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.
be an interior point on the side 
of an acute-angled triangle 
. Let the circumcircle of triangle 
intersect 
again at 
and the circumcircle of triangle 
intersect 
again at 
. Let 
, 
, and 
intersect the circumcircle of triangle again at 
, 
and 
, respectively. Let 
and 
be the incentres of triangles 
and 
, respectively. Prove that 
are concyclic.
be a median in an acute-angled triangle . A point 
is chosen on the line through 
tangent to the circumcircle of 
so that 
. The segments 
and meet at 
. Prove that the circumcenter of 
lies on the line .
be an integer. The sequence 
is defined via 
, 
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer 
. Prove that there are finitely many pairs 
of positive integers such that 
.
be a triangle with 
. Let be the intersection point of the internal bisector of angle 
and the circumcircle of . Let 
be the intersection point of the perpendicular bisector of with the external bisector of angle 
. Prove that the midpoint of the segment lies on the circumcircle of triangle 
.
![$a,b,c\in[0,1]$](http://latex.artofproblemsolving.com/8/b/d/8bd8f441b63556fdf318ad140bbd3e8e358e9e1d.png)
, assume 
, compute the range of![\[f(a,b,c)=(1-a^2)^2+(1-b^2)^2+(1-c^2)^2.\]](http://latex.artofproblemsolving.com/7/c/1/7c1611d9b80f5f694afed1894acd57a93aa5b31c.png)
with orthocenter , circumcenter and incenter 
, circum-radius 
, in-radius 
.
.
satisfying
for all 
.
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
, such that,
for all 
.
are primes, which satisfy:
is an integer
.
be circumscribed about circle 
, and let 
be the orthocenter of . The circle touches line at . The tangent to the circle 
at meets at 
. Let be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
and 
. Points 
and 
are on line segment 
so that 
and 
. Lines 
and 
intersect at 
. What is the area of 
?
now do find area of quaderilateral 
we simply subtract the areas of 
and 
from the area of rectangle 
and we get area=
now 
let 
so we have 
cross multiplying we get 
or 
now
now do find area of quaderilateral we simply subtract the areas of and from the area of rectangle and we get area= now let so we have cross multiplying we get or now
with on 
. So, 
. Let 
Thus, 
and 
. So we get 
. No apply 
to get the answer as 
.
and 
be real numbers . Prove that
be reals such that 
Prove that
Line joining 
.
.
.