,
and 
lie on 
and 
respectively such that 
.
and 
are the reflections of 
lies on segment 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle 
.
that satisfy the equation 
?
and 
Find the value of 
be a triangle with a circle 
with center 
tangent to sides 
at 
respectively. Suppose the circle with diameter 
intersects the circumcircle of 
is the reflection of 
lie on 
are parallel to 
lie on the circumcircle of 
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
be a sequence of positive real numbers such that for every 
,![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
,![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
be an acute triangle with 
and let 
be its orthocenter. Let 
such that 
and 
are on different sides of 
,
a point on the perpendicular bisector of 
such that 
and 
a point on the perpendicular bisector of 
such that 
and 
lie on the opposite side of 
.
and 
square is filled with numbers 
.
squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
lie on the perpendicular bisector of 
lie on a circle. Suppose 
are perpendicular to sides 
at points 
The tangent lines from points 
intersects at point 
Prove that: 
are parallel.
,
be four points in the plane, with 
and 
.![\[\frac{AB \cdot CD}{AC \cdot BD}, \]](http://latex.artofproblemsolving.com/f/3/c/f3cf9729b32b210faeea6e2d23da3582be2ae061.png)
and 
are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles 
such that for any positive real numbers 
and 
,
Proposed by Athanasios Kontogeorgis, Grecce, and Dorlir Ahmeti, Kosovo
Y by
Given triangle 
inscribed in circle 
and circumscribed about circle 
. A circle passing through 
and 
is tangent to at 
and intersects 
and 
at 
and 
respectively. Similarly, define 
and 
correspondingly. Let 
be the triangle formed by the radical axis of circles 
, 
, 
(as shown in the figure); 
is the triangle formed by the intersection of lines 
, 
, 
(as shown in the figure).
a) Prove that: Triangle and triangle are in perspective axially.
b) Let
be the point of concurrence of 
, 
, 
. Prove that: , 
, 
are collinear.
inscribed in circle 
and circumscribed about circle 
. A circle passing through 
and 
is tangent to at 
and intersects 
and 
at 
and 
respectively. Similarly, define 
and 
correspondingly. Let 
be the triangle formed by the radical axis of circles 
, 
, 
(as shown in the figure); 
is the triangle formed by the intersection of lines 
, 
, 
(as shown in the figure).a) Prove that: Triangle
and triangle are in perspective axially.b) Let
be the point of concurrence of 
, 
, 
. Prove that: , 
, 
are collinear.Attachments:
is NOT aligned with 
and 
.
.
