such that
be a positive integer. Find, in terms of 
with integer coefficients such that for every integer 
,
such that 
.
be an integer. Rowan and Colin play a game on an 
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of 
is selected inside acute 
so that 
and 
.
is chosen on ray 
so that 
.
be the midpoint of 
.
is tangent to the circumcircle of triangle 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
and 
be positive integers. The cells of an 
grid are colored amber and bronze such that there are at least 
amber cells and at least 
bronze cells. Prove that it is possible to choose 
chosen cells lie in the same row or column.
and 
are erected outside an acute triangle 
Suppose that ![\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]](http://latex.artofproblemsolving.com/3/3/7/3378f0c4409d14bc0afddf2616992c13f624be3b.png)
Prove that lines 
and 
are concurrent.
-
of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let 
and 
?
of identical dominoes is a maximal grid-aligned configuration on the board if 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: 
be the number of distinct maximal grid-aligned configurations obtainable from 
Y by
Let 
be a positive integer. Each cell of a 
square is painted in one of the 
colors (with some colors may be missing). We will call any two-cell rectangle a domino, and a domino is called colorful if its cells have different colors. Let 
be the total number of colorful dominoes in our square; 
be the maximum integer such that every partition of the square into dominoes contains at least colorful dominoes. Determine the maximum possible value of 
over all possible colourings of the square.
be a positive integer. Each cell of a 
square is painted in one of the 
colors (with some colors may be missing). We will call any two-cell rectangle a domino, and a domino is called colorful if its cells have different colors. Let 
be the total number of colorful dominoes in our square; 
be the maximum integer such that every partition of the square into dominoes contains at least colorful dominoes. Determine the maximum possible value of 
over all possible colourings of the square.
for illustration).![[asy]
size(400, 200);
pen gridPen = gray + linewidth(1); // Thicker gridlines
pen borderPen = black + linewidth(1); // Black border for the dominoes
pen dominoPen1 = lightgray;
pen dominoPen2 = lightblue;
pen dominoPen3 = lightgreen;
pen dominoPen4 = lightyellow;
// Function to draw an empty 8x8 grid at position (xShift, yShift)
void drawGrid(real xShift, real yShift) {
for (int i = 0; i <= 8; ++i) {
draw((i + xShift, 0 + yShift) -- (i + xShift, 8 + yShift), gridPen); // Vertical lines
draw((0 + xShift, i + yShift) -- (8 + xShift, i + yShift), gridPen); // Horizontal lines
}
}
// First square: Vertically placed dominoes
void verticalDominoes(real xShift, real yShift, pen fillColor) {
drawGrid(xShift, yShift); // Draw grid first, so it's in the background
for (int i = 0; i < 8; ++i) {
for (int j = 0; j < 8; j += 2) {
filldraw((i + xShift, j + yShift) -- (i + 1 + xShift, j + yShift) -- (i + 1 + xShift, j + 2 + yShift) -- (i + xShift, j + 2 + yShift) -- cycle, fillColor, borderPen);
}
}
}
// Second square: Horizontally placed dominoes
void horizontalDominoes(real xShift, real yShift, pen fillColor) {
drawGrid(xShift, yShift); // Draw grid first, so it's in the background
for (int i = 0; i < 8; i += 2) {
for (int j = 0; j < 8; ++j) {
filldraw((i + xShift, j + yShift) -- (i + 2 + xShift, j + yShift) -- (i + 2 + xShift, j + 1 + yShift) -- (i + xShift, j + 1 + yShift) -- cycle, fillColor, borderPen);
}
}
}
// Third square: Vertical dominoes in the first and last columns, horizontal dominoes in the inner columns
void shiftedVerticalAndHorizontalDominoes(real xShift, real yShift, pen fillColor) {
drawGrid(xShift, yShift); // Draw grid first, so it's in the background
// Fill the first and last columns with vertical dominoes
for (int j = 0; j < 8; j += 2) {
filldraw((0 + xShift, j + yShift) -- (1 + xShift, j + yShift) -- (1 + xShift, j + 2 + yShift) -- (0 + xShift, j + 2 + yShift) -- cycle, fillColor, borderPen); // First column
filldraw((7 + xShift, j + yShift) -- (8 + xShift, j + yShift) -- (8 + xShift, j + 2 + yShift) -- (7 + xShift, j + 2 + yShift) -- cycle, fillColor, borderPen); // Last column
}
// Fill the inner columns with horizontal dominoes
for (int i = 1; i < 7; i += 2) {
for (int j = 0; j < 8; ++j) {
filldraw((i + xShift, j + yShift) -- (i + 2 + xShift, j + yShift) -- (i + 2 + xShift, j + 1 + yShift) -- (i + xShift, j + 1 + yShift) -- cycle, fillColor, borderPen);
}
}
}
// Fourth square: Horizontal dominoes in the top and bottom rows, vertical dominoes in the inner rows
void shiftedHorizontalAndVerticalDominoes(real xShift, real yShift, pen fillColor) {
drawGrid(xShift, yShift); // Draw grid first, so it's in the background
// Fill the top and bottom rows with horizontal dominoes
for (int i = 0; i < 8; i += 2) {
filldraw((i + xShift, 0 + yShift) -- (i + 2 + xShift, 0 + yShift) -- (i + 2 + xShift, 1 + yShift) -- (i + xShift, 1 + yShift) -- cycle, fillColor, borderPen); // Top row
filldraw((i + xShift, 7 + yShift) -- (i + 2 + xShift, 7 + yShift) -- (i + 2 + xShift, 8 + yShift) -- (i + xShift, 8 + yShift) -- cycle, fillColor, borderPen); // Bottom row
}
// Fill the inner rows with vertical dominoes
for (int i = 0; i < 8; ++i) {
for (int j = 1; j < 7; j += 2) {
filldraw((i + xShift, j + yShift) -- (i + 1 + xShift, j + yShift) -- (i + 1 + xShift, j + 2 + yShift) -- (i + xShift, j + 2 + yShift) -- cycle, fillColor, borderPen);
}
}
}
// Draw the four grids with different colors and black borders
verticalDominoes(0, 0, dominoPen1); // First grid with lightgray dominoes (vertical)
horizontalDominoes(10, 0, dominoPen2); // Second grid with lightblue dominoes (horizontal)
shiftedVerticalAndHorizontalDominoes(0, -10, dominoPen3); // Third grid with mixed directions
shiftedHorizontalAndVerticalDominoes(10, -10, dominoPen4); // Fourth grid with mixed directions
[/asy]](http://latex.artofproblemsolving.com/3/a/2/3a2ef873cac73bafc19221470ab7e04dea14e4b2.png)
colorful dominoes in these partitions. However these partitions must contain at least 
colorful dominoes so 
?
I fixed my mistake
