and 
for 
.
.
for which 
is divisible by 
.
is divisible by 
satisfy,
Where 
}
be a fixed square and 
a variable point on segment 
.
is constructed such that 
is on segment 
and 
is on segment 
.
be the intersection point of lines 
and 
.
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.
Y by
Let 
be an acute triangle, let 
be a median, and let 
and 
be the altitudes. Let 
be the line perpendicular to passing through 
. Let 
be the intersection point of with , and let 
be the intersection point of with .
(a) Prove that is the midpoint of 
.
(b) Let
be the circumscribed circle of the triangle 
, and let 
and 
be any two circles that have two points 
and 
in common, and are tangent to the segment and the arc of not containing the point 
. Prove that points 
are collinear.
be an acute triangle, let 
be a median, and let 
and 
be the altitudes. Let 
be the line perpendicular to passing through 
. Let 
be the intersection point of with , and let 
be the intersection point of with .(a) Prove that
is the midpoint of 
.(b) Let
be the circumscribed circle of the triangle 
, and let 
and 
be any two circles that have two points 
and 
in common, and are tangent to the segment and the arc of not containing the point 
. Prove that points 
are collinear.This post has been edited 2 times. Last edited by parmenides51, Dec 21, 2022, 9:57 PM
