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 
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 Adventure10, Mango247
In a plane, the six different points 
are given such that triangles 
and 
are congruent, i.e. 
Let 
be the centroid of and 
be an intersection point of the circle with diameter 
and the circle with center 
and passing through 
Define 
and 
similarly. Prove that
![\[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]](//latex.artofproblemsolving.com/e/d/6/ed61fcba719f638475d07b9c78a39b4297240c0a.png)
are given such that triangles 
and 
are congruent, i.e. 
Let 
be the centroid of and 
be an intersection point of the circle with diameter 
and the circle with center 
and passing through 
Define 
and 
similarly. Prove that![\[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]](http://latex.artofproblemsolving.com/e/d/6/ed61fcba719f638475d07b9c78a39b4297240c0a.png)
we get the small circle is completely in the bigger, may we also say 
by the Pythagorean theorem and angle on the diameter.
are equilateral having same 
![$[BB'],[CC'],[AA']$](http://latex.artofproblemsolving.com/6/f/9/6f926b32d0bf8d1f9e19b951d6031d209cdc48f1.png)
in general. (G=G')
and hence done.
it isn't proven by me, did you do that?
.
,
and 
are the midpoints of 
respectively. The rest is obvious. Equality occurs iff 
(Note that this is true for any non-degenerate quadrilateral)
.
,
and the equation 
(which can be proven by the side bisector theorem), our desired inequality turns into
)
,
,
with ratio 
