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.
+1 w
, 
is the midpoint of 
. 
is an arbitrary point on 
. Let 
be the intersection of 
and 
. The line 
intersects with the circumcircle of 
, for the second time at 
. 
is the circumcenter of triangle 
. Prove that 
.
be an acute triangle, let 
and 
be its orthocentre and circumcentre, respectively,
and 
be the feet of the altitudes from 
to 
and from 
to 
, respectively.
be the midpoint of the segment 
, and let 
be the midpoint of the segment 
. The line, parallel to 
, crosses the sides and at 
and 
, respectively. The line 
again at 
, and the line 
meets the circle 
again at 
. Prove that
and 
are similar.
be the midpoint of on triangle . Point 
lies on segment such that 
and 
on line 
such that 
. Prove that 
.
be an odd prime number. Prove that for any integer 
, the equation 
has no solution in positive integers 
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that for all 
,
is a prime number, 
are natural numbers (
). It is known that 
. Prove that 
.
there exists a unique positive integers 
such that 
is a square of a rational number.
for which![\[2^p = 2^{q-2} + q!.\]](http://latex.artofproblemsolving.com/8/8/2/88270dc4aa4a54cc8b2e525b7d9d940c300e49ac.png)
. Determine the locus of center of rectangle inscribed in triangle such that one side of rectangle lies on side .
Prove that
