[/quote]
,
,
be a circumscribed quadrilateral with circumcircle 
such that 
,
is the intersection point of the diagonals 
and 
.
is taken on 
.
is the reflection of 
,
is tangent to the line 
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 
such that 
is a prime number.
such that for any prime 
and any positive integers 
both 
and 
are integers.
.
and 
be the number of vertices of the largest red and blue clique, respectively. Bob wins if at the end of the game 
.
denotes the complete graph with 
vertices and where there is an edge between any pair of vertices. A red (or blue, respectively) clique refers to a complete subgraph of which all the edges are red (or blue, respectively).
.
chessboard is comprised of 
angle with this side. When the beam hits a side of the table or a side of a cell containing a barrier, it reflects back according to the law of reflection. We say that the beam passes through a cell if it passes through the strict interior of the cell.
be a positive integer. Suppose that we can position the laser and place barriers in 
be a triangle. A circle passing through 
intersects segments 
at 
and 
and 
intersect at 
intersect at 
if and only if 
.
different colours.
such that the following holds:
and 
satisfy 
and 
.
contains no triangles and every vertex of 
be the set of all non-decreasing functions 
satisfying 
for all positive integers 
over all 
.
are collinear in this order, in such a way that the line 
,
.
be the midpoints of segments 
,
,
at point 
at point 
Show that 
Y by
Can someone help me with this problem? Let 
. We call a distribution the act of distributing the integers from 
to
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
We say that
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
, the number of chicken-fox-viper distributions?
. We call a distribution the act of distributing the integers from 
to
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.We say that
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
, the number of chicken-fox-viper distributions?
?
,
are the draws of players 
and 
.
,
.
would satisfy 
with equal probability 
,
.
,
and player 
has 
