AoPS Academy
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)
such that: 
be an odd prime number. Prove that there exist sequence such that: 
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.
+1 w
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 
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 
, how many orderly colorings are there?
is selected inside acute 
so that 
and 
. Point 
is chosen on ray 
so that 
. Let 
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 amber cells and bronze cells such that no two of the 
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.
-by- grid. He colors 
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 equal equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of 
?
-by- board of unit squares for some odd positive integer . We say that a collection 
of identical dominoes is a maximal grid-aligned configuration on the board if consists of 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let 
be the number of distinct maximal grid-aligned configurations obtainable from by repeatedly sliding dominoes. Find the maximum value of as a function of .
![$\mathcal{M}=\mathbb{Q}[x,y,z]$](http://latex.artofproblemsolving.com/5/a/c/5ac5be746053d9ceabb9223751de427fa1006bc3.png)
be the set of three-variable polynomials with rational coefficients. Prove that for any non-zero polynomial 
there exists non-zero polynomials 
such that![\[ R(x^2y,y^2z,z^2x) = P(x,y,z)Q(x,y,z). \]](http://latex.artofproblemsolving.com/1/8/a/18a597b189c198ad261fcd88f139e7cc7a2796a2.png)
denote the polynomials of the form 
![$x^iy^kz^j \in \mathbb{Q}[x^2y,y^2z,z^2x]$](http://latex.artofproblemsolving.com/6/7/5/67541da99a38e330dc744e29f6c4618c1993f0b7.png)
iff 
the monomials for whom the degree of 
is 
, for sake of convenience denote by 
, then choose 
for this particular choice of 
we have, 
hence having the degree of divisible by 
, so ![$P \cdot Q \equiv \mathbb{Q}[x^3,y,z]$](http://latex.artofproblemsolving.com/0/8/b/08b7c5dc585aee1aec55e7ff7ec22064838a290d.png)
now keep interchanging variables to get to ![$\mathbb{Q}[x^N,y^N,z^N]$](http://latex.artofproblemsolving.com/e/5/8/e582c8ef30c4a7a64244a7072a368817f63d035c.png)
for a large 
which has been shown to be reachable by Claim 1.
.
can be written as 
,
.
so that 
,
,
.
,
,
.
then the condition of the last claim holds. For this, we use the following variation of the problem.![\[ R(x^3,y,z) = P(x,y,z)Q(x,y,z). \]](http://latex.artofproblemsolving.com/0/5/5/05552e11eba5d90420d6250f54e4efc4a4e48e28.png)
and 
, let 
.
and set 
then 
.
or three from different, both of which have 
has 
.
,
and 
,
too and it works.
,
,
,
,
.
;
,
for a big enough 
be the sum of monomials 
of 
that have 
.
form a subset 
and say that ![$\mathbb{S}[x,y,z]$](http://latex.artofproblemsolving.com/8/c/2/8c2959657cf1e9eba6de6cd44c16c9e6589c2cbb.png)
are the polynomials, all of which monomials are in ![$\mathbb{S}[x,y,z] = \mathcal{S}$](http://latex.artofproblemsolving.com/f/5/1/f51b0cebe52c21336f36afa96a3b593e6ce1b4f5.png)
.
so that 
.
.![\[ \forall 1 \leq r \leq 8, \sum_{i+j \equiv r \pmod{9}}^{} P_i(x,y,z)Q_j(x,y,z) = 0 \]](http://latex.artofproblemsolving.com/d/c/9/dc978c51ee3e9b106b4a30100b9587ced4fb359a.png)
Which is the same as saying that ![\[ \forall 1 \leq r \leq 8, \sum_{i+j \equiv r \pmod{9}}^{} P_i(x,y,z)z^kQ_j(x,y,z)z^s= 0 \]](http://latex.artofproblemsolving.com/7/d/f/7df807e56ca264616e0a4f41a98a36731632b49c.png)
where 
,
and 
.
is fixed and the equation holds. Note that k can be negative.
where each and 
are in 
.
for large enough 
is in 
in powers less than 9, we can ensure that we get polynomials by multiplying the solution for 
.
and the iff condition follows, for the other direction we get a system of equations which clearly have solutions. (2022 C4 anyone?)
can be written in that way, then 
can also be written went hard, next the only way left to proceed is to find a characterization of monomials that can be represented in the said form, which leads to a solution!