AoPS Academy
be the circumcircle of a triangle 
. A circle passing through points 
and 
meets the sides 
and 
at 
and 
, respectively. The lines 
and 
meet again at 
and 
, respectively. The tangent lines of at and meet the line 
at 
and 
, respectively. Prove that the lines 
and 
meet at .
is the orthocenter of an acute triangle 
, and let 
be the midpoint of 
. Suppose 
meets 
and 
at 
respectively. 
meets 
at 
, and the line through perpendicular to meets 
at 
. Prove that 
are collinear.
prime number, there exists only one positive number 
that makes the equation 
a perfect square of a positive integer
that 
'strong' and 
'strong'
![$f:[1,8]\to R$](http://latex.artofproblemsolving.com/c/e/c/cec97ea0372e37977e8a77e8e77518a2344c1c43.png)
that:
and the distribution function is 
then
and find the expression of 
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
be a continuous function . Suppose![\[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\]](http://latex.artofproblemsolving.com/c/0/c/c0c48c4259d4e98562c21a23cf4a6841704b8e82.png)
and all 
. Then show that there exists a constant 
such that 
, 
, 
, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
is differentiable and 
for all 
. Show that for some 
, 
.
and 
are its altitudes. On the rays 
respectively, let 
such that 
. Find all values of 
such that 
for any non-triangle 
