[/quote]
,
such that![\[
f(f(x)) + x f(xy) = x + f(y)
\]](http://latex.artofproblemsolving.com/9/1/5/9157eb92f89f6c96333ce105a88ebed02c5307ef.png)
for all positive real numbers 
and 
.
be positive real numbers such that 
.![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?
?
and 
Prove that
be a function satisfying
for all integers 
and 
.
and 
such that 
for all integers 
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
such that the equation 
has an integer solution 
.
(Adit Aggarwal, India.)
such that: 
.
.
and 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
can be inscribed within a circle.
have,where 
be the set of the positive real numbers. Find all functions 
such that 
for all positive real numbers 
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
with 
and the circle 
of radius 
Let 
be the midpoint of 
The line 
intersects 
Let 
Let 
be the intersection of 
and 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
,
.
Y by Adventure10, Mango247
Let 
be a cyclic quadrilateral inscirbed in circle 
. Let the tangent to at 
meet 
at 
, and the tangent to at 
meet 
at 
. Circle with as its center and passing meets at 
, and 
meets again at 
. The circle with as its center and passing meets at 
. Let 
. Prove that 
are collinear if and only if 
.
be a cyclic quadrilateral inscirbed in circle 
. Let the tangent to at 
meet 
at 
, and the tangent to at 
meet 
at 
. Circle with as its center and passing meets at 
, and 
meets again at 
. The circle with as its center and passing meets at 
. Let 
. Prove that 
are collinear if and only if 
.
)
,
is the angle bisector of 
.
bisects 
.
,
.
so 
.
and by Fact-5 we are done.
