https://artofproblemsolving.com/wiki/index.php?title=2007_Cyprus_MO/Lyceum/Problem_6&feed=atom&action=history
2007 Cyprus MO/Lyceum/Problem 6 - Revision history
2024-03-29T09:21:03Z
Revision history for this page on the wiki
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Leonariso at 14:39, 8 May 2007
2007-05-08T14:39:13Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:39, 8 May 2007</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></div></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></div></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>\displaystyle ABCD</math> is a [[square]] of side length 2 and <math>FG</math> is an [[arc]] of the [[circle]] with centre the [[midpoint]] <math>K</math> of the side <math><del class="diffchange diffchange-inline">\overline{</del>AB<del class="diffchange diffchange-inline">}</del></math> and [[radius]] 2. The length of the segments <math>\displaystyle FD=GC=x</math> is</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><math>\displaystyle ABCD</math> is a [[square]] of side length 2 and <math>FG</math> is an [[arc]] of the [[circle]] with centre the [[midpoint]] <math>K</math> of the side <math>AB</math> and [[radius]] 2. The length of the segments <math>\displaystyle FD=GC=x</math> is</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><math> \mathrm{(A) \ } \frac 14\qquad \mathrm{(B) \ } \frac{\sqrt{2}}2\qquad \mathrm{(C) \ } 2-\sqrt{3}\qquad \mathrm{(D) \ } \sqrt{3} - 1\qquad \mathrm{(E) \ } \sqrt{2} - 1</math></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><math> \mathrm{(A) \ } \frac 14\qquad \mathrm{(B) \ } \frac{\sqrt{2}}2\qquad \mathrm{(C) \ } 2-\sqrt{3}\qquad \mathrm{(D) \ } \sqrt{3} - 1\qquad \mathrm{(E) \ } \sqrt{2} - 1</math></div></td></tr>
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Leonariso
https://artofproblemsolving.com/wiki/index.php?title=2007_Cyprus_MO/Lyceum/Problem_6&diff=15044&oldid=prev
Leonariso: img
2007-05-08T14:38:33Z
<p>img</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:38, 8 May 2007</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">{{image}}</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"><div style="float:right"></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">[[Image:2007 CyMO-6.PNG|250px]]</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"></div></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><math>\displaystyle ABCD</math> is a [[square]] of side length 2 and <math>FG</math> is an [[arc]] of the [[circle]] with centre the [[midpoint]] <math>K</math> of the side <math>\overline{AB}</math> and [[radius]] 2. The length of the segments <math>\displaystyle FD=GC=x</math> is</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><math>\displaystyle ABCD</math> is a [[square]] of side length 2 and <math>FG</math> is an [[arc]] of the [[circle]] with centre the [[midpoint]] <math>K</math> of the side <math>\overline{AB}</math> and [[radius]] 2. The length of the segments <math>\displaystyle FD=GC=x</math> is</div></td></tr>
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Leonariso
https://artofproblemsolving.com/wiki/index.php?title=2007_Cyprus_MO/Lyceum/Problem_6&diff=14993&oldid=prev
I_like_pie at 00:41, 7 May 2007
2007-05-07T00:41:31Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 00:41, 7 May 2007</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>BC}</math>, respectively)  </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>BC}</math>, respectively)  </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>Draw radii <math>\overline{KF},\ \overline{KG}</math>, both with length <math>2</math>. <math>\displaystyle AK = BK = 1</math>, so we form <math>30-60-90</math> [[right triangle]]s. <math>AF = BG = \sqrt{3}</math>, and so <math>DF = CG = 2 - \sqrt{3} \<del class="diffchange diffchange-inline">Rightarrow </del>\mathrm{C}</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>Draw radii <math>\overline{KF},\ \overline{KG}</math>, both with length <math>2</math>. <math>\displaystyle AK = BK = 1</math>, so we form <math>30-60-90</math> [[right triangle]]s. <math>AF = BG = \sqrt{3}</math>, and so <math>DF = CG = 2 - \sqrt{3} \<ins class="diffchange diffchange-inline">Longrightarrow </ins>\mathrm{C}</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>==See also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td></tr>
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I_like_pie
https://artofproblemsolving.com/wiki/index.php?title=2007_Cyprus_MO/Lyceum/Problem_6&diff=14968&oldid=prev
Azjps: Don't have img so I can't solve this problem, but I believe this is the correct solution
2007-05-06T20:31:38Z
<p>Don't have img so I can't solve this problem, but I believe this is the correct solution</p>
<p><b>New page</b></p><div>{{image}}<br />
<math>\displaystyle ABCD</math> is a [[square]] of side length 2 and <math>FG</math> is an [[arc]] of the [[circle]] with centre the [[midpoint]] <math>K</math> of the side <math>\overline{AB}</math> and [[radius]] 2. The length of the segments <math>\displaystyle FD=GC=x</math> is<br />
<br />
<math> \mathrm{(A) \ } \frac 14\qquad \mathrm{(B) \ } \frac{\sqrt{2}}2\qquad \mathrm{(C) \ } 2-\sqrt{3}\qquad \mathrm{(D) \ } \sqrt{3} - 1\qquad \mathrm{(E) \ } \sqrt{2} - 1</math><br />
<br />
==Solution==<br />
(<math>F,\ G</math> are on <math>\overline{AD},\ \overline{<br />
BC}</math>, respectively) <br />
<br />
Draw radii <math>\overline{KF},\ \overline{KG}</math>, both with length <math>2</math>. <math>\displaystyle AK = BK = 1</math>, so we form <math>30-60-90</math> [[right triangle]]s. <math>AF = BG = \sqrt{3}</math>, and so <math>DF = CG = 2 - \sqrt{3} \Rightarrow \mathrm{C}</math>.<br />
<br />
==See also==<br />
{{CYMO box|year=2007|l=Lyceum|num-b=5|num-a=7}}<br />
<br />
[[Category:Introductory Geometry Problems]]</div>
Azjps