[/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 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
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 
.
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 an equilateral triangle. Let 
be a point inside of it such that the square root of the distance of to one of the sides is equal to the sum of the square roots of the distances of to the other two sides. Find the geometric place of .
be an equilateral triangle. Let 
be a point inside of it such that the square root of the distance of to one of the sides is equal to the sum of the square roots of the distances of to the other two sides. Find the geometric place of .
.
be its contact points with 
respectively, i.e. 
the projections of 
respectively.
.
.
.
.
and 
we get the required 
.
is 
.
