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 an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
be an odd prime. Define 
over the positive integers as follows:
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
reducible over the rational numbers, given that it has integer coefficients?
such that 
.
students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix 
. Prove that 
.
be an acute triangle and let be an interior point for it such that 
. Prove that
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
is the convex quaddrilateral, 
are incircles of the 
tangent to 
at 
The internal homothety center of is 
If 
either condition 
or 
makes is a parallelogram and the proposition becomes trivial. Therefore, WLOG, let 
and 
Then 
or 
which means that the convex is tangential with an excircle 
in the (smaller) angle 
is the external homothety center of 
and 
the internal homothety center of 
so that the internal homothety center 
of is on 
therefore 
Let be the A-excircle of the 
The internal homothety center of 
is on their common internal tangent 
is the external homothety center of and 
the internal homothety center of 
the internal homotherty center of is on 
it is identical with 
But then the other tangent 
of 
through is also tangent of and the convex is tangential with the excircle in the angle Consequently, 
or 
which means that 
,there are two points 
,the incenters of triangles 
are 
(j=1,2),then 
are concurrent.
is a quadrilateral , there is a circle tangent to it whose centre is constructed by intersecting the internal angle bisectors of 
and external angle bisectors of 
if and only if 
.
respectively on 
away from such that 
. Then 
and therefore the bisectors intersect at the circumcentre of 
.
, we just need to show that there is such a circle ( internal bisectors of 
) iff the concurrence .
be the incentres of 
, 
, 
.
and line 
, 
, where 
are the inradii . Therefore , 
iff 
, at the same time 
cuts the segment 
at the ratio 
, we see that what's next is trivial .
and 
.if the two tangency points are symmetical with respect to the midpoint of the diagonal,then we have
,letthe incenter of the triangle and be 
respectively, and the intersection of two diagonals are collinear.
