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.
such that
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
, prove that 
is the circumcenter of 
, and 
are points on segments 
and 
respectively with 
. Let 
be a point such that 
and 
.
intersect and at points 
and 
respectively. Let the line passing through and perpendicular to 
intersect and at points 
and 
respectively. Prove that points 
, and are concyclic.
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)
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
![$P(x) \in \mathbb{Q}[x]$](http://latex.artofproblemsolving.com/9/1/0/9104130500a15ab4bffb68bc493faa4ea2891610.png)
be a polynomial with rational coefficients and degree 
. Prove there is no infinite sequence 
of rational numbers such that 
for all 
.
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
aand 
Prove that
Let 
aand Prove that
Let 
aand 
Prove that 
with the center 
touches the sides 
at 
respectively. The line 
meets the circumcircle of at 
. The line 
meets the line 
at 
and the line 
meets the circumcircle of at 
. Define 
similarly. Prove that the lines 
are concurrent.
intersect at point 
. Point 
lies between the parallel lines 
and 
such that 
, and line 
separates points and . Prove that 
.
diagonals intersect at 
and 
Let 
be the foot of the perpendicular from 
to and be the second intersection point of the circumcircle of triangle 
and line segment 
. Find 
.
be the foot of perpendicular from 
onto 
; from the given segments relation, 
, since 
is right-angled and isosceles. Hence 
.
are altitudes, so 
, while p.o.p. 
w.r.t. 
gives 
, consequently 
is cyclic, and 
.
be the foot of perpendicular from onto ; from the given segments relation, , since is right-angled and isosceles. Hence . are altitudes, so , while p.o.p. w.r.t. gives , consequently is cyclic, and .
. So 
??
aand 
Prove that 
