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.
be the midpoint of 
in 
. Let 
be the intersection of 
with 
. Let 
and 
be the intersection of lines through parallel to 
and 
with and . Let 
be the intersection of with 
. Let 
and 
be the intersection of 
and 
with the tangent to at 
. Prove that 
is cyclic.
be the incenter of 
, and be the midpoint of . Suppose the 
-median line and the -angle bisector of intersect at point 
. Show that the circumcircle of 
is tangent to the perpendicular bisector of 
.
+1 w
satisfying the condition![\[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\]](http://latex.artofproblemsolving.com/e/d/3/ed333c1123c4cb6a1e687dc572c5d847671caded.png)
and 
.
be positive integers. Given 
positive integers 
and finite sets 
, it is known that for any non-empty subset 
of 
,![\[\left| \bigcup_{i \in I} A_i \right| \geq \sum_{i \in I} c_i + d.\]](http://latex.artofproblemsolving.com/8/d/b/8dbf309aeab4c7c519288f4ef96cd5029467ce85.png)
Prove that there exist sets 
satisfying the following conditions: [list]
, 
and 
; of ,![\[\left| \bigcup_{i \in I} B_i \right| \geq \sum_{i \in I} c_i + d.\]](http://latex.artofproblemsolving.com/3/4/8/3488127dbd5ae2a5819795232bf10d0e9ace2c38.png)
[/list]
be a cyclic quadrilateral such that 
. Let 
, and let 
be the incenters of triangles 
and 
, respectively. Let and be the midpoints of and 
, respectively. Prove that the segment 
bisects the segment 
.
, 
is the circumcircle and 
is the nine-point circle. The tangents to at 
and 
intersect at 
. Let 
be a point on such that 
and does not lie on 
. Construct a circle 
symmetric to with respect to , and let intersect at points 
and 
. Let 
be the isogonal conjugate of with respect to . Prove that 
are concyclic.
be the set of positive integers. Find all functions 
satisfying the following two conditions:
for all positive integers 
and 
; and and 
and real numbers 
, 
, 
, 
that are not all zero such that ![\[\left|\sum_{i = 0}^m c_i f(n + i)\right| \leq M\]](http://latex.artofproblemsolving.com/e/3/b/e3b6b57a44d5527b298642b6ca1c764b0c4832f8.png)
for all positive integers .
and have answers
and 
.
over all triples 
of distinct real numbers such that
is called perfect if the sum of its positive divisors 
is twice , that is, 
. For example, 
is a perfect number since the sum of its positive divisors is 
, which is twice . Prove that if is a positive perfect integer, then:![\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]](http://latex.artofproblemsolving.com/0/c/0/0c08f527e87f7aae4ca5226f19bd084249551fe3.png)
where the sums are taken over all prime divisors 
of .
is a complex matrix with 
prove that is self-adjoint, i.e., that 
,
,
,
,
.
and its absolute and relative error, known that its absolute error is equal or lower than 
