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 
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 mathematicsy
Let 
be an acute-angled triangle inscribed in the circle 
. Let 
be the diameter of . The points 
are chosen on 
such that 
. The lines 
cut again at 
. Prove that 
.
Tran Quang Hung, Vietnam
be an acute-angled triangle inscribed in the circle 
. Let 
be the diameter of . The points 
are chosen on 
such that 
. The lines 
cut again at 
. Prove that 
.Tran Quang Hung, Vietnam
This post has been edited 2 times. Last edited by parmenides51, Dec 16, 2022, 9:50 AM
,
is the perpendicular bisector of 
.
and therefore also 
and 
.
,
.
