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.
is given with 
. Line 
tangents in a point 
the circumcirle of . A circle centered in with radius 
cuts 
in the point 
and the line in points 
(such that 
and 
are in the same halfplane with respect to ). Prove that the line 
passes through the incenter of .
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
such that their isogonal conjugate 
in 
has 
parallel to 
.
in this set lies on a rectangular hyperbola, and locate its center.
be an acute scalene triangle , 
be the orthogonal projection of 
on , 
and 
are the midpoints of 
and 
respectively. Let and are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on .
touches sides 
and at points 
and 
, respectively. Let be the intersection of lines 
and 
. The reflections of with respect to 
and 
are 
and 
, respectively. Prove that lines 
and 
are concurrent at a point on line 
, where 
and 
are the incenter and circumcenter of triangle .
![\[ a^2+b^2+c^2=a^2b^2.\]](http://latex.artofproblemsolving.com/9/7/3/973246fd8d710bccf27fefe9ee8a81952122e085.png)
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
satsfying 
and:![\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]](http://latex.artofproblemsolving.com/b/c/1/bc18d6ad72589f012dcab7836228b3ed7ee061c2.png)
be positive integer.Prove the following equality.![\[ \sum_{r=1}^n (-1)^r\ _n C_r\ \frac{1}{r}=-\sum_{r=1}^n \frac{1}{r} \]](http://latex.artofproblemsolving.com/4/0/2/40263db7dae2474de3540974d1d3246590367983.png)
Where 
is quadratic mean 
, 
is harmonic mean 
, 
is arithmetic mean 
, and 
is geometric mean ![$\sqrt[n]{a_1a_2\cdots a_n}$](http://latex.artofproblemsolving.com/5/5/6/556f366fbe0e31be453f8b9c4ea6da4572cc66b4.png)
.
be a square. Let 
be two points on sides 
, respectively such that 
, and let be a point on 
such that 
. Prove that ![\[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]](http://latex.artofproblemsolving.com/6/c/7/6c70e8998c74639e81491cabec266fed92bb3678.png)
is the besictor of angle 
and 
, then 
is cyclic. Since 
, we have that is a isosceles trapezium with 
. Let 
and 
be the foot of height from 
and to , respectively. Notice that 
so 
and as 
is cyclic, thus 
.
then 
and 
is cyclic, therefore
is given where 
are positive reals. Prove that:
and 
Prove that
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
also 
so 
now 
. Now,
.
is the internal bisector of 
is the external bisector of 
.
.
