,
is the midpoint of 
and 
is a point on the side 
.
,
,
are positive integers 
.
is also an integer and equals 
.
,
and 
be two isogonal conjugate with respect to triangle 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of 
and 
bisect two arcs 
and not containing 
.
and 
.
of triangle 
.
be the reflection of 
be the foot of the perpendicular from 
,
be the midpoint of the segment 
.
intersect the circle 
.
is tangent to the circumcircle of triangle 
.
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
,
,
,
,
.
there exists an 
all of whose digits are odd.
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
denotes the number of positive divisors of the positive integer 
.
such that for all real numbers 
and 
,![\[f(f(xy)+y)=(x+1)f(y).\]](http://latex.artofproblemsolving.com/d/1/f/d1f369109456464bc8b0102763211802770d983d.png)
be positive real numbers such that 
.![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
formed by using the digits 
once each such that the number 
is divisible by 
for 
.
be an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of 
positive integers with the property that the product of any two distinct members of 
and 
in the set 
,
makes an angle of 
with side 
and an angle of 
with side 
.
Y by
Two circles 
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.Thanks.
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