[/quote]
,
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.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
lies on side 
lies on side 
.
and 
be points on segments 
and 
,
is parallel to 
be a point on line 
,
.
be the point on line 
,
.
,
such that 
satisfying 
has an incenter 
Suppose 
.
.
meets 
again at point 
.
and 
.
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
and 
.
.
and 
,
is a perfect square. For example, if the number on the blackboard is 
,
,
,
,
.
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
is perpendicular to 
.
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
and 
)
Prove that if lines 
are similar.
Y by Adventure10, Mango247
The vertices of a given square are clockwise lettered 
. On the side 
is situated a point 
such that 
. Starting from an arbitrarily chosen point 
on segment 
and going clockwise around the perimeter of the square, a series of points 
is marked on the perimeter such that 
for each 
. It will be clear that when is chosen in 
or in , then some 
will coincide with . Does this possibly also happen if is chosen otherwise?
. On the side 
is situated a point 
such that 
. Starting from an arbitrarily chosen point 
on segment 
and going clockwise around the perimeter of the square, a series of points 
is marked on the perimeter such that 
for each 
. It will be clear that when is chosen in 
or in , then some 
will coincide with . Does this possibly also happen if is chosen otherwise?
