https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_1&feed=atom&action=history
2016 IMO Problems/Problem 1 - Revision history
2024-03-29T10:47:59Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_1&diff=204563&oldid=prev
Tomasdiaz at 05:35, 19 November 2023
2023-11-19T05:35:17Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 05:35, 19 November 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;">==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><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>Triangle <math>BCF</math> has a right angle at <math>B</math>. Let <math>A</math> be the point on line <math>CF</math> such that <math>FA=FB</math> and <math>F</math> lies between <math>A</math> and <math>C</math>. Point <math>D</math> is chosen so that <math>DA=DC</math> and <math>AC</math> is the bisector of <math>\angle{DAB}</math>. Point <math>E</math> is chosen so that <math>EA=ED</math> and <math>AD</math> is the bisector of <math>\angle{EAC}</math>. Let <math>M</math> be the midpoint of <math>CF</math>. Let <math>X</math> be the point such that <math>AMXE</math> is a parallelogram. Prove that <math>BD,FX</math> and <math>ME</math> are concurrent.</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>Triangle <math>BCF</math> has a right angle at <math>B</math>. Let <math>A</math> be the point on line <math>CF</math> such that <math>FA=FB</math> and <math>F</math> lies between <math>A</math> and <math>C</math>. Point <math>D</math> is chosen so that <math>DA=DC</math> and <math>AC</math> is the bisector of <math>\angle{DAB}</math>. Point <math>E</math> is chosen so that <math>EA=ED</math> and <math>AD</math> is the bisector of <math>\angle{EAC}</math>. Let <math>M</math> be the midpoint of <math>CF</math>. Let <math>X</math> be the point such that <math>AMXE</math> is a parallelogram. Prove that <math>BD,FX</math> and <math>ME</math> are concurrent.</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;">==Solution==</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 style="font-weight: bold; text-decoration: none;">==See Also==</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 style="font-weight: bold; text-decoration: none;">{{IMO box|year=2016|before=First Problem|num-a=2}}</ins></div></td></tr>
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Tomasdiaz
https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_1&diff=113495&oldid=prev
Piphi: Added LaTeX
2019-12-27T00:00:08Z
<p>Added LaTeX</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 00:00, 27 December 2019</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>Triangle BCF has a right angle at B.Let A be the point on line CF such that FA=FB and F lies between C <del class="diffchange diffchange-inline">and A</del>. Point D is chosen <del class="diffchange diffchange-inline">such </del>that DA=DC and AC is the bisector of <del class="diffchange diffchange-inline">∠DAB</del>. Point E is chosen <del class="diffchange diffchange-inline">such </del>that EA=ED and AD is the bisector of <del class="diffchange diffchange-inline">∠EAC</del>. Let M be the midpoint of CF . Let X be the point such that AMXE is a parallelogram <del class="diffchange diffchange-inline">(where AM||EX and AE||MX)</del>. Prove that <del class="diffchange diffchange-inline">the lines </del>BD,FX and ME are concurrent.</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>Triangle <ins class="diffchange diffchange-inline"><math></ins>BCF<ins class="diffchange diffchange-inline"></math> </ins>has a right angle at <ins class="diffchange diffchange-inline"><math></ins>B<ins class="diffchange diffchange-inline"></math></ins>. Let <ins class="diffchange diffchange-inline"><math></ins>A<ins class="diffchange diffchange-inline"></math> </ins>be the point on line <ins class="diffchange diffchange-inline"><math></ins>CF<ins class="diffchange diffchange-inline"></math> </ins>such that <ins class="diffchange diffchange-inline"><math></ins>FA=FB<ins class="diffchange diffchange-inline"></math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>F<ins class="diffchange diffchange-inline"></math> </ins>lies between <ins class="diffchange diffchange-inline"><math>A</math> and <math></ins>C<ins class="diffchange diffchange-inline"></math></ins>. Point <ins class="diffchange diffchange-inline"><math></ins>D<ins class="diffchange diffchange-inline"></math> </ins>is chosen <ins class="diffchange diffchange-inline">so </ins>that <ins class="diffchange diffchange-inline"><math></ins>DA=DC<ins class="diffchange diffchange-inline"></math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>AC<ins class="diffchange diffchange-inline"></math> </ins>is the bisector of <ins class="diffchange diffchange-inline"><math>\angle{DAB}</math></ins>. Point <ins class="diffchange diffchange-inline"><math></ins>E<ins class="diffchange diffchange-inline"></math> </ins>is chosen <ins class="diffchange diffchange-inline">so </ins>that <ins class="diffchange diffchange-inline"><math></ins>EA=ED<ins class="diffchange diffchange-inline"></math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>AD<ins class="diffchange