[/quote]
,
be a fixed positive integer. Find the greatest real constant 
that has the following property: Any 
real numbers, not necessarily distinct, that lie in the interval ![$[100,101]$](http://latex.artofproblemsolving.com/8/e/3/8e3fe24e3b899bbf2128eebe08756e1ff963aaa0.png)
,
and 
such that
.
, 
and 
.
.
-
board.
board consisting of 
unit segments. A configuration of several 
is a divisor of 
.
,
and 
are equal, and they intersect at 
.
and 
intersect at point 
lying inside triangle 
,
and 
intersect at point 
lying inside triangle 
.
.
denote the set of positive rational numbers. Find, with proof, all functions 
such that, for all positive rational numbers 
and 
we have ![\[f(x)=f(x+y)+f(x+x^2f(y)).\]](http://latex.artofproblemsolving.com/9/e/a/9ea71d2ee277dc3c689ba2d791b451e29353edb3.png)
.
be the intersections of the perpendicular bisectors of 
and 
.
and 
and the point 
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
Y by
Radek challenges you to a duel. He draws a triangle 
with a small circle inside. Afterwards he marks the incenter of and connects it with the vertices of . This results in a split of into three triangles. Now, in each move you may choose one of triangles, mark its incenter and connect it with vertices of chosen triangle (which would result in exchanging chosen triangle with three new ones). Your task is to, after a finite number of moves, mark some incenter inside the small circle. Can you always do that, regardless of what triangle and circle was drawn by Radek?
with a small circle inside. Afterwards he marks the incenter of and connects it with the vertices of . This results in a split of into three triangles. Now, in each move you may choose one of triangles, mark its incenter and connect it with vertices of chosen triangle (which would result in exchanging chosen triangle with three new ones). Your task is to, after a finite number of moves, mark some incenter inside the small circle. Can you always do that, regardless of what triangle and circle was drawn by Radek?This post has been edited 2 times. Last edited by parmenides51, Nov 20, 2021, 2:07 PM
