]|after=[[2017 USAJMO Problems<del class="diffchange diffchange-inline">|2017 USAJMO</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 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">*[[USAJMO Problems and Solutions]]</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>
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Etmetalakret
https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&diff=196442&oldid=prev
Etmetalakret at 20:37, 5 August 2023
2023-08-05T20:37:05Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:37, 5 August 2023</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;">{{USAJMO box|year=2016|before=[[2015 USAJMO Problems|2015 USAJMO]]|after=[[2017 USAJMO Problems|2017 USAJMO]]}}</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>{{MAA Notice}}</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>{{MAA Notice}}</div></td></tr>
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Etmetalakret
https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&diff=92379&oldid=prev
Mathgeek2006: latex fix
2018-02-26T06:24:38Z
<p>latex fix</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 06:24, 26 February 2018</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 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>Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math>{1, 2,<del class="diffchange diffchange-inline">...</del>,N}</math>, one can still find <math>2016</math> distinct numbers among the remaining elements with sum <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>Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math><ins class="diffchange diffchange-inline">\</ins>{1, 2,<ins class="diffchange diffchange-inline">\dots</ins>,N<ins class="diffchange diffchange-inline">\</ins>}</math>, one can still find <math>2016</math> distinct numbers among the remaining elements with sum <math>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>[[2016 USAJMO 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>[[2016 USAJMO Problems/Problem 4|Solution]]</div></td></tr>
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Mathgeek2006
https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&diff=78165&oldid=prev
Benq at 20:42, 21 April 2016
2016-04-21T20:42:41Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:42, 21 April 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>{{MAA Notice}}</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>{{MAA Notice}}</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 style="font-weight: bold; text-decoration: none;">{{USAJMO newbox|year= 2016 |before=[[2015 USAJMO]]|after=[[2017 USAJMO]]}}</ins></div></td></tr>
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Benq
https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&diff=78151&oldid=prev
Benq: Created page with "==Day 1== ===Problem 1=== The isosceles triangle <math>\triangle ABC</math>, with <math>AB=AC</math>, is inscribed in the circle <math>\omega</math>. Let <math>P</math> be a v..."
2016-04-21T18:56:50Z
<p>Created page with "==Day 1== ===Problem 1=== The isosceles triangle <math>\triangle ABC</math>, with <math>AB=AC</math>, is inscribed in the circle <math>\omega</math>. Let <math>P</math> be a v..."</p>
<p><b>New page</b></p><div>==Day 1==<br />
===Problem 1===<br />
The isosceles triangle <math>\triangle ABC</math>, with <math>AB=AC</math>, is inscribed in the circle <math>\omega</math>. Let <math>P</math> be a variable point on the arc <math>\stackrel{\frown}{BC}</math> that does not contain <math>A</math>, and let <math>I_B</math> and <math>I_C</math> denote the incenters of triangles <math>\triangle ABP</math> and <math>\triangle ACP</math>, respectively.<br />
<br />
Prove that as <math>P</math> varies, the circumcircle of triangle <math>\triangle PI_BI_C</math> passes through a fixed point.<br />
<br />
[[2016 USAJMO Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Prove that there exists a positive integer <math>n < 10^6</math> such that <math>5^n</math> has six consecutive zeros in its decimal representation.<br />
<br />
[[2016 USAJMO Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
Let <math>X_1, X_2, \ldots, X_{100}</math> be a sequence of mutually distinct nonempty subsets of a set <math>S</math>. Any two sets <math>X_i</math> and <math>X_{i+1}</math> are disjoint and their union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldots, 99\}</math>. Find the smallest possible number of elements in <math>S</math>.<br />
<br />
[[2016 USAJMO Problems/Problem 3|Solution]]<br />
<br />
==Day 2==<br />
===Problem 4===<br />
Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math>{1, 2,...,N}</math>, one can still find <math>2016</math> distinct numbers among the remaining elements with sum <math>N</math>.<br />
<br />
[[2016 USAJMO Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Let <math>\triangle ABC</math> be an acute triangle, with <math>O</math> as its circumcenter. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to line <math>\overleftrightarrow{BC}</math>, and points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>H</math> to the lines <math>\overleftrightarrow{AB}</math> and <math>\overleftrightarrow{AC}</math>, respectively.<br />
<br />
Given that <cmath>AH^2=2\cdot AO^2,</cmath>prove that the points <math>O,P,</math> and <math>Q</math> are collinear.<br />
<br />
[[2016 USAJMO Problems/Problem 5|Solution]]<br />
<br />
===Problem 6===<br />
<br />
Find all functions <math>f:\mathbb{R}\rightarrow \mathbb{R}</math> such that for all real numbers <math>x</math> and <math>y</math>,<br />
<cmath>(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.</cmath><br />
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[[2016 USAJMO Problems/Problem 6|Solution]]<br />
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