Complicated FE
by XAN4, Apr 23, 2025, 11:53 AM
Find all solutions for the functional equation 
, in which 
: 
Note: the solution is actually quite obvious -
, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
, in which 
: 
Note: the solution is actually quite obvious -
, but the proof is important.Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
positive integers 
such that for every index 
,
,
is a multiple of 
.
with integer coefficients such that ![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
be an acute triangle with circumcircle 
.
be the midpoint of 
and let 
be the midpoint of 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
and 
be the points in which the median and the angle bisector, respectively, at 
.
and 
meets 
and 
,
the point in which the perpendicular at 
produced.
is perpendicular to 
.
,
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
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