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.
Y by Adventure10, Mango247
Point 
is chosen at the diagonal 
of parallelogram 
The straight line 
intersects the side 
and the straight line 
at points 
and 
respectively. Let 
be the circle with centre and radius 
and 
be the circumcircle of triangle 
Denote by 
and 
the intersection points of circles and 
Prove that the circle is inscribed into the angle 
is chosen at the diagonal 
of parallelogram 
The straight line 
intersects the side 
and the straight line 
at points 
and 
respectively. Let 
be the circle with centre and radius 
and 
be the circumcircle of triangle 
Denote by 
and 
the intersection points of circles and 
Prove that the circle is inscribed into the angle 
be the center of the parallelogram and 
the reflection of 
about 
Since 
Let 
be the infinity point of this direction. Since 
it follows that the pencil 
is harmonic 
By Newton's theorem, we have then 
