[/quote]
,
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
.
,
.
.
and 
to 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
can be inscribed within a circle.
be a triangle whose incircle is tangent to 
,
,
at 
,
and 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
.
intersect the circumcircle of 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
.
,
.
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form 
,
and 
are not necessarily distinct. Determine 
.
.
and 
.
,
.
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
and 
.
?
be the number of positive integers that divide the integer 
.
,
?
is a square of an integer.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
Y by Mango247, Mango247, Mango247
Let 
be a square, point 
lies on the side 
, 
be the center of the square, 
be the midpoint of 
, 
. It is known that 
is isosceles. In what ratio does the point divide the side of the square ?.
(Klurman O.)
be a square, point 
lies on the side 
, 
be the center of the square, 
be the midpoint of 
, 
. It is known that 
is isosceles. In what ratio does the point divide the side of the square ?.(Klurman O.)
This post has been edited 3 times. Last edited by parmenides51, Mar 16, 2021, 9:58 PM
?
