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 
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.
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
with 
.
lie on 
respectively, such that 
.
to meet the circumcircle of 
respectively. Let the circumcircles of 
meet at points 
.
to 
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
,
.
to meet the circumcircle of 
at 
,
.
and 
.
Equality holds when 
Equality holds when 
Equality holds when 
Y by Mango247
A semicircular arc and a diameter 
with a length of 
are given. Let 
be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point 
at the distance 
from the midpoint of the diameter , 
. A line through 
and intersects the semicircle at point 
. Through point we draw another line at right angle against 
that intersects the semicircle at point 
. Through point we draw a line 
, parallel to 
and then a line 
, through parallel to 
. The lines and intersect at point 
. Show that the distance between and is equal to 
with a length of 
are given. Let 
be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point 
at the distance 
from the midpoint of the diameter , 
. A line through 
and intersects the semicircle at point 
. Through point we draw another line at right angle against 
that intersects the semicircle at point 
. Through point we draw a line 
, parallel to 
and then a line 
, through parallel to 
. The lines and intersect at point 
. Show that the distance between and is equal to 
