AoPS Academy
:![\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]](http://latex.artofproblemsolving.com/f/b/6/fb6d5ebb4f448d8eeb73cce3288fea3a56d7f528.png)
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
of non-negative real numbers such that the following three conditions are all satisfied:
;
;
of such that 
[/list]
is a permutation of 
, and they are both permutations of 
. Note that any list is a permutation of itself.
converge?
is a polynomial with integer coefficients, with 
odd. Suppose that 
for all 
Prove that 
is nonzero for all 
and 
define
if 
& 
if 
for a positive integer r, such that
where 
and 
are positive integers.
and respectively can be
![$f:[0,1]->R$](http://latex.artofproblemsolving.com/4/4/6/4462365b0a7312aba97a0ac040c492240c3884f8.png)
one function diferențiale whirt 
integable and 
,
.Prove rhat :
l
, let 
be the centroid, and let 
be the center of the inscribed circle. Let 
and 
be the angles at the vertices 
and 
, respectively. Suppose that the segment 
is parallel to 
and that 
. Find .
is defined on the interval 
. A point 
is said to be a fixed point of if 
. Suppose that is a differentiable function such that 
and 
. Show that if 
, then has a fixed point in the interval 
.
where , , and 
are nonnegative integers.
be a sequence given recrusively such that 
and 
for 
is a positive integer.
is a square of an integer. be a given positive integer. How many solutions are there in ordered positive integer pairs 
to the equation
and 
.
be a triangle with 
and 
its incenter. Let 
be the midpoint of the side 
and 
the projection of 
The line 
intersects the circle with center 
at 
and 
Prove that 
Call the circle with radius 
.
obviously 
is tangency point of A_excircle with 
feet of 
on 
and 
are cyclic so 
so 
is cyclic and 
note that 
.
,
,
.
.
.
,
and 
.