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be reals such that 
Prove that
of the triangle 
touch its side 
at 
. Let the line 
intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.
and 
are concyclic.
be a triangle with 
. is the midpoint of 
. Let the angle bisector of 
cut 
at 
and 
be the centroid of . 
meets at 
and 
is the projection of 
onto 
. Prove that 
lie on a common circle.
be 
. Points 
and 
are two moving points on the circle 
: 
, satisfying 
. Find the maximum area of 
.
are complex numbers with absolute value 
. For 
define 
. Given that the sequence 
converges. Show that,
.
such that there exists an integer polynomial with no integer roots and a constant 
such that
has at least 
distinct prime divisors for all positive integers 
.
is an integer, then there exist invertible matrices 
with non-zero entries such that:![\[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\]](http://latex.artofproblemsolving.com/a/f/9/af9ff29771952abe37a00f3830c72e8c517e1cd6.png)
be a closed and countable set. Prove that if the analytic function 
is bounded, then 
is constant.
be an integer with 
Over all real polynomials 
of degree 
what is the largest possible number of negative coefficients of 
fibonacci numbers (
) find all n such that:
and 
![$f:[1,8]\to R$](http://latex.artofproblemsolving.com/c/e/c/cec97ea0372e37977e8a77e8e77518a2344c1c43.png)
that:
be a trapezoid inscribed in circle 
, 
. Let is the symmetric point of across , 
intersects at . Let is midpoint of and 
is orthocenter of triangle . On take a point 
so that 
. Prove that: 
coincide
be a trapezoid inscribed in circle , . Let is the symmetric point of across , intersects at . Let is midpoint of and is orthocenter of triangle . On take a point so that . Prove that: coincide
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