[/quote]
,
,
and 
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)
and positive integers 
satisfying the equation ![\[a^2+b^2+16c^2 = 9k^2 + 1.\]](http://latex.artofproblemsolving.com/5/4/5/545f8edbdebe4e759c5a70bd3d489e5d53c9147c.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 
.
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 
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 
is tangent to 
.
,
,
.
meets the line 
meets 
and at 
.
and 
.
.
.
be complex numbers satisfying 
.
.
Y by PikaPika999, Rounak_iitr
m:}
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
.
Prove that the point lies on the circumcircle of 
.
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
.Prove that the point
lies on the circumcircle of 
.This post has been edited 1 time. Last edited by youochange, Apr 6, 2025, 11:28 AM
Reason: Y
Reason: Y
.
and 
.
is the midpoint of arc 
,
.
and 
,
.
are collinear and 
.
.
.
are concyclic, we need to prove that 
.
,
is the midpoint of 
,
is cyclic.
with 
.
is harmonic (the tangents from 
and 
intersect on 
)
onto 
.
.
,
,
is cyclic.
.
,
and 
.
and thus 
which means that 
is cyclic. Taking into account the result that 
is cyclic. 
is the intersection of 
and 
,
and 
are colinear.
,
inversion
is a point in the plane let after inversion it be 
where 
H
will be a random point on 
, 
and 
again in 
,
.
.
