[/quote]
,
and in 
steps and has to reach 
is valid but 
is not valid.
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
such for any 
:
.
satisfy,
Where 
}
be an integer. We are given a balance and 
weights of weight 
.
to satisfy that the equation 
has at least 1 roots 
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
of the equation![\[n^3=p^2-p-1\]](http://latex.artofproblemsolving.com/f/f/8/ff88093bb0aa93afcfdff3ae25149ec5fbacb61f.png)
is a prime number and 
?
be points on a plane such that 
Let 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
be reals such that 
. Prove that
specifies the greatest common divisor of 
and 
.
Y by
On the plane three different points 
, and 
are chosen. It is known that however one chooses another point 
on the plane, the point 
is always either closer to than the point 
or closer to than the point . Prove that the point lies on the line segment 
.
, and 
are chosen. It is known that however one chooses another point 
on the plane, the point 
is always either closer to than the point 
or closer to than the point . Prove that the point lies on the line segment 
.
and 
.
