[/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 
and 
to 
respectively. Prove that 
.
consisting of a prime number 
and two positive integers 
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
Let 
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least 
.
Determine the minimum value of the red number at the end of this procedure.
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least 
.Determine the minimum value of the red number at the end of this procedure.
