[/quote]
,
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
passing through point 
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
be a positive integer. Show that any polynomial ![$Q\in\mathbb{Z}[X]$](http://latex.artofproblemsolving.com/e/8/2/e8286b6f2b9790f39c88849a1415d86f85c0b92c.png)
with all positive coefficients such that
is a perfect square for any 
must be a constant polynomial.
be an acute triangle and let 
be the foot of the 
be the midpoint of 
with diameter 
meets 
at 
,
with diameter 
at 
.
are concyclic.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
having 
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
are taken such that the point 
.
and 
,
and 
.
.
Y by anantmudgal09, Adventure10, Mango247
Determine whether there exists a polynomial 
with two variables, with integer coefficients, and two points 
and 
in the plane, satisfying the following conditions:
(i)
is an integer point (i.e 
and 
are integers);
(ii)
;
(iii)
for all integer points 
in the plane other than ;
(iv)
for all integer points 
in the plane other than 
.
Massimo Gobbino, Italy
with two variables, with integer coefficients, and two points 
and 
in the plane, satisfying the following conditions:(i)
is an integer point (i.e 
and 
are integers);(ii)
;(iii)
for all integer points 
in the plane other than ;(iv)
for all integer points 
in the plane other than 
.Massimo Gobbino, Italy
,
.
everywhere.
.
.
,
as a product of polynomials which are either squares or sum of squares. So we just need to choose the factors right.
,
(HUGE in this case means that 
and 
and we can easily prove that this function is suitable
.
and 
still.
(since I showed it doesn't equal 0). Note that 
.
only at 
(remember that everything is an integer point). So 
,
.
,
and 
.
and clearly 
for every integer point 
implies 
,
integers). Also, we have 
;
(with equality iff 
)
,
.
,
where 
is insanely small. Of course, 
.
,
and so it's the smallest possible value 
.
![$f\in\mathbb{Z}[x_1,x_2]$](http://latex.artofproblemsolving.com/7/0/6/70669195ea3d3b7b3de7094de530ea49203b565f.png)
which attains its minimum over 
at 
,
is negative in ![$]0;1[$](http://latex.artofproblemsolving.com/a/5/1/a518a9434ce77fb8067ab8f94e64a8a0355325e5.png)
,
and set![\[f(x,y)=(x^2+y^2)+\text{googolplex}\cdot (y^2-y).\]](http://latex.artofproblemsolving.com/6/8/d/68dd78fd25727b09e73eb9980936c54e440e7013.png)
for 
,
for any other integer point. However, 
is absurdly negative because 
.
as follows: 
and let us set 
.
be positive real numbers such that 
and 
.
and 
.
has integer coefficients and for all non zero lattice points 
we have 
and since 
and 
is a prime number of the form 
with exponent 
.
and as 
and thus, 
,
and minimized on nonnegative reals at some 
.
for all 
.
,
intersects the diamond 
at some point 
.
,
