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.
st for every two subsets 
of 
, there exists a polynomial 
with integer coefficients st either 
or 
where the equations are considered mod n.
is the set of reminders of 
where 
mod n.
be a positive integer such that 
.
.
be Rutthee sequence if 
be a positive integer and
for all 
and there are some positive integer 
sastisfied 
such that 
and 
also find all possible value of in Rutthee sequence
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
be two positive integers, such that 
. Find all the integer values that 
can take, where ![\[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]](http://latex.artofproblemsolving.com/c/b/2/cb27ccbcde6d90f98e10f8f6876c36ae8972af7f.png)
be positive real integer such that 
Prove that
such that 
and 
real number on the board such that each number is more than 
.
on the board and write 
.
and 
.
?
of positive integers satisfying the equation 
be a natural number . Define 
be cut number at 
if
Prove that if 
be cut number at 
prove that 
.
be triangle and point 
be 
altitude of 
. be a point on 
and 
such that 
and 
. Point is the intersection of 
and 
. Point 
be intersection of 
and . Line 
intersect circumcircle of again at 
.
.
bisect .
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino. cell 
Note: tetromino can be rotated but not flipped
Prove that
Let 
Prove that
Prove that
be reals such that 
. Prove that
Let 
. Prove that
it follows that
Then
and ,
we have
it follows that
.![\[
a^2 + b^2 + c^2 = a^4 + b^4 + c^4 = 2,
\]](http://latex.artofproblemsolving.com/2/e/f/2ef5e28eb01d8ec580b9d68407793485306b7fdf.png)
and it is evident that![\[
a^2,\ b^2,\ c^2 \leq \frac{4}{3}.
\]](http://latex.artofproblemsolving.com/3/8/0/3800858b389c2e63fa979776f37b8d0b2c2a2cf3.png)
![\[
\frac{1}{a^4+1}+\frac{1}{b^4+1} \geq \frac{4}{a^4+1+b^4+1} = \frac{4}{4-c^4} \geq 1,
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7aa9b57b2d695e908f72904596476be885089.png)
![\[
\frac{1}{a^4+1}+\frac{1}{b^4+1}+\frac{1}{c^4+1} = \frac{1}{50}\sum \left( 27(2-a^2)-\frac{(4-3a^2)(3a^2-1)^2}{a^4+1} \right) \leq \frac{27}{50}\sum (2-a^2)=\frac{54}{25}.
\]](http://latex.artofproblemsolving.com/f/2/d/f2d67667408d5455e1bed4a8a516f650be2cab1e.png)
![\[
\frac{1}{a^4+1}+\frac{1}{b^4+1} \leq \frac{54}{25}-\frac{1}{c^4+1}\leq \frac{9}{5}.
\]](http://latex.artofproblemsolving.com/6/d/6/6d6b2dbebcb82cae8d3ff745dac98f29ae2cd794.png)
then 
and 
at 
at 