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 
there exists infinitely many positive integers 
for which 
and 
are not relatively primes.
let 
and 
denote the feet of the altitudes from 
and 
, respectively. Let line 
intersect circumcircle at points 
. Similarly, let line 
intersect circumcircle at points 
. Prove that the radical axis of circles 
and 
passes through the orthocenter of triangle
and a circle 
in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line 
and a point 
, constructs a new line passing through parallel to . (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle whose sides are parallel to 
in some order.![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
?
with 
.![$P(x,y) \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/b/3/7/b37942b05292a493c8e350322adc122205bc778d.png)
?
be a quadrilateral inscribed in a circle , and let be a point on the extension of 
such that 
and 
are tangent to . The tangent at intersects at 
and the line 
at 
. Let 
be the second point of intersection between 
and . Prove that 
, , are collinear.
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
