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.
, 
, and 
of a hexagon 
inscribed in a circle 
intersect at a point 
, and the acute angle between any two of them is 
. Let 
be the radius of the circle tangent to segments 
and 
and internally tangent to ; the radii 
, 
, 
, 
, and 
are defined similarly. Prove that![\[
r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}.
\]](http://latex.artofproblemsolving.com/0/d/6/0d66d6685bbc97c7553e9ed3ba14ea0dc3328b49.png)
such that 
for every 
.
+1 w
and 
be polynomials with integer coefficients. Suppose that the integers 
and 
are roots of , and that 
. Prove that the equation 
has no integer solutions.
. Find 
.
be three positive real numbers with 
. Find the minimum value of the expression 
is strictly increasing : 
such that:
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
such that for all 
. 
.
be positive real numbers such that 
. Prove that
.
satisfying the following conditions: divides 
,
divides 
,
.
and 
are constructed in the extirior of a right-angled triangle 
with 
.Let 
be the midpoint of 
.Given that 
and 
.Find the length of .
, but I might be wrong with calculations.
are equilateral 
![$[CG] = [CA] = [GA]$](http://latex.artofproblemsolving.com/a/b/4/ab421e7805fb68ff78fa7ef170bc6193534bf750.png)
, ![$[FB] = [AB] = [AF]$](http://latex.artofproblemsolving.com/c/6/f/c6fe82d238ee035deb72c832bd709a8eab1b177f.png)
and 
by SAS ![$[GB] = [FC]$](http://latex.artofproblemsolving.com/4/d/4/4d49ac5d96d4e5de516d1ad4566e59c96f8ad94d.png)
.
are G-, F-medians of 
![$[GM]^2 = \tfrac{1}{4}\left(2[GB]^2 + 2[GC]^2 - [BC]^2\right)$](http://latex.artofproblemsolving.com/e/1/f/e1fb86879f913c0da9bf27f79fb4d5c977e1ac8f.png)
and ![$[FM]^2 = \tfrac{1}{4}\left(2[FB]^2 + 2[FC]^2 - [BC]^2\right)$](http://latex.artofproblemsolving.com/9/b/e/9bed683d4e3a6886c553e24ef14d12ad349dbb38.png)
![$[FM]^2 - [GM]^2 = \tfrac{1}{2}\left([FB]^2 - [GC]^2\right) = \tfrac{1}{2}\left([AB]^2 - [CA]^2\right) = \tfrac{1}{2} [BC]^2 =$](http://latex.artofproblemsolving.com/a/e/4/ae4bfe990229b05c41464d1bba8ed0f5941eeae3.png)
![$[BC]^2 = 144$](http://latex.artofproblemsolving.com/1/d/7/1d729c3d6702a814cca0ff398eb2165df65bd570.png)
![$[BC] = 12$](http://latex.artofproblemsolving.com/f/c/a/fca981613c8213607683ec4f336bfad3402be878.png)
.
and are constructed in the extirior of a right-angled triangle with .Let be the midpoint of .Given that and .Find the length of .
.
Let 
.
and 
. Prove that
be the midpoint of ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and 
be the midpoint of ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
, 
and 
.
is right at its vertex 
hence
, 
formula (cosine formula ) gives
, so we have
we get
we get
