AoPS Academy
, are there infinitely many positive integers 
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
+2 w
be a triangle with circumcircle 
, circumcenter 
, and orthocenter 
. Assume that 
and that 
. Let 
and 
be the midpoints of sides 
and 
, respectively, and let 
and 
be the feet of the altitudes from 
and 
in 
, respectively. Let 
be the intersection of line 
with the tangent line to at 
. Let 
be the intersection point, other than , of with the circumcircle of 
. Let 
be the intersection of lines 
and 
. Prove that 
.
be the foot of perpendicular from to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle . A circle 
with centre 
passes through and , and it intersects sides and at 
and 
respectively. Let be the foot of altitude from to 
, and let be the midpoint of . Prove that the circumcentre of triangle 
is equidistant from and .
+2 w
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
exterior of circle 
, we draw tangents 
, 
and the secant 
. The line passing through point parallel to intersects chords , 
at points , respectively. Prove that 
.
exterior of circle , we draw tangents , and the secant . The line passing through point parallel to intersects chords , at points , respectively. Prove that .
