[/quote]
,
grid 
)
black cells.Detemine the number of balanced colorings.
inscribed in 
with 
.
of 
onto 
that is not on 
intersects 
at 
and 
intersect at 
is an isosceles triangle.
for which there exist real numbers 
satisfying 
,
and
for 
.
be an integer. Find the smallest integer 
with the property that there exists a set of 
other distinct elements of the set.
-
and 
.
,
,
are positive integers 
.
is also an integer and equals 
.
,
such that for all positive real numbers 
and 
:
and 
such that 
and 
are perpendicular and parallel to bisector of 
and 
bisect two arcs 
and not containing 
.
and 
.
of triangle 
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 
,
.
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)
Y by
In parallelogram 
, 
. Construct a circle with center 
and radius 
, intersecting the extensions of 
and 
at 
and 
respectively. Construct a circle with center 
and radius 
, intersecting the extensions of 
and 
at 
and 
. Let 
be the intersection of 
and 
, 
be the intersection of 
and 
, 
(
) be the intersection of and the circle with center , and 
(
) be the intersection of 
and the circle with center . Show that , , and are concyclic.
, 
. Construct a circle with center 
and radius 
, intersecting the extensions of 
and 
at 
and 
respectively. Construct a circle with center 
and radius 
, intersecting the extensions of 
and 
at 
and 
. Let 
be the intersection of 
and 
, 
be the intersection of 
and 
, 
(
) be the intersection of and the circle with center , and 
(
) be the intersection of 
and the circle with center . Show that , , and are concyclic.
