[/quote]
,
denote the maximum number of 
-
board such that each 
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
,![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all 
and 
,
such that 
or 
,![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
.The solution to 
is only 
for some 
,
plane: 
.
for which these circles have two intersection points, then find the equation of the line passing through these points.
on the perimeter of a circle with radius 1. Let the length of the cord 
.
.
satisfies 
.
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
be real numbers such that 
.
.
Y by Rounak_iitr
Let 
be the number of positive integers with 
digits 
in base (where 
for all 
and 
) such that the polynomial
![\[d_9x^9+d_8x^8+\cdots+d_1x+d_0\]](//latex.artofproblemsolving.com/6/6/7/667e5a3d7f6c4f984e1278b5f76559c3b50c9690.png)
is irreducible in 
. Prove that 
is even.
(A polynomial is irreducible in if it cannot be factored into two non-constant polynomials with rational coefficients.)
be the number of positive integers with 
digits 
in base (where 
for all 
and 
) such that the polynomial![\[d_9x^9+d_8x^8+\cdots+d_1x+d_0\]](http://latex.artofproblemsolving.com/6/6/7/667e5a3d7f6c4f984e1278b5f76559c3b50c9690.png)
is irreducible in 
. Prove that 
is even.(A polynomial is irreducible in
if it cannot be factored into two non-constant polynomials with rational coefficients.)
then 
for all 
then 
with 
as a polynomial is irreducible iff its reverse is.
else 
divides the polynomial, and 
works. This is because if 
was somehow not irreducible, i.e. 
for some nonconstant polynomials ![$g(x), h(x) \in \mathbb{Q}[x]$](http://latex.artofproblemsolving.com/7/b/5/7b596909c781d727c0920f07a722b1118e714905.png)
,
contradicting the fact that 
as 
,
.
,
is irreducible, then its reverse is also irreducible.
.
implies the polynomial is divisible by 
is palindromic, then 
is a root so the polynomial is reducible.
multiplied by 
,
.
for which 
.
is reducible as 
.
and vice-versa.
.
,
.
,
.
,
is the leading coefficient of 
,
,
to get two polynomials. But if indeed 
,
is a factor, contradicting the irreducibility of 
,
.
,
.
and it can easily be checked that one is irreducible if and only if the other is
