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.
denote the set of positive integers. Let 
be a function satisfying the following property: for 
, the equation![\[
f(mn)^2 = f(m^2)f(f(n))f(f(mf(n)))
\]](http://latex.artofproblemsolving.com/e/c/2/ec2ddb40ced3a06ad9cd2f2138fae90040d64bae.png)
holds if and only if 
and 
are coprime., determine all the possible values of 
.
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is a positive divisor of other than itself.
consists of positive integers, each of which has at least three proper divisors. For each 
, the integer 
is the sum of the three largest proper divisors of 
.
.
+6 w
grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
+13 w
which is known to both players. On the 
-th turn of the game (starting with 
) the following happens:
If is odd, Alice chooses a non-negative real number 
such that![\[
x_1 + x_2 + \cdots + x_n \leq \lambda n.
\]](http://latex.artofproblemsolving.com/d/1/7/d177864767df8222d05b215760bcf367d4031bb5.png)
If is even, Bazza chooses a non-negative real number such that![\[
x_1^2 + x_2^2 + \cdots + x_n^2 \leq n.
\]](http://latex.artofproblemsolving.com/4/1/f/41f497975597be624be77efca930b287c888c104.png)
, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players. for which Alice has a winning strategy, and all those for which Bazza has a winning strategy.
+1 w
dente the set of positive integers. Let 
be a function satisfying the following property: for 
, the equation
holds if and only if and are coprime., determine all the possible values of .
+1 w
which is known to both players. On the th turn of the game (starting with 
) the following happens: is odd, Alice chooses a nonnegative real number 
such that![\[
x_1 + x_2 + \cdots + x_n \le \lambda n.
\]](http://latex.artofproblemsolving.com/f/5/e/f5e4f751946952d021736c6eafa0b7ec02d02055.png)
[*]If is even, Bazza chooses a nonnegative real number such that![\[
x_1^2 + x_2^2 + \cdots + x_n^2 \le n.
\]](http://latex.artofproblemsolving.com/f/f/4/ff40463336f4c5dc7381b14e71abe7d45b78a1c0.png)
[/list], the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players. for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
+2 w
of positive integers such that for every 
there exists 
with 
.
and 
be circles with centres 
and , respectively, such that the radius of is less than the radius of . Suppose and intersect at two distinct points 
and 
. Line 
intersects at 
and at 
, so that 
lie on in that order. Let 
be the circumcentre of triangle 
. Line 
meets again at 
and meets again at 
. Let 
be the orthocentre of triangle 
. parallel to is tangent to the circumcircle of triangle 
.
be a triangle with incentre 
, and let be the circumcircle of triangle 
. Let 
be a point in the interior of segment 
such that 
. Suppose that the angle bisector of 
intersects at points 
and 
such that and lie on the same side of , and that the angle bisector of 
intersects at points 
and 
such that and lie on the same side of .
.
with the following property: for all positive divisors 
of , we have 
or 
is prime.
be a quadrilateral with 
parallel to 
and 
. Lines 
and intersect at a point . Point distinct from lies on the circumcircle of triangle such that 
. Point distinct from lies on the circumcircle of triangle 
such that 
. Lines 
and 
intersect at 
.
is parallel to . be a positive integer. There are 
knights sitting at a round table. They consist of pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some time.
be an acute triangle with 
, and let be the circumcircle of . Points and lie on so that 
and meet on the external angle bisector of 
. Suppose that the tangents to at 
and intersect at a point 
on the same side of as , and that 
and 
intersect at 
and 
, respectively. Let 
be the centre of the excircle of triangle 
opposite the vertex .
bisects .
and the recursive relation
for all natural number 
. Find the general form of .
is the best I could come up with. Probably no better general form than this.
and the recursive relation for all natural number 
try to find 
