AoPS Academy
be an acute triangle and 
it's incenter. Let 
, 
and 
be the projections of onto 
, 
and 
respectively. 
and 
is the projection of onto 
. Let 
be the middle of the arc not containing 
and 
the second intersection of 
and the circumcircle of 
. If 
is the midpoint of 
, 
, 
and 
, prove that 
is cyclic.
is recursively defined such that 
is not a power of 
, and for all nonnegative integers 
:
is even, then 
is the largest odd factor of is odd, then 
where 
is the smallest prime factor of such that 
for all 
.
such that 
is the square of an integer.
+2 w let 
denote the midpoint of side . Let the circle passing through point and tangent to side at point intersect sides and at points and , respectively. Let the line segments 
and 
intersect in point 
. Let 
be the second point of intersection of the circumcircles of triangles 
and 
. Prove that points 
and are collinear.
, where 
. Verify by direct substitution that 
satisfies the quadratic equation 
and deduce that the value of is 
.
. Show that 
, show that the value of 
is 
.
be the set of all 
matrices with at most two entries in each row equal to 
and all other entries equal to 
.
.
denotes the determinant of the matrix .
be the nearest integer to 
.
Also, generalise your result.
such that 
and 
. Find the value of 
make the notation 
for the midpoints of 
respectively, 
Additionally, we know that 
is the common perpendicular of 
and 
is the common perpendicular of 
Show that 
