[/quote]
,
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
?
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of 
are concyclic.
and 
Prove that
Let 
Prove that
we are given :
and one side length of the triangle equals 
,
.
be a trapezoid with parallel sides 
and 
,
.
and 
intersects 
again at 
respectively. Prove that 
are concurrent.
and 
Prove that
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
,
is an integer. For example, 
,
and 
there are elements 
such that 
.
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
such that 
.
are positive integers pairwise distinct and 
.![$\sqrt[n]{p},\sqrt[n]{q},\sqrt[n]{r}$](http://latex.artofproblemsolving.com/3/1/3/3133b60c921168020d149a9d28ef69b756a704d9.png)
form terms of a same arithmetic progression.
. Prove that
Let 
be reals such that 
. Prove that
Y by Funcshun840
An integer number 
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
less than is the difference between two numbers on the sheet, and no other positive number is.
Find all the feirense numbers.
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
less than is the difference between two numbers on the sheet, and no other positive number is.Find all the feirense numbers.
