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is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
+1 w
be real numbers . Prove that : There exist positive integer 
such that 
Where 
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.
, for which
in the array can be changed into either 
or 
so that the row-sums and column-sums remain unchanged. (Note that is the least integer greater than or equal to , while is the greatest integer less than or equal to .)
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex 
a non-negative real number 
, for 
, such that 
. Bob then selects a sequence of distinct vertices 
such that 
and 
are connected by an edge for every 
. (Note that the length 
is not fixed and the first selected vertex always has to be 
.) Bob wins if![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
, determine the largest possible value of 
for which Bobby has a winning strategy.
, 
, 
, 
, 
, 
be positive integers and let 
.
divides 
and 
. Prove that is composite.
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side at , and to the lines and at 
and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
opposite the vertex 
is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond .)
that 
where 
is the canonical form of with 
.
where is the canonical form of is true for all positive integer . that 
.
, we get that 
for all 
.
where 
. Modulo 
gives us 
, impossible since all 
s are distinct.
. This gives 
for some prime number 
. that not smaller than 
is 
.
that where is the canonical form of with . where is the canonical form of is true for all positive integer . that ., we get that for all . where . Modulo gives us , impossible since all s are distinct.. This gives for some prime number . that not smaller than is .
and 
, and them being the only zeroes of this function. One can also say 
doesn't take any negative numbers.
, where 
, and assume this number satisfies the given condition
We get that 
. Rewriting 
gives 
It should be obvious that and 
. Since 
(Or else 
), we have . All that is left to find the smallest prime that satisfies 
, which is 
such that :
such that 
)))))