[/quote]
,
be positive integers. If for each positive integer 
we have that 
Prove that 
.
,
is fantabulous, then all the elements are fantabulous. Does it follow that the number 
is fantabulous?
,
:
,
is a cyclic quadrilateral with 
.
and 
be the feet of perpendiculars from 
respectively. Prove that 
.
consisting of a prime number 
and two positive integers 
and 
such that 
and 
are both powers of 
be integers. Suppose that for each 
there exists an integer 
such that 
is divisible by 
.
for some integer 
,
on the whiteboard. In each move, she selects two distinct numbers, 
and 
,
.
is a perfect square.
,
is a prime.
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
,
to be the largest value in row 
,
.
to be the largest value in column 
.
?
we consider a 
rectangular frame, consisting of the 
boundary squares of a 
for which Renate has a winning strategy.
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
.
Y by Adventure10
Let 
be a point on the tangent to circle 
through the point 
on the circle. A point 
outside the circle is chosen so that segment 
intersects the circle in two distinct points. Let 
be the circle tangent to at and to at some point 
, where and 
are on the opposite sides of the line . Let 
be the circumcentre of triangle 
. Show that lies on the circumcircle of triangle 
.
be a point on the tangent to circle 
through the point 
on the circle. A point 
outside the circle is chosen so that segment 
intersects the circle in two distinct points. Let 
be the circle tangent to at and to at some point 
, where and 
are on the opposite sides of the line . Let 
be the circumcentre of triangle 
. Show that lies on the circumcircle of triangle 
.
remains constant but 
changes and so does 
.
is not necessarily cyclic.
