[/quote]
,
is mundane if there exist positive integers 
and 
less than 
such that 
is a positive integer. Find, with proof, all prime numbers that are not mundane.
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 
.
intersect at point 
,
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 
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)
Y by
Let 
be a triangle whose angle at 
is right, and let 
be the foot of the altitude from . A variable point 
traces the interior of the minor arc 
of the circle . The internal bisector of the angle 
crosses 
at 
. The line through and perpendicular to crosses the line 
at 
. Determine the locus of the point where the line 
crosses the line 
.
* * *
be a triangle whose angle at 
is right, and let 
be the foot of the altitude from . A variable point 
traces the interior of the minor arc 
of the circle . The internal bisector of the angle 
crosses 
at 
. The line through and perpendicular to crosses the line 
at 
. Determine the locus of the point where the line 
crosses the line 
.* * *
This post has been edited 2 times. Last edited by oVlad, Nov 2, 2022, 4:00 PM
.
)
,
is a cyclic quadrilateral.
,
,
being the tangent of 
to 
)
,
of the circle centered at 
