,
.
be 
g
is the greatest number on at least one of the two arcs 
.
such that regardless of the values of 
are written on a board. 
and 
plays the following game: They take turns choosing a number from the board and deleting them. 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
is a fifth degree polynomial. 
,
,
,
.
is differentiable and 
for all 
.
,
.
a prime number, with 
and 
,
vector space.
is a group?
and the distribution function is 
then
and find the expression of 
![\[
\int_0^1 \int_{x^2}^x (x^2 + y^2)^{-1/2} \, dy \, dx
\]](http://latex.artofproblemsolving.com/e/a/9/ea9f3a093b1d4c0fa39e5b6d42afa2f71dace1e0.png)
has no roots in the disc of unity.
Y by
For any three points 
, 
, 
define 
to be the semiperimeter of 
. Let 
be a triangle and let 
be on side 
such that
![\[\frac{s(ABD)}{BD}=\frac{s(ACD)}{CD}\]](//latex.artofproblemsolving.com/9/4/f/94f0bcb2cf847452d6e6197d5bb9b6f0ffd72f6d.png)
Let 
be a point on 
. Let 
and 
be on 
and 
such that 
and 
. Prove that there exists a line 
parallel to such that the circumcircles of 
and 
intersect at two fixed points.
, 
, 
define 
to be the semiperimeter of 
. Let 
be a triangle and let 
be on side 
such that![\[\frac{s(ABD)}{BD}=\frac{s(ACD)}{CD}\]](http://latex.artofproblemsolving.com/9/4/f/94f0bcb2cf847452d6e6197d5bb9b6f0ffd72f6d.png)
Let 
be a point on 
. Let 
and 
be on 
and 
such that 
and 
. Prove that there exists a line 
parallel to such that the circumcircles of 
and 
intersect at two fixed points.
and 
having the same radius. Let 
denote the centres of these circles. We now have 
.
.
denote the midpoint of 
.
is cyclic. Define 
as the reflections of 
;
is cyclic. Conveniently sweeping certain signed length problems under the rug, we may observe that ![\[AP' + AQ' = 2s(ABD) = AB+AD+BD\]](http://latex.artofproblemsolving.com/5/2/3/523162f2d1cc486df5fe026e7b42dcc5a0c29397.png)
implies that 
.
on 
so that 
and note that we also have 
.
