such that 
on 
.
intersects the circumcircle of 
at 
intersects the circumcircle of 
at 
.
does not depend on the choice of 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
coins, all of which show 
on one side and 
on the opposite side. The coins are laid out in a row on the table. Peer and Solveig alternate taking turns. On his turn, Peer may turn over one or more coins, so long as he does not turn over two adjacent coins. On her turn, Solveig picks precisely two adjacent coins and turns them over. When the game begins, all the coins are showing 
for which Solveig can keep Peer from winning.
Y by Adventure10, Mango247
Let 
be a point in the interior of 
. The lengths of the sides of is 
, and the distance from to the sides of is 
. Show that the circumradius 
of satisfies ![\[\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.\]](//latex.artofproblemsolving.com/f/5/5/f55add952445e30cc31fb38a1cf29c0b3f8b79ac.png)
When does equality hold?
be a point in the interior of 
. The lengths of the sides of is 
, and the distance from to the sides of is 
. Show that the circumradius 
of satisfies ![\[\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.\]](http://latex.artofproblemsolving.com/f/5/5/f55add952445e30cc31fb38a1cf29c0b3f8b79ac.png)
When does equality hold?This post has been edited 1 time. Last edited by k2c901_1, Aug 14, 2005, 3:05 PM
![$2S=ap+bq+cr\ge3(abcpqr)^{\frac{1}{3}}\Rightarrow\frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}\ge \frac{(a^2+b^2+c^2)\sqrt[3]{abc}}{12S}$](http://latex.artofproblemsolving.com/e/6/3/e63485bdfb7987fecd14649feb0d1be368fe399f.png)
.
.
