[/quote]
,
and 
.
.
prime and 
a positive integer. Determine all pairs 
satisfying the equation: 
be positive real numbers with 
.
be the midpoint 
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
and 
satisfy 
,
is an even number.
,
and 
and a minimum value of 
over the interval ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
,
,
for which the equality holds.
.
point on AC, AB respectively such that 
,
.
,
.
,
.![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
and 
we define 
as the set of all products of an element of 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and 
is good.
,
,
,
and 
are integers.)
Y by
Let 
be an equilateral triangle.M and N are the midpoints of 
and 
respectively. Externally to the triangle an isosceles right triangle APC is constructed, with the angle 
.If point 
is the intersection of 
and 
, show that 
is a bisector of the angle 
be an equilateral triangle.M and N are the midpoints of 
and 
respectively. Externally to the triangle an isosceles right triangle APC is constructed, with the angle 
.If point 
is the intersection of 
and 
, show that 
is a bisector of the angle 
