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 , 
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)
.
. If the product of the 2 numbers is also a member of this group, how many selections can be done?
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 
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
.
.
has 
,
,
.
.
,
possible selections.