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 
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
Take any point 
on the side 
of a triangle 
and draw the following chain of lines: 
parallel to 
; 
parallel to ; 
parallel to 
; 
parallel to 
; and 
parallel to , Here, 
lie on ; 
lie on and 
on > Show that 
is parallel to .
on the side 
of a triangle 
and draw the following chain of lines: 
parallel to 
; 
parallel to ; 
parallel to 
; 
parallel to 
; and 
parallel to , Here, 
lie on ; 
lie on and 
on > Show that 
is parallel to .
so 
another parallelogram.
and 
are paralleogram beside
are homothetic then 
and 
and 
with 
yield 
is parallelogram
=
=
=
=
=
(
the result follows.
and 
notice we have 
then 
and we also have 
and 
then 
a parallelogram
,
