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)
be 
.
meet at point 
.
on line 
which is not on the segment 
be 
.
meet 
respectively. Let 
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 chord of a circle 
and 
a diameter of perpendicular to at 
with 
. A circle 
centered at 
with radius 
intersects at points 
and 
. The line 
intersects at 
and 
at 
; and the extension of 
meets at 
. Prove that is perpendicular to 
be a chord of a circle 
and 
a diameter of perpendicular to at 
with 
. A circle 
centered at 
with radius 
intersects at points 
and 
. The line 
intersects at 
and 
at 
; and the extension of 
meets at 
. Prove that is perpendicular to 
we get that 
is cyclic so 
intersect 
Since 
and 
we get 
.
is a cyclic kite 
I think that inversion is faster: of pole 
sends the circle 
to 
,
.
,
is parallel to 
,
so points 
are concyclic.Hence 
L
.
.
be the center of 
.
concyclic, where 
.
