AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
be positive integers. Find all pairs of non-negative integers 
and 
that always satisfy the following condition: white dots and black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly white dots and black dots.
divides 
for positive integers 
then
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.
is the radical axis of and .
where 
and 
are positive integers that do not exceed 
?
denote the set of positive integers
such that for all 
it holds that 
be a positive integer. There are circles drawn on a chalkboard such that any two circles intersect at 
distinct points and no 
circles pass through the same point. Turbo the snail slides along the circles in the following manner, leaving snail goo behind. Initially he moves on one of the circles in clockwise direction. He keeps sliding along until he reaches an intersection with another circle. Then, he continues his journey on this new circle and also changes the direction he is moving in. We define a snail orbit to be the covering of the whole surface of a circle with turbo's goo, and specifically only a single layer of it. Prove that for every odd there exists at least one configuration of circles with a single snail orbit, and find all such that there is exactly one of the aforementioned configuration type.
such that for any two infinite sequences of positive integers 
and 
, such that 
for all 
, there always exists a real number 
such that 
for all 
.
, , 
, 
, 
, only. Note that 
, and 
, so that 
.
satisfys the condition.
Note that 
rectangle can be cut to 
rectangle if 
. Partition the grid to four of the 
rectangle, as following.![[asy]
draw((0, 0) -- (45, 0));
draw((45, 0) -- (45, 45));
draw((45, 45) -- (0, 45));
draw((0, 45) -- (0, 0));
draw((0, 23) -- (23, 23));
draw((22, 0) -- (22, 23));
draw((23, 45) -- (23, 22));
draw((22, 22) -- (45, 22));
fill((22, 22) -- (22, 23) -- (23, 23) -- (23, 22) -- cycle, black);
[/asy]](http://latex.artofproblemsolving.com/c/d/a/cdaf5762b57ff4a04c17001a7b3f73a0cde7b496.png)
or 
, each of the rectangle can be cut to 
, or rectangle, so we are done.
. The idea is to color each of the square, either black or white. Moreover, let (, ) denotes the square with the th row, th column. Then color the grid such that the square is black if and only if is odd and 
, or even and 
. The example with 
replaced to 
are shown below.![[asy]
unitsize(1cm);
for(int i = 0 ; i <= 5 ; i = i + 1) draw((0, i) -- (5, i));
for(int i = 0 ; i <= 5 ; i = i + 1) draw((i, 0) -- (i, 5));
for(int i = 0 ; i <= 4 ; i = i + 2) fill((i, 5) -- (i + 1, 5) -- (i + 1, 3) -- (i, 3) -- cycle, black);
for(int i = 1 ; i <= 3 ; i = i + 2) fill((i, 3) -- (i + 1, 3) -- (i + 1, 1) -- (i, 1) -- cycle, black);
for(int i = 0 ; i <= 4 ; i = i + 2) fill((i, 0) -- (i + 1, 0) -- (i + 1, 1) -- (i, 1) -- cycle, black);
[/asy]](http://latex.artofproblemsolving.com/9/3/e/93e8fdfd44e05ba88096780859aafcc7aace2184.png)
black squares and same for white. but since the centre (, ) is colored white, the number of the black square is ![\[
23\cdot 23 + 22\cdot 22 = 1013.
\]](http://latex.artofproblemsolving.com/c/f/5/cf58b758901a6796c8c4facd6641d3502ad05ad4.png)
So the number of white square is 
, contradiction.
grid has had the central unit square removed. For which positive integers 
and 