be the set of positive real numbers. Determine all functions 
and 
![\[f(x+f(xy))+y=f(x)f(y)+1\]](http://latex.artofproblemsolving.com/c/6/a/c6a968a17f6fd5fd637c1ee5d855bdff11e5423a.png)
and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
.
meet at point 
.
on line 
which is not on the segment 
be 
.
meet 
respectively. Let 
be the midpoint of 
lie on a circle.
red cards, 
white cards, and 
blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
Y by Adventure10
Let 
be a triangle, such that its incenter 
and circumcenter 
are distinct. For all points 
in the interior of the triangle let 
be the sum of distances from to the three (possibly extended) sides of the triangle.
Prove: If two distinct points
in the interior of the triangle satisfy 
, then 
is perpendicular to 
.
be a triangle, such that its incenter 
and circumcenter 
are distinct. For all points 
in the interior of the triangle let 
be the sum of distances from to the three (possibly extended) sides of the triangle.Prove: If two distinct points
in the interior of the triangle satisfy 
, then 
is perpendicular to 
.
