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.
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
, in which 
: 
, but the proof is important.
and 
Prove that
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
be non-negative real numbers such that 
Prove that
be a scalene triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. The circumcircle of triangle 
intersects segment 
at a second point 
(other than ). Prove that 
.
+1 w
and be positive integers such that 
. Define 
for 
. Prove that if all the numbers 
are integers, then 
is divisible by an odd prime.
![\[ \frac{\displaystyle\sum_{k=0}^r \binom nk \binom{n-2k}{r-k}}{\displaystyle\sum_{k=r}^n \binom nk \binom{2k}{2r}\left(\frac{3}{4}\right)^{n-k}\left(\frac{1}{2}\right)^{2k-2r}}\]](http://latex.artofproblemsolving.com/9/d/e/9dedd490eaf9c6d22aaf93bbd74d82375adcfa85.png)
\
.
we have ![$\overline{PQ}^2=(s-t)^2+(s-2t)^2=16\ \cdots [*]$](http://latex.artofproblemsolving.com/9/0/b/90bce8b149c5390735ea8904f06bf9966a4fa6e9.png)
be 
we have 
![$[*],$](http://latex.artofproblemsolving.com/3/9/8/39807cad2f85a5dd22f9556255337b1b459c8d56.png)
we obytain 
which is ashwinrk_jan obtained.
which yetti obtained.
is equal to
and the other end-point on the line 
.