and 
.
and 
.
Equality holds when 
or 
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
.
Where 
Y by Adventure10
Q4) Each box in a 
table can be colored black or white . How many different colorings of the table are there?
Q5) Determine all positive integer a such that the equation
has two prime roots, i.e. both roots are prime mumbers.
Q6) Let
be the roots of the equation 
and let 
be the roots of the equation 
, where 
are some positive real mumbers. Suppose that 
is an integer . Determine 
.
Q 7) Let P be the common point of 3 internal bisectors of a given ABC. The line passing through P and perpendicular to CP intersects AC and BC at M and N, respectively. If AP=3cm, BP=4cm, compute the value of
.
Q8) If
and 
are both perfect squares, find .
Q 9) Let
be the positive integer such that 
. Prove that 
is an perfect integer.
table can be colored black or white . How many different colorings of the table are there?Q5) Determine all positive integer a such that the equation
has two prime roots, i.e. both roots are prime mumbers.Q6) Let
be the roots of the equation 
and let 
be the roots of the equation 
, where 
are some positive real mumbers. Suppose that 
is an integer . Determine 
.Q 7) Let P be the common point of 3 internal bisectors of a given ABC. The line passing through P and perpendicular to CP intersects AC and BC at M and N, respectively. If AP=3cm, BP=4cm, compute the value of
.Q8) If
and 
are both perfect squares, find .Q 9) Let
be the positive integer such that 
. Prove that 
is an perfect integer.
patterns with 
block colored black, there are 
colorings available. If rotations/reflections are considered identical, then with 
black there is 
black there are 
and 
colorings.
.
be the roots, since 
,
is the only possible answer.
.
for some integer 
.
.
is an integer.
for all perfect squares 
.
as 
and equality occurs when 
.
colorings.
,
.
correctly. I changed the solution. Nice catch!
white corresponds to a different coloring with 
