[/quote]
,
,
:
,
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 
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
intersects segment 
at a second point 
(other than 
.
and 
be positive integers such that 
.
for 
.
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)
\
.
is equal to
Y by
The side of the square 
has a length equal to 
. On the sides 
¸and 
take respectively the arbitrary points 
and 
so that the perimeter of the triangle 
is equal to 
.
a) Determine the measure of the angle
.
b) If the point
is the foot of the perpendicular taken from point 
to the line 
, determine the locus of the points .
has a length equal to 
. On the sides 
¸and 
take respectively the arbitrary points 
and 
so that the perimeter of the triangle 
is equal to 
.a) Determine the measure of the angle
.b) If the point
is the foot of the perpendicular taken from point 
to the line 
, determine the locus of the points .
