[/quote]
,
prime number, there exists only one positive number 
that makes the equation 
a perfect square of a positive integer
,
is a power of 
.
where 
is a positive integer.
determine the minimum and maximum value of![\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]](http://latex.artofproblemsolving.com/6/f/2/6f2427edf9a5e8e8f2e12276d0a841765e8322f1.png)
and 
of a circle intersect at 
,
.
and 
are the intersections of the segments 
and 
and 
.
such that 
is a perfect square.
and 
lie on segments 
,
and 
intersect at point 
.
and 
intersect at a second point 
.
intersect at a second point 
.
lies on segment 
such that 
.
and 
are similar.
the set of all positive real numbers.
which have the property:![\[f(x)f(y)=2f(x+yf(x))\]](http://latex.artofproblemsolving.com/4/f/f/4ffbde35ff082585a188c1a4368d39338d1b07a4.png)
and 
.
such that :
for which there exists a sequence 
of positive integers satisfying the following conditions:[list]
for all integers 
for all integers 
the sequence 
is a permutation of
[/list]
is a positive integer.
exist that satisfy 
for all real 
,
?
and 
Prove that
Let 
Prove that
.
Y by Mango247
Consider a tetrahedron 
whose altitude 
passes through the intersection point of altitudes of triangle 
.
a) Prove that the opposite edges of this tetrahedron are perpendicular.
b) Prove that the lines on which the altitudes of this tetrahedron lie pass through one point.
c) Find out whether the converse is true: “If the lines on which lie the altitudes of the tetrahedron pass through one point, then the bases of the altitudes of this the tetrahedron are the points of intersection of the heights of the corresponding faces. ''
whose altitude 
passes through the intersection point of altitudes of triangle 
.a) Prove that the opposite edges of this tetrahedron are perpendicular.
b) Prove that the lines on which the altitudes of this tetrahedron lie pass through one point.
c) Find out whether the converse is true: “If the lines on which lie the altitudes of the tetrahedron pass through one point, then the bases of the altitudes of this the tetrahedron are the points of intersection of the heights of the corresponding faces. ''
