[/quote]
,
positive real numbers such that 
. Prove that :
and 
are taken on the legs 
,
and 
.
and 
.
be real numbers. Prove that
be real numbers satisfying 
. Find the maximum value of![\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]](http://latex.artofproblemsolving.com/7/b/2/7b2ec0a87ca05d3cb29da888a4dfa73e10f238f8.png)
grid. Initially the whole board is black. On 
subgrid and color it white. On 
'
subgrid and colors it black. 
be a convex quadrilateral such that 
.
,
and 
at 
and 
,
.
be the n-th prime number in increasing order for 
.
satisfying 
for all 
and 
intersect at points 
and 
(not at 
and 
meet 
and 
respectively, and the lines 
and 
meet at 
.
all lie on one fixed circle.
penguin and each one is numbered using numbers 
.
is called 
if 
where 
is 
prime divisor of 
(
where 
)
is the midpoint of arc 
is the midpoint of arc 
and 
.
is projection of 
onto 
is the second intersection of 
and 
is tangent to 
.
with center 
on segments 
,
are following condition: 
.
.
and 
and 
are concyclic.
with diameter 
is given. The point 
intersects 
is tangent to 
.
,
.
meets the line 
meets 
and at 
.
and 
.
.
.
be complex numbers satisfying 
.
.
Y by Adventure10, Mango247
Let there be an acute scalene triangle 
with circumcenter 
. Denote 
be the reflection of with respect to 
, respectively. The circumcircle of 
meets 
, 
, the circumcircle of at points 
, respectively, and they are all distinct from 
. Prove that the lines 
are concurrent.
with circumcenter 
. Denote 
be the reflection of with respect to 
, respectively. The circumcircle of 
meets 
, 
, the circumcircle of at points 
, respectively, and they are all distinct from 
. Prove that the lines 
are concurrent.
then 
is cyclic.
.
and line 
is the image of 
is the image of 
,
are collinear and we are done
is cyclic. Hence, 
and 
,
and 
concur.
so 
is cyclic. Let 
so 
