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.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
Find all the integers satisfying 
be a triangle with incentre 
and circumcircle 
. Let 
and 
be the second intersection points of with 
and 
, respectively. The chord 
meets 
at a point 
, and 
at a point 
. Let 
be the intersection point of the line through parallel to 
and the line through parallel to 
. Suppose that the tangents to at 
and 
meet at a point 
. Prove that the three lines 
and 
are either parallel or concurrent.
is an odd number, let 
be a positive integer. If 
then 
satisfying
be the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
, there exists some positive integer 
that makes the following equation have no integer root 
.
: 
, 
and 
are altitudes, 
and 
are midpoints of sides and 
, respectively. At what angle do the straight lines 
and 
intersect?
.
.
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
and 
Prove that
Prove that
and 
Prove that
.
.
.
.