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 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
Y by
Let 
be the midpoint of the arc 
, that does not contain the vertex 
, of the circle circumscribed around triangle 
. Suppose 
is a line parallel to the line 
passing through the point , and 
is a line parallel to 
passing through the point . Let the circle passing through point 
and point , tangent to , intersect the line at point 
, different than point . Let a circle passing through point 
and point , tangent to , intersect the line at point 
, different from point .
a) Prove that
b) Prove that the points
and lie on a circle.
Note: Grade 10 had (a), grades 11-12 had (b)
be the midpoint of the arc 
, that does not contain the vertex 
, of the circle circumscribed around triangle 
. Suppose 
is a line parallel to the line 
passing through the point , and 
is a line parallel to 
passing through the point . Let the circle passing through point 
and point , tangent to , intersect the line at point 
, different than point . Let a circle passing through point 
and point , tangent to , intersect the line at point 
, different from point .a) Prove that
b) Prove that the points
and lie on a circle.Note: Grade 10 had (a), grades 11-12 had (b)
This post has been edited 5 times. Last edited by parmenides51, Aug 20, 2021, 9:18 AM
Reason: coorected wording thanks to Nersik.883
Reason: coorected wording thanks to Nersik.883
