AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be positive integers. If for each positive integer 
we have that 
Prove that 
.
, if any element of the set 
is fantabulous, then all the elements are fantabulous. Does it follow that the number 
is fantabulous?
, in which 
: 
, but the proof is important.
is a cyclic quadrilateral with 
. Let 
and 
be the feet of perpendiculars from 
and 
to 
respectively. Prove that 
.
consisting of a prime number 
and two positive integers 
and 
such that 
and 
are both powers of .
+1 w
be integers. Suppose that for each 
there exists an integer 
such that 
is divisible by 
. Prove that 
for some integer .
, find the maximal area of the triangle.
on the whiteboard. In each move, she selects two distinct numbers, 
and 
, erases them, and replaces them with 
. She repeats this process until only one number, , remains. What are all the possible values of ?
, 
is a perfect square.
, so that the number 
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 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.
?
we consider a 
rectangular frame, consisting of the 
boundary squares of a rectangle.
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 
. How many cats of each breed does he have?
is circumcenter of 
. Then lies on perpendicular bisector of 
as 
and 
so lies on 
. Also lies on perpendicular bisector of 
and 
too, so is point of intersection of and 
. lies on perpendicular bisector to 
and as 
so lies on perpendicular bisector to 
, lies on perpendicular bisector to and lies on perpendicular bisector to 
, and 
so is midpoint of . So 
is median of 
and 
lies on median. lies on other median of , so is centroid of 
are on side 
of triangle 
,
are circumcenters of 
.