diffchange-inline"></math> </ins>is the bisector of <ins class="diffchange diffchange-inline"><math>\angle{EAC}</math></ins>. Let <ins class="diffchange diffchange-inline"><math></ins>M<ins class="diffchange diffchange-inline"></math> </ins>be the midpoint of <ins class="diffchange diffchange-inline"><math></ins>CF<ins class="diffchange diffchange-inline"></math></ins>. Let <ins class="diffchange diffchange-inline"><math></ins>X<ins class="diffchange diffchange-inline"></math> </ins>be the point such that <ins class="diffchange diffchange-inline"><math></ins>AMXE<ins class="diffchange diffchange-inline"></math> </ins>is a parallelogram. Prove that <ins class="diffchange diffchange-inline"><math></ins>BD,FX<ins class="diffchange diffchange-inline"></math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>ME<ins class="diffchange diffchange-inline"></math> </ins>are concurrent.</div></td></tr>
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Piphi
https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_1&diff=106167&oldid=prev
Alapan1729 at 06:29, 8 June 2019
2019-06-08T06:29:55Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 06:29, 8 June 2019</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">Pr♦❜❧❡♠ ✶✳ ❚r✐❛♥❣❧❡ </del>BCF <del class="diffchange diffchange-inline">❤❛s ❛ r✐❣❤t ❛♥❣❧❡ ❛t B✳ ▲❡t </del>A <del class="diffchange diffchange-inline">❜❡ t❤❡ ♣♦✐♥t ♦♥ ❧✐♥❡ </del>CF <del class="diffchange diffchange-inline">s✉❝❤ t❤❛t </del>FA = FB <del class="diffchange diffchange-inline">❛♥❞ </del>F <del class="diffchange diffchange-inline">❧✐❡s ❜❡t✇❡❡♥ </del>A <del class="diffchange diffchange-inline">❛♥❞ C✳ P♦✐♥t </del>D <del class="diffchange diffchange-inline">✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t </del>DA = DC <del class="diffchange diffchange-inline">❛♥❞ </del>AC <del class="diffchange diffchange-inline">✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ∠DAB✳ P♦✐♥t </del>E <del class="diffchange diffchange-inline">✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t </del>EA = ED <del class="diffchange diffchange-inline">❛♥❞ </del>AD <del class="diffchange diffchange-inline">✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ∠EAC✳ ▲❡t </del>M <del class="diffchange diffchange-inline">❜❡ t❤❡ ♠✐❞♣♦✐♥t ♦❢ CF✳ ▲❡t </del>X <del class="diffchange diffchange-inline">❜❡ t❤❡ ♣♦✐♥t s✉❝❤ t❤❛t </del>AMXE <del class="diffchange diffchange-inline">✐s ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ✭✇❤❡r❡ </del>AM <del class="diffchange diffchange-inline">k </del>EX <del class="diffchange diffchange-inline">❛♥❞ </del>AE <del class="diffchange diffchange-inline">k MX✮✳ Pr♦✈❡ t❤❛t ❧✐♥❡s BD✱ FX✱ ❛♥❞ </del>ME <del class="diffchange diffchange-inline">❛r❡ ❝♦♥❝✉rr❡♥t✳</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">Triangle </ins>BCF <ins class="diffchange diffchange-inline">has a right angle at B.Let </ins>A <ins class="diffchange diffchange-inline">be the point on line </ins>CF <ins class="diffchange diffchange-inline">such that </ins>FA=FB <ins class="diffchange diffchange-inline">and </ins>F <ins class="diffchange diffchange-inline">lies between C and </ins>A<ins class="diffchange diffchange-inline">. Point </ins>D <ins class="diffchange diffchange-inline">is chosen such that </ins>DA=DC <ins class="diffchange diffchange-inline">and </ins>AC <ins class="diffchange diffchange-inline">is the bisector of ∠DAB. Point </ins>E <ins class="diffchange diffchange-inline">is chosen such that </ins>EA=ED <ins class="diffchange diffchange-inline">and </ins>AD <ins class="diffchange diffchange-inline">is the bisector of ∠EAC. Let </ins>M <ins class="diffchange diffchange-inline">be the midpoint of CF . Let </ins>X <ins class="diffchange diffchange-inline">be the point such that </ins>AMXE <ins class="diffchange diffchange-inline">is a parallelogram (where </ins>AM<ins class="diffchange diffchange-inline">||</ins>EX <ins class="diffchange diffchange-inline">and </ins>AE<ins class="diffchange diffchange-inline">||MX). Prove that the lines BD,FX and </ins>ME <ins class="diffchange diffchange-inline">are concurrent.</ins></div></td></tr>
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Alapan1729
https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_1&diff=106166&oldid=prev
Alapan1729: Created page with "Pr♦❜❧❡♠ ✶✳ ❚r✐❛♥❣❧❡ BCF ❤❛s ❛ r✐❣❤t ❛♥❣❧❡ ❛t B✳ ▲❡t A ❜❡ t❤❡ ♣♦✐♥t ♦♥ ❧✐♥❡ CF s✉❝❤..."
2019-06-08T06:18:54Z
<p>Created page with "Pr♦❜❧❡♠ ✶✳ ❚r✐❛♥❣❧❡ BCF ❤❛s ❛ r✐❣❤t ❛♥❣❧❡ ❛t B✳ ▲❡t A ❜❡ t❤❡ ♣♦✐♥t ♦♥ ❧✐♥❡ CF s✉❝❤..."</p>
<p><b>New page</b></p><div>Pr♦❜❧❡♠ ✶✳ ❚r✐❛♥❣❧❡ BCF ❤❛s ❛ r✐❣❤t ❛♥❣❧❡ ❛t B✳ ▲❡t A ❜❡ t❤❡ ♣♦✐♥t ♦♥ ❧✐♥❡ CF s✉❝❤ t❤❛t FA = FB ❛♥❞ F ❧✐❡s ❜❡t✇❡❡♥ A ❛♥❞ C✳ P♦✐♥t D ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t DA = DC ❛♥❞ AC ✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ∠DAB✳ P♦✐♥t E ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t EA = ED ❛♥❞ AD ✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ∠EAC✳ ▲❡t M ❜❡ t❤❡ ♠✐❞♣♦✐♥t ♦❢ CF✳ ▲❡t X ❜❡ t❤❡ ♣♦✐♥t s✉❝❤ t❤❛t AMXE ✐s ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ✭✇❤❡r❡ AM k EX ❛♥❞ AE k MX✮✳ Pr♦✈❡ t❤❛t ❧✐♥❡s BD✱ FX✱ ❛♥❞ ME ❛r❡ ❝♦♥❝✉rr❡♥t✳</div>
Alapan1729