https://artofproblemsolving.com/wiki/index.php?title=2015_IMO_Problems/Problem_4&feed=atom&action=history 2015 IMO Problems/Problem 4 - Revision history 2020-09-28T05:27:40Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2015_IMO_Problems/Problem_4&diff=72394&oldid=prev Bandera: Page created and Bandera's solution added 2015-10-06T03:58:25Z <p>Page created and Bandera&#039;s solution added</p> <p><b>New page</b></p><div>==Problem==<br /> <br /> Triangle &lt;math&gt;ABC&lt;/math&gt; has circumcircle &lt;math&gt;\Omega&lt;/math&gt; and circumcenter &lt;math&gt;O&lt;/math&gt;. A circle &lt;math&gt;\Gamma&lt;/math&gt; with center &lt;math&gt;A&lt;/math&gt; intersects the segment &lt;math&gt;BC&lt;/math&gt; at points &lt;math&gt;D&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt;, such that &lt;math&gt;B&lt;/math&gt;, &lt;math&gt;D&lt;/math&gt;, &lt;math&gt;E&lt;/math&gt;, and &lt;math&gt;C&lt;/math&gt; are all different and lie on line &lt;math&gt;BC&lt;/math&gt; in this order. Let &lt;math&gt;F&lt;/math&gt; and &lt;math&gt;G&lt;/math&gt; be the points of intersection of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt;, such that &lt;math&gt;A&lt;/math&gt;, &lt;math&gt;F&lt;/math&gt;, &lt;math&gt;B&lt;/math&gt;, &lt;math&gt;C&lt;/math&gt;, and &lt;math&gt;G&lt;/math&gt; lie on &lt;math&gt;\Omega&lt;/math&gt; in this order. Let &lt;math&gt;K&lt;/math&gt; be the second point of intersection of the circumcircle of triangle &lt;math&gt;BDF&lt;/math&gt; and the segment &lt;math&gt;AB&lt;/math&gt;. Let &lt;math&gt;L&lt;/math&gt; be the second point of intersection of the circumcircle of triangle &lt;math&gt;CGE&lt;/math&gt; and the segment &lt;math&gt;CA&lt;/math&gt;.&lt;br /&gt;&lt;br /&gt;<br /> Suppose that the lines &lt;math&gt;FK&lt;/math&gt; and &lt;math&gt;GL&lt;/math&gt; are different and intersect at the point &lt;math&gt;X&lt;/math&gt;. Prove that &lt;math&gt;X&lt;/math&gt; lies on the line &lt;math&gt;AO&lt;/math&gt;.&lt;br /&gt;&lt;br /&gt;<br /> Proposed by Silouanos Brazitikos and Evangelos Psychas, Greece<br /> <br /> ==Solution==<br /> <br /> &lt;u&gt;'''Lemma'''&lt;/u&gt; (On three chords). ''If two lines pass through different endpoints of two circles' common chord, then the other two chords cut by these lines on the circles are parallel.''&lt;br /&gt;<br /> &lt;u&gt;Proof&lt;/u&gt; The second and the third chords are anti-parallel to the first (common) chord with respect to the given lines, so they are parallel to each other. &lt;math&gt;\Box&lt;/math&gt;&lt;br /&gt;<br /> To solve this problem, it is sufficient to apply the lemma 5 times. Indeed, let the lines &lt;math&gt;FD, GE, FK, GL&lt;/math&gt; meet &lt;math&gt;\Omega&lt;/math&gt; second time at &lt;math&gt;H, I, M, N&lt;/math&gt; respectively. One of the circles that figure in lemma is always &lt;math&gt;\Omega&lt;/math&gt;, while the other is one of three other circles from the problem statement. Applying the lemma to the lines &lt;math&gt;FDH&lt;/math&gt; and &lt;math&gt;GEI&lt;/math&gt;, &lt;math&gt;FKM&lt;/math&gt; and &lt;math&gt;BDC&lt;/math&gt;, &lt;math&gt;FDH&lt;/math&gt; and &lt;math&gt;BKA&lt;/math&gt;, &lt;math&gt;GLN&lt;/math&gt; and &lt;math&gt;CEB&lt;/math&gt;, &lt;math&gt;GEI&lt;/math&gt; and &lt;math&gt;CLA&lt;/math&gt;, we get &lt;math&gt;DE \parallel IH&lt;/math&gt;, &lt;math&gt;KD \parallel MC&lt;/math&gt;, &lt;math&gt;KD \parallel AH&lt;/math&gt;, &lt;math&gt;LE \parallel NB&lt;/math&gt;, &lt;math&gt;LE \parallel AI&lt;/math&gt;, respectively. From this, &lt;math&gt;BC \parallel IH&lt;/math&gt;, &lt;math&gt;MC \parallel AH&lt;/math&gt;, &lt;math&gt;NB \parallel AI&lt;/math&gt;. Therefore, &lt;math&gt;AN=IB=HC=AM&lt;/math&gt;. This means that &lt;math&gt;N&lt;/math&gt; and &lt;math&gt;M&lt;/math&gt; are symmetric wrt &lt;math&gt;AO&lt;/math&gt;, a diameter of &lt;math&gt;\Omega&lt;/math&gt; through &lt;math&gt;A&lt;/math&gt;. So are &lt;math&gt;F&lt;/math&gt; and &lt;math&gt;G&lt;/math&gt;, as &lt;math&gt;AF=AG&lt;/math&gt;. Therefore, the lines &lt;math&gt;FM&lt;/math&gt; and &lt;math&gt;GN&lt;/math&gt; are symmetric wrt &lt;math&gt;AO&lt;/math&gt; and meet on it.<br /> <br /> ==See Also==<br /> <br /> {{IMO box|year=2015|num-b=3|num-a=5}}<br /> <br /> [[Category:Olympiad Geometry Problems]]</div> Bandera