[/quote]
,
such that 
be a triangle with a circle 
tangent to sides 
at 
respectively. Suppose the circle with diameter 
intersects the circumcircle of 
is the reflection of 
over 
lie on 
are parallel to 
lie on the circumcircle of 
cause there are 10 ways to choose 1 element that will be repeated. Similarly for 2 same elements it would be 
the answer would be ![$(2^{10}-1)-([A_1+A_3+A_5+A_7+A_9]-[A_2+A_4+A_6+A_8+A_{10}].$](http://latex.artofproblemsolving.com/7/3/1/73130cab8af6ffb5c2638548fdf6af3ecc70a6ed.png)
But this number turned out to be 
But I realized that 
So I was really confused of why i got the right answer somehow in my calculations.
![\[f(xf(y) + x) = xy + f(x)\]](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)
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.
,
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
of positive real numbers satisfies ![\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]](http://latex.artofproblemsolving.com/0/3/9/03909a5f60d944063182ec97cf73f283178e8d4d.png)
for every positive integer 
.
for every 
.
and 
then find the greatest possible value of 
be positive real numbers that fulfill![\[
\frac{x^2}{y^2}+\frac{4x^2-3xy-4y^2}{2xy-5y^2}=2
\]](http://latex.artofproblemsolving.com/6/c/c/6ccba85deb3e6aa205abd3bc8e9e9bb68f4df6fd.png)
Hence find the value of 
Y by Rounak_iitr
Let 
be the center of the circumcircle of acute 
. Let 
be a point on arc 
so that 
are on opposite sides of side . Point 
is on chord 
such that 
bisects 
and 
. The circle 
passing through 
intersect side 
at 
. Line 
meets at 
and line 
meets side AB at 
. Prove that 
.
be the center of the circumcircle of acute 
. Let 
be a point on arc 
so that 
are on opposite sides of side . Point 
is on chord 
such that 
bisects 
and 
. The circle 
passing through 
intersect side 
at 
. Line 
meets at 
and line 
meets side AB at 
. Prove that 
.
