[/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 
.
,
are equal, and they intersect at 
.
intersect at point 
,
and 
intersect at point 
.
.
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)
Y by RUSH2008, Adventure10
Let 
be the circumcenter of the isosceles triangle 
(
). Let 
be a point of the segment 
and 
the symmetric of with respect to the midpoint of 
. If 
cuts at 
and the circle that passes through 
and cuts 
in 
, show that 
.
be the circumcenter of the isosceles triangle 
(
). Let 
be a point of the segment 
and 
the symmetric of with respect to the midpoint of 
. If 
cuts at 
and the circle that passes through 
and cuts 
in 
, show that 
.This post has been edited 1 time. Last edited by parmenides51, Jul 26, 2019, 9:23 PM
Reason: name typo
Reason: name typo
,
and 
and 
,
and 
.
- midpoint of 
- parallelogram, 
.
