[/quote]
,
find the number of ordered real triples 
such that![\[\begin{aligned}
\begin{cases}
x+y^2+z^2=a,\\
x^2+y+z^2=a,\\
x^2+y^2+z=a.
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/9/8/6/986593653bf128bf7b3d497abed78e1459134c5f.png)
numbers, none of which appears more than 
times. Prove that they can be divided into 
groups of 
numbers each in such a way that the sums of the numbers in these groups are pairwise distinct.
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
such that ![\[ \dfrac{a^2}{2ab^2-b^3+1} \]](http://latex.artofproblemsolving.com/0/d/2/0d2f10786d22af2ea94ebeed6b15e723f4042734.png)
is a positive integer.
?
be a triangle and let 
and 
be variable points on sides 
and 
,
.
denote the point on 
such that 
is parallel to 
,
denote the point on 
such that 
is parallel to 
.
vary, the circumcircle of 
passes through a fixed point, other than 
.
satisfy that 
.
![\[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\]](http://latex.artofproblemsolving.com/7/9/e/79e388eba20412905964dc2b4ebb491f3c79143b.png)
and the sums ![\[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\]](http://latex.artofproblemsolving.com/3/c/4/3c44766edcc9e5269f0d3396386e41676cbdf42c.png)
For any pair define 
and 
,
.
,
such that the sequences 
and 
are unbounded while the sequence 
is bounded?
,
?
,
.
lies on 
such that 
.
passing trough 
at 
.
.
be a positive integer. Find, in terms of 
with integer coefficients such that for every integer 
such that 
.
.
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
