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.
and 
play a game, given an integer 
, writes down 
first, then every player sees the last number written and if it is 
then in his turn he writes 
or 
, but his number cannot be bigger than . The player who writes wins. For which values of does win?
be the circumcircle of a scalene triangle 
. Let 
be a circle internally tangent to in . Tangents from touch in 
and 
, such that lies in the interior of 
. Similarly, tangents from 
touch in 
and 
, such that lies in the interior of .
and 
concur on the bisector of 
.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
, 
and 
are the midpoints of sides 
and 
respectively. Point lies inside the triangle. Let 
. Prove that 
.
, 
, and 
of a hexagon 
inscribed in a circle 
intersect at a point 
, and the acute angle between any two of them is 
. Let 
be the radius of the circle tangent to segments 
and 
and internally tangent to ; the radii 
, 
, 
, 
, and 
are defined similarly. Prove that![\[
r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}.
\]](http://latex.artofproblemsolving.com/0/d/6/0d66d6685bbc97c7553e9ed3ba14ea0dc3328b49.png)
, let 
be the circumcentre, 
be the orthocentre, and 
be the centroid.
satisfy the following condition:
is constant.
. Find .
such that numbers 
are both perfect squares.
such that ![\[\left(g(m)+n\right)\left(g(n)+m\right)\]](http://latex.artofproblemsolving.com/5/a/c/5ac9fce315e8a524a3a01b345cfe76bc76056d68.png)
is a perfect square for all 
be an acute triangle inscribed in a circle of center . If the altitudes 
intersect at and the circumcenter of 
is 
, prove that 
is a parallelogram.
is a perfect square number.
is good if there exists a permutation 
of 
such that 
is perfect square for all 
. Determine all the good numbers in the set 
.
of positive integers such that 
and 
are both powers of 
.
be prime numbers such that their sum isn't divisible by 
. Find the all 
positive integer quadruples satisfy:
is a positive integer, then 
divides 
. Now, see that the 
which is or 
; but, as divides 
then divides .
then is clear than 
, 
, 
and is even, or 
, 
, and positive integer.
, then 
or ; otherwise, and 
are and 
and divides . So, if and 
, sp 
. So 
and 
, 
and 
or 
and 
. If and 
then and .
.
and 
for every 
-
:
