[/quote]
,
be an acute-angled triangle in which 
and 
.
lie on segment 
and point 
lie on segment 
such that 
,
and 
.
be the circumcenter of triangle 
,
the orthocenter of triangle 
the point of intersection of the lines 
and 
.
consecutive even numbers, such that the square of their product divides the sum of their squares.
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
such that 
.
such that 
.
.
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
lies between 
and 
lies between 
is acute and let 
and 
and 
.
,
.
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
Find all possible values of 
.
be a positive integer and 
be a positive integer coprime to 
.
,
,
Find, in terms of 
,
such that 
is divisible by 
.
table are initially white. Alice and Bob play a game. First Alice paints 
Y by
Hello guys I have made my own functional equation problem.
Find all functions
such that ![\[f(x)(f(yf(x)+1))=f(x)+f(y)\]](//latex.artofproblemsolving.com/1/0/6/1069c29ad9eeb7fe8597d7bac806006a12ff50c9.png)
for all positive reals 
and 
and also satisfies the property that ![\[\mathbb{R}^+\subseteq\frac{\text{Im}(f)}{\text{Im}(f)},\]](//latex.artofproblemsolving.com/1/0/2/102f23eabefae24b158fbd7516b58b1e879c1b74.png)
or in other words, the set of positive reals is a subset of the set ![\[\left\{\frac{x}{y}\mid x,y\in\text{Im}(f)\right\}\]](//latex.artofproblemsolving.com/f/d/4/fd4e9cf0bfc63603529ded7679e2811560219338.png)
PS: This is my first positive real to positive real fe I have made
Find all functions
such that ![\[f(x)(f(yf(x)+1))=f(x)+f(y)\]](http://latex.artofproblemsolving.com/1/0/6/1069c29ad9eeb7fe8597d7bac806006a12ff50c9.png)
for all positive reals 
and 
and also satisfies the property that ![\[\mathbb{R}^+\subseteq\frac{\text{Im}(f)}{\text{Im}(f)},\]](http://latex.artofproblemsolving.com/1/0/2/102f23eabefae24b158fbd7516b58b1e879c1b74.png)
or in other words, the set of positive reals is a subset of the set ![\[\left\{\frac{x}{y}\mid x,y\in\text{Im}(f)\right\}\]](http://latex.artofproblemsolving.com/f/d/4/fd4e9cf0bfc63603529ded7679e2811560219338.png)
PS: This is my first positive real to positive real fe I have made
This post has been edited 2 times. Last edited by rama1728, Nov 25, 2021, 6:50 AM
Reason: .
Reason: .
,
be the assertion 
can take any positive value we want, 
from 
,
and for some 
,
,
.
,
.
for all 
.
so 
which simplifies as (setting 
and 
)
This is obviously not a solution (just take 
then 
