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.
positive real numbers such that 
. Prove that :
, points 
and 
are taken on the legs 
and 
, respectively, so that 
and 
. Find the acute angle between line segments 
and 
.
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 
's move, he selects 
subgrid and color it white. On 
's move, he selects a 
subgrid and colors it black. wants to make the whole board white. Can he do it?
be a convex quadrilateral such that 
. Let and be the midpoints of and 
, respectively. The lines 
and meet the line 
at 
and 
, respectively. Prove that 
.
be the n-th prime number in increasing order for 
. Prove that there exists a sequence of distinct prime numbers 
satisfying 
for all 
and 
intersect at points and . Distinct points 
and 
(not at or ) are selected on . The lines 
and 
meet again at 
and 
respectively, and the lines 
and 
meet at 
. Prove that, as and vary, the circumcentres of triangles 
all lie on one fixed circle.
penguin and each one is numbered using numbers 
. penguin 
is called 
if 
where 
is 
prime divisor of 
(
(where k is number of 's prime divisors). what is maximum ratio of 
where is number of in antarctica?(
)
. 
is the midpoint of arc not containing . 
is the midpoint of arc 
is the midpoint of 
and is the midpoint of 
. 
is projection of 
onto . 
is the second intersection of 
and 
is tangent to 
.
with center is incircle of triangle . is projection of onto Points 
on segments 
, 
are following condition: 
. is the midpoint of 
. is the second intersection of 
and . is the second intersection of 
and .
are concyclic.
with diameter 
is given. The point lies in the interior of the circle, but not on . The line 
intersects in and . The tangent to at intersects the line through perpendicular to , at . The point lies on , and is such that 
is tangent to and 
.
, and are collinear.
has centre 
, and is a diameter of . Let be a point of such that 
. Let be the midpoint of the arc which does not contain . The line through parallel to 
meets the line 
at . The perpendicular bisector of 
meets at 
and at 
. Prove that is the incentre of the triangle 
lies inside triangle such that 
and 
. Point is the midpoint of segment . Point lies on segment with 
. Prove that 
.
be complex numbers satisfying 
. Show that there exist some among the complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
be a triangle. The perpendicular bisector of intersects at and the extension of at . The bisector of 
intersects at . If 
and 
, find the angles of .
be real numbers. Prove that
