[/quote]
,
denote the set of all real numbers. Find all functions 
such that ![\[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]](http://latex.artofproblemsolving.com/8/5/c/85c37ee472f918eace12c8cd7bef6766e617d2aa.png)
(where 
are the digits and 
)
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
pairwise distinct positive integer numbers 
,![\[
\frac{a_i^2 + a_j^2}{a_i + a_j}
\]](http://latex.artofproblemsolving.com/3/5/3/3537ddec6d793d3831ba82a1e13b98d43b1c29bc.png)
be the incircle of triangle 
.
is parallel to side 
and tangent to 
is parallel to side 
and tangent to 
,
satisfy all conditions below:
for all 
for all 
for all 
that are solutions of the equation 
.
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
are such that 
,
of prime positive integers 
such that number 
equals to a square of an integer.
be real number greater than 
.![\[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\]](http://latex.artofproblemsolving.com/5/9/e/59e1a33017116963165f744b49b8515f126c64ef.png)
.
for which:
for all 
.
Y by AnSoLiN
Let 
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of respectively. Find the largest real number 
such that the inequality 
holds for each polygon constructed with these conditions.
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of respectively. Find the largest real number 
such that the inequality 
holds for each polygon constructed with these conditions.
s![[asy]
size(7cm);
real a = 1.0, b = 1.0;
// int m = 5, n = 5;
// real w = a*m, h = b*n;
// path out = (-1, -1) -- (-1, h+1) -- (w+1, h+1) -- (w+1, -1) -- cycle;
// fill(out, RGB(0, 255, 255));
// draw(out, white);
void fill_cell(int row, int column, Label entry) {
fill((row*a, column*b) -- (row*a+a, column*b) -- (row*a+a, column*b+b) -- (row*a, column*b+b) -- cycle, RGB(80, 80, 255));
label(entry, (row*a+a/2, column*b+b/2), p = fontsize(20pt));
}
void outside_cell(int row, int column, Label entry) {
fill((row*a, column*b) -- (row*a+a, column*b) -- (row*a+a, column*b+b) -- (row*a, column*b+b) -- cycle, RGB(255, 255, 255));
label(entry, (row*a+a/2, column*b+b/2), p = fontsize(20pt));
}
void draw_side_u(int row, int column) {
draw((row*a+a, column*b+b) -- (row*a, column*b+b), RGB(0, 0, 255)+linewidth(2));
}
void draw_side_d(int row, int column) {
draw((row*a, column*b) -- (row*a+a, column*b), RGB(0, 0, 255)+linewidth(2));
}
void draw_side_l(int row, int column) {
draw((row*a, column*b) -- (row*a, column*b+b), RGB(0, 0, 255)+linewidth(2));
}
void draw_side_r(int row, int column) {
draw((row*a+a, column*b+b) -- (row*a+a, column*b), RGB(0, 0, 255)+linewidth(2));
}
outside_cell(0, 0, '1');
outside_cell(0, 1, '3');
outside_cell(-1, 2, '3');
outside_cell(0, 2, '0');
outside_cell(1, 2, '3');
outside_cell(0, 3, '2');
outside_cell(-1, 4, '3');
outside_cell(0, 4, '0');
outside_cell(1, 4, '3');
outside_cell(0, 5, '3');
outside_cell(-2, 0, '0');
outside_cell(-2, 1, '0');
outside_cell(-2, 2, '1');
outside_cell(-2, 3, '0');
outside_cell(-2, 4, '1');
outside_cell(-2, 5, '0');
outside_cell(-2, 6, '0');
outside_cell(-1, 0, '0');
outside_cell(-1, 1, '2');
outside_cell(-1, 3, '3');
outside_cell(-1, 5, '2');
outside_cell(-1, 6, '0');
outside_cell(0, 6, '1');
outside_cell(1, 0, '0');
outside_cell(1, 1, '2');
outside_cell(1, 3, '3');
outside_cell(1, 5, '2');
outside_cell(1, 6, '0');
outside_cell(2, 0, '0');
outside_cell(2, 1, '0');
outside_cell(2, 2, '1');
outside_cell(2, 3, '0');
outside_cell(2, 4, '1');
outside_cell(2, 5, '0');
outside_cell(2, 6, '0');
draw_side_u(-1, 4);
draw_side_d(-1, 4);
draw_side_l(-1, 4);
draw_side_u(1, 4);
draw_side_d(1, 4);
draw_side_r(1, 4);
draw_side_u(0, 5);
draw_side_l(0, 5);
draw_side_r(0, 5);
draw_side_u(0, 0);
draw_side_l(0, 1);
draw_side_r(0, 1);
draw_side_u(-1, 2);
draw_side_d(-1, 2);
draw_side_l(-1, 2);
draw_side_u(1, 2);
draw_side_d(1, 2);
draw_side_r(1, 2);
draw_side_l(0, 3);
draw_side_r(0, 3);
outside_cell(0, 8, '');
[/asy]](http://latex.artofproblemsolving.com/f/8/6/f86400a93eb07d41f07853b90ed51b57b34c4a7d.png)
shapes more, and get a higher 
ratio. Simple calculation yields that we approach 
.
