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
Given 
is cicrumcircle of 
and 
is inside .Contruct perpendicular line at 
of 
,four line intersect at 
as you seen in fig.Prove that 
Easy to see that from this we have the midpoints of
and 
is colinear.(Same as proof of Newton line)
If you can prove midpoints of and is colinear but not use ,plz show me.Thanks.
is cicrumcircle of 
and 
is inside .Contruct perpendicular line at 
of 
,four line intersect at 
as you seen in fig.Prove that 
Easy to see that from this we have the midpoints of
and 
is colinear.(Same as proof of Newton line)If you can prove midpoints of
and is colinear but not use ,plz show me.Thanks.Attachments:
are tangent to a central conic 
with pedal circle 
(the reflection of T about O). See the previous lemma in the problem 
as the problem states, then 
circumscribed about 
circumscribed about a circle with center 
Midpoints go to midpoints under the parallel projection 
Newton line 
of 
of 
Since 
(well-known), then 
and the conclusion follows.
