[/quote]
,
,
is the midpoint of 
,
the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
.
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
Y by
A cylindrical container has height 
cm and radius 
cm . It rests on a circular hoop which has also radius cm and the hoop is fixed in a horizontal plane. The container rests with its axis horizontal and with each of its circular rims touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop at two points. Find, with proof, the locus of the centre of one of the cylinder's circular ends.
cm and radius 
cm . It rests on a circular hoop which has also radius cm and the hoop is fixed in a horizontal plane. The container rests with its axis horizontal and with each of its circular rims touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop at two points. Find, with proof, the locus of the centre of one of the cylinder's circular ends.