,
.![[asy]
size(8cm);
real a = 1.0, b = 1.0;
// int m = 5, n = 5;
// real w = a*m, h = b*n;
// path out = (-1, -1) -- (-1, h+1) -- (w+1, h+1) -- (w+1, -1) -- cycle;
// fill(out, RGB(0, 255, 255));
// draw(out, white);
void fill_cell(int row, int column, Label entry) {
fill((row*a, column*b) -- (row*a+a, column*b) -- (row*a+a, column*b+b) -- (row*a, column*b+b) -- cycle, RGB(80, 80, 255));
label(entry, (row*a+a/2, column*b+b/2), p = fontsize(20pt));
}
void outside_cell(int row, int column, Label entry) {
fill((row*a, column*b) -- (row*a+a, column*b) -- (row*a+a, column*b+b) -- (row*a, column*b+b) -- cycle, RGB(255, 255, 255));
label(entry, (row*a+a/2, column*b+b/2), p = fontsize(20pt));
}
void draw_side_u(int row, int column) {
draw((row*a+a, column*b+b) -- (row*a, column*b+b), RGB(0, 0, 255)+linewidth(2));
}
void draw_side_d(int row, int column) {
draw((row*a, column*b) -- (row*a+a, column*b), RGB(0, 0, 255)+linewidth(2));
}
void draw_side_l(int row, int column) {
draw((row*a, column*b) -- (row*a, column*b+b), RGB(0, 0, 255)+linewidth(2));
}
void draw_side_r(int row, int column) {
draw((row*a+a, column*b+b) -- (row*a+a, column*b), RGB(0, 0, 255)+linewidth(2));
}
outside_cell(0, 0, '2');
outside_cell(1, 0, '2');
outside_cell(2, 0, '2');
outside_cell(3, 0, '2');
outside_cell(4, 0, '2');
outside_cell(0, 1, '2');
outside_cell(-1, 2, '3');
outside_cell(0, 2, '0');
outside_cell(1, 2, '3');
outside_cell(0, 3, '2');
outside_cell(-1, 4, '3');
outside_cell(0, 4, '0');
outside_cell(1, 4, '3');
outside_cell(0, 5, '2');
outside_cell(-1, 6, '3');
outside_cell(0, 6, '0');
outside_cell(1, 6, '3');
outside_cell(0, 7, '3');
outside_cell(4, 1, '2');
outside_cell(3, 2, '3');
outside_cell(4, 2, '0');
outside_cell(5, 2, '3');
outside_cell(4, 3, '2');
outside_cell(3, 4, '3');
outside_cell(4, 4, '0');
outside_cell(5, 4, '3');
outside_cell(4, 5, '2');
outside_cell(3, 6, '3');
outside_cell(4, 6, '0');
outside_cell(5, 6, '3');
outside_cell(4, 7, '3');
outside_cell(-2, -1, '0');
outside_cell(-2, 0, '0');
outside_cell(-2, 1, '0');
outside_cell(-2, 2, '1');
outside_cell(-2, 3, '0');
outside_cell(-2, 4, '1');
outside_cell(-2, 5, '0');
outside_cell(-2, 6, '1');
outside_cell(-2, 7, '0');
outside_cell(-2, 8, '0');
outside_cell(-1, -1, '0');
outside_cell(-1, 0, '1');
outside_cell(-1, 1, '2');
outside_cell(-1, 3, '3');
outside_cell(-1, 5, '3');
outside_cell(-1, 7, '2');
outside_cell(-1, 8, '0');
outside_cell(0, -1, '1');
