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be the set of positive real numbers. Determine all functions 
such that for all positive real numbers 
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 
. The tangents to at 
meet at point 
. For a point 
on line 
which is not on the segment , let the midpoint of 
be 
. Lines 
meet again at points 
respectively. Let 
be the midpoint of . Prove that the points 
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 
be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line 
is perpendicular to the axis of the parabola.
and be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line is perpendicular to the axis of the parabola.
for some 
. Let 
be the points of intersection of circle centered at 
and the parabola such that 
lie on the same side of the axis of the parabola and quadrilateral 
is convex. First coordinate of points will be denoted with small letter. For other two: 
where 
. We wish to prove 
.
gives 
so Viete's formulas imply 
.
Adding and substracting equations yields 
Thus 
and 
Returning to 
we have 
since lie on the same side of the axis of the parabola so 
.
