[/quote]
,
has a unique solution in integers with 
and 
.
as 
fir 
.
for all 
.
.
intersect 
respectively. Lines through 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
are positive real numbers.Prove that
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
with 
such that the equation 
has an integer solution 
.
such tha
(Adit Aggarwal, India.)
be positive reals such that 
.
be reals such that 
Prove that
.
of triangle 
meets 
at 
.
be a point on the (internal) angle bisector of 
such that 
.
meet line 
and 
.
is isosceles.
such that, for all positive integers 
and 
and 
Show that
with center 
.
the incenter of triangle 
the contact points of the incircle with sides 
of triangle 
the foot of the perpendicular line from 
.
passes from the antidiametric point 
is a diametre of the circle 
Y by
Given a non-isosceles triangle 
that is inscribed in 
. The incircle 
is tangent to 
at 
respectively. A line through 
parallel to 
intersects at 
, and 
intersects again at 
. Let 
is the midpoint of . 
be the second intersection of 
with . 
intersects again at 
. Prove that the line passing through perpendicular to 
bisects 
.
that is inscribed in 
. The incircle 
is tangent to 
at 
respectively. A line through 
parallel to 
intersects at 
, and 
intersects again at 
. Let 
is the midpoint of . 
be the second intersection of 
with . 
intersects again at 
. Prove that the line passing through perpendicular to 
bisects 
.
