AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
. The interior bisector of 
intersects 
at 
and the minor arc of 
in circumcircle of at 
. Suppose that 
is a point on the minor arc of such that 
. 
is a point on the line segment of 
such that 
. 
is the circumcenter of . Prove that 
.
+1 w
be reals such that 
and 
Find the value of 
be a triangle with circumcentre . The points 
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
. (By interior we mean, for example, that the point lies on the line and is between 
and 
on that line.)
be the circumcentre of triangle 
. Prove that the lines 
and are perpendicular.
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
and we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good., , no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.) be an acute triangle. Let be the midpoint of side , and let and 
be the feet of the altitudes from and , respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point , and that lies on the circumcircle of . Prove that line 
is perpendicular to line .
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
with conditions,
is a prime number.
is a perfect square.
is also perfect square.
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
such that:
for all 
such that:
for all
.
