AoPS Academy
and 
are given in space, such that 
. Let 
, 
and 
be the midpoints of 
, 
and 
. Prove that:
is a right triangle;
and 
are perpendicular.
with circumcircle 
, and two arbitrary points 
on . Let 
, 
, 
be points on lines 
, 
, 
, respectively, such that 
, 
, and 
concur at a point 
. Let 
be a point on line such that 
, 
, , are concyclic. Similarly, let 
be a point on line such that , , , are concyclic, and let 
be a point on line such that , , , are concyclic. Prove that 
, 
, 
concur at a single point.
, we denote by 
the sum of its digits. Find all integers 
such that there exist 
with 
for all 
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
be non-negative real numbers such that 
Prove that
such that![\[(a^2-b^2)(b^2-c^2) = abc.\]](http://latex.artofproblemsolving.com/7/1/8/7188261c40a45f5e15533100432d36abbf515efd.png)
. If this is still too hard assume in 
that 
is a prime.
+1 w
, 
and 
are non-negatives such that 
. Prove that:![\[\sum_{cyc}\sqrt{a^2+b^2}\leq\frac{11(a^2+b^2+c^2)+7(ab+ac+bc)}{3\sqrt2(a+b+c)}\]](http://latex.artofproblemsolving.com/4/3/3/433e1cd5ff26aa0dc20d67a78856870d0e46c439.png)
![\[\sum_{cyc}\sqrt{a^2+7ab+b^2}\leq\frac{2(a^2+b^2+c^2)+7(ab+ac+bc)}{a+b+c}\]](http://latex.artofproblemsolving.com/2/b/e/2bec411802cbe089dffc05bf1d92ec6936cfce3c.png)
![\[\sum_{cyc}\sqrt{4a^2+ab+4b^2}\leq\frac{5(a^2+b^2+c^2)+4(ab+ac+bc)}{a+b+c}\]](http://latex.artofproblemsolving.com/b/7/d/b7dd891c0d169bf766725197b6ca0eaff79ad7a8.png)
be an integer. Show that, if 
is an integer, then it is a perfect square.[/quote]
. If is an integer, then 
is at least rational, so that 
must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs 
for which this happens are 
and 
, and that, for every such pair 
, the "next" such pair can be calculated as
The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?
such that there exist four complex numbers 
with 
, and for any complex number 
, if 
, then![\[|az^3+bz^2+cz+d|\le M.\]](http://latex.artofproblemsolving.com/a/9/b/a9b4f124bf07c5a5aaa2d2e46c249ac44b7aaf57.png)
. By ROUF we can easily get 
. This inequality is strict. The best upper bound I get is 
, for 
