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 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.
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
such that 
where 
, and the statement is still true when 
? such that 
where 
, and the statement is still true when ?.
be a scalene triangle with circumcircle 
and circumcircle 
. Denote 
and 
as the midpoints of 
and 
, respectively. The altitude from 
to intersects again at 
. The line 
meets a second time at 
, and 
intersects at 
. The perpendicular bisector of intersects 
at 
and 
at 
. Let the circumcircle of 
intersect again at 
, and let the line 
intersect at 
.
is the reflection of over , prove that the intersection of 
and 
lies on the circumcircle of 
.
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
Approach: I'm trying to factor 
to get 
.
such that 
divides 
. be a positive integer and 
be positive real numbers such that 
Prove that 
and 
in the convex quadrilateral 
are not acute. Points 
and 
are marked on the sides 
and 
respectively. Prove that the perimeter of 
is not less than the double length of the diagonal 
.
and in the convex quadrilateral are not acute. Points and are marked on the sides and respectively. Prove that the perimeter of is not less than the double length of the diagonal .
and 
be midpoints of sides 
and 
respectively.
and 
.
and 
.
. So 
. Hence we obtain 
. Thus 
and thus 
which is a contradiction as angle isn't acute. So we obtain . Similarly and our claim is proven.
which gives us that perimeter of is greater than or equal to 
as desired.
![$a,b,c \in [-\frac{1}{2}, \frac{1}{2}]$](http://latex.artofproblemsolving.com/5/0/b/50bc691baa4e4bbef868e4799929c8c93d1b6f1a.png)
.
Let ![$a,b,c \in [-2,2]$](http://latex.artofproblemsolving.com/8/0/6/8066d4353091772c401e3e34f5499a8fa2bd9991.png)
.
