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
a) In a triangle 
, the lenghts of the sides are less than 
. Prove that the lenght of the altitude corresponding to the side 
is less than 
.
b) In a tetrahedron
, at least 
edges have their lenghts less than .Prove that the volume of the tetrahedron is less than 
.
, the lenghts of the sides are less than 
. Prove that the lenght of the altitude corresponding to the side 
is less than 
.b) In a tetrahedron
, at least 
edges have their lenghts less than .Prove that the volume of the tetrahedron is less than 
.
be the median related to the line 
and 
isn't necesarily less than 2. Denote 
the projection of the point 
on the line 
.In the triangle 
the sides are less than 2 , therefore 
Analogously the altitude 
from vertice 
in the triangle 
is less than 
therefore ![$ V[ABCD]=\frac{1}{3}\cdot h\cdot [BCD]<\frac{a(16-a^2)}{24}$](http://latex.artofproblemsolving.com/b/2/b/b2b2c909093a5a5374923c6bf312610258866518.png)
.
and 
, 
from where the conclusion results.
I made a stupid mistake 
