,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
be a triangle with circumcircle 
,
,
.
and that 
.
and 
be the midpoints of sides 
and 
,
and 
be the feet of the altitudes from 
and 
in 
,
be the intersection of line 
with the tangent line to 
.
be the intersection point, other than 
.
be the intersection of lines 
and 
.
.
be the foot of perpendicular from 
with centre 
and 
respectively. Let 
,
is equidistant from 
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
Y by
Fix a nonnegative integer 
to define a sequence of integers 
by letting 
be the smallest integer (strictly) greater than 
making 
into a perfect square. Let 
be the set of positive integers not expressible as the difference of two terms of the sequence 
Prove that 
is finite and determine its size in terms of 
to define a sequence of integers 
by letting 
be the smallest integer (strictly) greater than 
making 
into a perfect square. Let 
be the set of positive integers not expressible as the difference of two terms of the sequence 
Prove that 
is finite and determine its size in terms of 
This post has been edited 2 times. Last edited by oVlad, May 14, 2023, 10:18 AM
and proving that 
by using simple bounding. Once you have done this, the number 
will become important and you will get that all differences of the form 
and 
are achievable. This essentially tells you that all large enough odd numbers and all large enough even numbers are expressible; thereby solving the problem. (for the actual construction one needs some more rigour but it's not that hard to recreate; one can prove these are the smallest possible differences and nothing below these can be constructed)