outside_cell(0, 8, '1');
outside_cell(1, -1, '1');
outside_cell(1, 1, '3');
outside_cell(1, 3, '3');
outside_cell(1, 5, '3');
outside_cell(1, 7, '2');
outside_cell(1, 8, '0');
outside_cell(2, -1, '1');
outside_cell(2, 1, '1');
outside_cell(2, 2, '2');
outside_cell(2, 3, '0');
outside_cell(2, 4, '2');
outside_cell(2, 5, '0');
outside_cell(2, 6, '2');
outside_cell(2, 7, '0');
outside_cell(2, 8, '0');
outside_cell(3, -1, '1');
outside_cell(3, 1, '3');
outside_cell(3, 3, '3');
outside_cell(3, 5, '3');
outside_cell(3, 7, '2');
outside_cell(3, 8, '0');
outside_cell(4, -1, '1');
outside_cell(4, 8, '1');
outside_cell(5, -1, '0');
outside_cell(5, 0, '1');
outside_cell(5, 1, '2');
outside_cell(5, 3, '3');
outside_cell(5, 5, '3');
outside_cell(5, 7, '2');
outside_cell(5, 8, '0');
outside_cell(6, -1, '0');
outside_cell(6, 0, '0');
outside_cell(6, 1, '0');
outside_cell(6, 2, '1');
outside_cell(6, 3, '0');
outside_cell(6, 4, '1');
outside_cell(6, 5, '0');
outside_cell(6, 6, '1');
outside_cell(6, 7, '0');
outside_cell(6, 8, '0');
draw_side_u(0, 7);
draw_side_l(0, 7);
draw_side_r(0, 7);
draw_side_l(4, 1);
draw_side_r(4, 1);
draw_side_u(3, 2);
draw_side_d(3, 2);
draw_side_l(3, 2);
draw_side_u(5, 2);
draw_side_d(5, 2);
draw_side_r(5, 2);
draw_side_l(4, 3);
draw_side_r(4, 3);
draw_side_u(3, 4);
draw_side_d(3, 4);
draw_side_l(3, 4);
draw_side_u(5, 4);
draw_side_d(5, 4);
draw_side_r(5, 4);
draw_side_l(4, 5);
draw_side_r(4, 5);
draw_side_u(3, 6);
draw_side_d(3, 6);
draw_side_l(3, 6);
draw_side_u(5, 6);
draw_side_d(5, 6);
draw_side_r(5, 6);
draw_side_u(4, 7);
draw_side_l(4, 7);
draw_side_r(4, 7);
draw_side_d(0, 0);
draw_side_l(0, 0);
draw_side_u(1, 0);
draw_side_d(1, 0);
draw_side_u(2, 0);
draw_side_d(2, 0);
draw_side_u(3, 0);
draw_side_d(3, 0);
draw_side_d(4, 0);
draw_side_r(4, 0);
draw_side_l(0, 1);
draw_side_r(0, 1);
draw_side_u(-1, 2);
draw_side_d(-1, 2);
draw_side_l(-1, 2);
draw_side_u(1, 2);
draw_side_d(1, 2);
draw_side_r(1, 2);
draw_side_l(0, 3);
draw_side_r(0, 3);
draw_side_u(-1, 4);
draw_side_d(-1, 4);
draw_side_l(-1, 4);
draw_side_u(1, 4);
draw_side_d(1, 4);
draw_side_r(1, 4);
draw_side_l(0, 5);
draw_side_r(0, 5);
draw_side_u(-1, 6);
draw_side_d(-1, 6);
draw_side_l(-1, 6);
draw_side_u(1, 6);
draw_side_d(1, 6);
draw_side_r(1, 6);
[/asy]](http://latex.artofproblemsolving.com/9/f/4/9f450a3122eccac959e3bb1726874ceb84960782.png)
.
is the number of (good, bad) neighbor pairs, we get 
.
.
