[/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 
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 
,
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.
Y by Adventure10
Let 
be an acute triangle. Denote by 
and 
the feet of the altitudes from vertices 
and 
, respectively. Let 
be a point inside the triangle such that the line 
is tangent to the circumcircle of the triangle 
and the line 
is tangent to the circumcircle of the triangle 
. Show that the line 
is perpendicular to 
.
be an acute triangle. Denote by 
and 
the feet of the altitudes from vertices 
and 
, respectively. Let 
be a point inside the triangle such that the line 
is tangent to the circumcircle of the triangle 
and the line 
is tangent to the circumcircle of the triangle 
. Show that the line 
is perpendicular to 
.
the third altitude, 
.
; similarly, 
,
and 
hence 
or 
.
，
，
，
,
.
and 
.
.
.
.![[asy]defaultpen(linewidth(0.7));
pair A=origin, B=(12,0), C=(12,5), D=(0,5), P=(6,2.5);
draw(A--B--C--D--A--C^^B--D);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$",D, NW);
label("$P$", P, N);[/asy]](http://latex.artofproblemsolving.com/6/8/9/68954d502c5415a2c5323fdac045f34b83c94b7a.png)
![[asy]defaultpen(linewidth(0.7));
pair P=origin, B=(8,0), A=(0,3), C=(0,-14), D=(-13,0);
draw(A--B--C--D--A--C^^B--D);
label("$A$", A, N);
label("$B$", B, E);
label("$C$", C,S);
label("$D$",D, W);
label("$P$", P, NE, fontsize(10));
markscalefactor=0.1;
draw(rightanglemark(A,P,D));[/asy]](http://latex.artofproblemsolving.com/b/c/0/bc052d430e6830f58d40f44c04fcfe9a611c705a.png)
<=> 
<=> 
and with the right angle we get the two similar triangles.Where's the mistake?(if 
and 
are above 
and 
Carnot theorem.
altitude.
.
But this is clearly true as 
.
