[/quote]
,
and 
Prove that
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
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.
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 
.
.
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 that line 
is not parallel to line 
is perpendicular to 
.
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
Y by Adventure10
The inscribed circle 
of the non-isosceles acute-angled triangle 
touches the side 
at the point 
. Suppose that 
and 
are the centres of inscribed circle and circumcircle of triangle respectively. The circumcircle of triangle 
intersects 
at the points 
and 
. Prove that 
is equal to the radius 
of .
of the non-isosceles acute-angled triangle 
touches the side 
at the point 
. Suppose that 
and 
are the centres of inscribed circle and circumcircle of triangle respectively. The circumcircle of triangle 
intersects 
at the points 
and 
. Prove that 
is equal to the radius 
of .
is the reflection of 
on the perpendicular bisector of 
,
.
is cyclic. So 
.
in 
.
we will be done since this implies 
being cyclic and therefore 
.
the intersection of 
.
.
.
.
which is just simple angle chase.
So I moved your post here and deleted the old thread.
be the intersection of the 
.
are symmetric in 
,
.
we get 
so 
as required.
.
.
.
.
is cyclic so 
which implies 
.
which gives 
as 
hence 
is an isosceles trapezoid, so 
.
