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 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
Y by rightways, Adventure10, Mango247, and 1 other user
Given the rays 
and 
.Inside the smaller angle 
selected points 
and 
, such that 
and 
The circle, which concern the rays 
and 
, intersects the second circle, which concern the rays 
and at the points 
and 
. Prove that
and 
.Inside the smaller angle 
selected points 
and 
, such that 
and 
The circle, which concern the rays 
and 
, intersects the second circle, which concern the rays 
and at the points 
and 
. Prove that
This post has been edited 3 times. Last edited by ts0_9, May 20, 2012, 7:17 PM
when 
is the bisector of 
resp. as we might do and an inversion with 
are on 
inversion and swap about the bisector swaps the two circles and hence 
and 
.
