https://artofproblemsolving.com/wiki/index.php?title=2008_IMO_Problems/Problem_6&feed=atom&action=history 2008 IMO Problems/Problem 6 - Revision history 2021-06-16T11:56:25Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2008_IMO_Problems/Problem_6&diff=31200&oldid=prev Vo Duc Dien: Hints for problem 6 IMO 2008 2009-04-10T23:17:59Z <p>Hints for problem 6 IMO 2008</p> <p><b>New page</b></p><div>Here are some hints:<br /> <br /> Let B be the top vertex of triangle ABC, O and K are the centers of the incircles of triangles ABC and ADC with radii R and r, respectively. S is the center of the circumcircle tangential to the extensions of AB, AC and DC. And let<br /> <br /> E be the foot of the projection of O to AB.<br /> U be the foot of the projection of O to AC.<br /> V be the foot of the projection of K to AC.<br /> M be the foot of the projection of S to DC.<br /> L be the foot of the projection of K to DC.<br /> L be the intercept of DC and AB.<br /> <br /> We have<br /> <br /> /_EOB = /_LSC<br /> /_AOU = /_SOC<br /> /_ASO = /_KSC<br /> /_ASK = /_OSC<br /> /_LSB = /_KCO<br /> UV = BC – AB<br /> AU = VC<br /> OA. AK. cos(/_OAK) = OC. KC. cos(/_OCK)<br /> OK**2 = (R + r)**2 + UV**2<br /> SK**2 = (R' + r)**2 + ML**2<br /> <br /> Use sin (90-x) = cos x and cos(90-x) = sinx<br /> and characteristic of triangle a**2 = b**2 + c**2 - 2.b.c.cosine(angle) to solve.<br /> <br /> Vo Duc Dien</div> Vo Duc Dien