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be a triangle with circum-circle 
. Let 
be a point in the interior of triangle which is also on the bisector of 
. Let 
meet in 
respectively. Suppose 
is the point of intersection of 
with 
; and 
is the point of intersection of 
with 
. Prove that 
is parallel to 
.
such that 
is a perfect square but none of the integers 
are perfect squares.
be a triangle with incenter 
and intouch triangle 
. Let 
be the foot of the perpendicular from 
onto 
. Assume that 
, 
intersect the sides 
, 
in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in 
, 
respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of .
be an odd positive integer satisfying 
with exactly 2 distinct prime factors. Prove that is a prime number or the square of a prime number.
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
. 
: sum of elements of
is drawn externally on side of . Point is taken on 
. Find 
if 
.
for all 
.
be a triangle where
B=55 and C = 65. D is the mid-point of BC. Circumcircle of ACD and ABD cuts AB and AC at point F and E respectively. Center of circumcircle of AEF is O. FDO = ?
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
and 
.
Equality holds when 
or 
or 
,
,
.
be the angle bisector of angle 
such that 
lies on side 
at 
.
?
of the numbers 
satisfy 
and 
?
is cyclic.
is concyclic.