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.
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 
.(Andrew Brahms, USA)
be positive real numbers such that 
. Prove 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 ?
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
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 with 
such that the equation 
has an integer solution 
. such that
and 
(Adit Aggarwal, India.)
such that: 
. Find the maximum: 
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
have,where and are natural numbers.
be the set of the positive real numbers. Find all functions 
such that 
for all positive real numbers and .
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 centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let 
be the intersection of 
and Prove that 
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 
, there will be a colour coloring exactly one integer from the set 
.
, let point 
be the midpoint of the hypotenuse and lie point lie on the side such that 
. Let segments 
and 
intersect at point . Prove that the line 
touches the circumscribed circle of the triangle 
.
, let point be the midpoint of the hypotenuse and lie point lie on the side such that . Let segments and intersect at point . Prove that the line touches the circumscribed circle of the triangle .
be a perpendicular foot from to .
and 
., then 
.![\[
ME: MC=\frac{1}{2}=KE: AC,
\]](http://latex.artofproblemsolving.com/b/3/6/b360dd47154676fa49527d9fba3af32459b9818d.png)
which implies that 
.![\[
\angle{KME}=\angle{AMC}\implies \angle{AMB}=\angle{KMC}.
\]](http://latex.artofproblemsolving.com/f/2/b/f2b75aeb20d8c41301eb80f532ea0c5b8b89fd5f.png)
Also, since , and 
lie on the same circle with centre , then 
.![\[
180^\circ-\angle{KCM}-\angle{KMC}=180^\circ-\angle{ABM}-\angle{AMB} \implies \angle{MAB}=\angle{MKC}.
\]](http://latex.artofproblemsolving.com/f/a/a/faa0bf95734ff4c4f6445ec8a5d98229df9d098a.png)
Thus, 
, which means the line touches the circumscribed circle of the triangle .
?
and 
Prove that
