[/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 
.
,
.
.
be the feet of the altitudes from 
to 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
,
,
at 
,
and 
be the midpoints of 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
.
,
lie on a line in this order and satisfy 
.
,
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 
.
be a cyclic quadrilateral with 
.
not containing 
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 Adventure10, Mango247
the edges of triangle 
are 
respectively,
,
is the bisector of 
,point 
is on segment 
.
(1)find the property,
,
have,so that there exists point 
on 
satisfy 
,and 
(2)when such exist,express 
with
are 
respectively,
,
is the bisector of 
,point 
is on segment 
.(1)find the property
,
,
have,so that there exists point 
on 
satisfy 
,and 
(2)when such
exist,express 
with This post has been edited 1 time. Last edited by horizon, Feb 25, 2012, 9:38 AM
.
,
satisfying the following condtions: 
,
and 
:
so we conclude that:
is the angle bisector of 
so:
.
,
.
,
and 
,
