AoPS Academy
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
+1 w
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
+1 w
be an acute triangle with 
and let 
be its orthocenter. Let 
be a point on the perpendicular bisector of 
such that 
and 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 
w.r.t 
. Prove that 
and are collinear.
we have ![$\overline{PQ}^2=(s-t)^2+(s-2t)^2=16\ \cdots [*]$](http://latex.artofproblemsolving.com/9/0/b/90bce8b149c5390735ea8904f06bf9966a4fa6e9.png)
be 
we have 
![$[*],$](http://latex.artofproblemsolving.com/3/9/8/39807cad2f85a5dd22f9556255337b1b459c8d56.png)
we obytain 
which is ashwinrk_jan obtained.
which yetti obtained.
and the other end-point on the line 
.