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 .
+1 w
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
So i think that, as an Olympiad problem, this problem deserves a place here, but if it's more convenient to place it in the intermediate forum, it doesn't matter 
so every number of the form 
can be written in that form, and only such numbers do. hence the answer to the first problem is 
minus the last term. so these sums are equal to 
So numbers that can be written in both ways are those numbers of the form 
so the rest is easy.
so every number of the form can be written in that form, and only such numbers do. hence the answer to the first problem is minus the last term. so these sums are equal to So numbers that can be written in both ways are those numbers of the form so the rest is easy.
not , I assume, you missed the case 
so every number of the form can be written in that form, and only such numbers do. hence the answer to the first problem is minus the last term. so these sums are equal to So numbers that can be written in both ways are those numbers of the form so the rest is easy. not , I assume, you missed the case gives 0, which is not a positive integer.
