2023 Silk Road

Part of Silk Road

Subcontests

(4)

1

polynomials of three variables with rational coefficients

\mathcal{M}=\mathbb{Q}[x,y,z]

be the set of three-variable polynomials with rational coefficients. Prove that for any non-zero polynomial

P\in \mathcal{M}

there exists non-zero polynomials

Q,R\in \mathcal{M}

R(x^2y,y^2z,z^2x) = P(x,y,z)Q(x,y,z).

1

directed graph labeled with residues

p

be a prime number. We construct a directed graph of

p

vertices, labeled with integers from

0

p-1

. There is an edge from vertex

x

y

x^2+1\equiv y \pmod{p}

f(p)

denotes the length of the longest directed cycle in this graph. Prove that

f(p)

can attain arbitrarily large values.

1

colorful dominoes on a painted square

n

be a positive integer. Each cell of a

2n\times 2n

square is painted in one of the

4n^2

colors (with some colors may be missing). We will call any two-cell rectangle a domino[/I], and a domino is called colorful[/I] if its cells have different colors. Let

k

be the total number of colorful dominoes in our square;

l

be the maximum integer such that every partition of the square into dominoes contains at least

l

colorful dominoes. Determine the maximum possible value of

4l-k

over all possible colourings of the square.

1

triangles with equal areas

ABCD

be a trapezoid with

AD\parallel BC

M

is chosen inside the trapezoid, and a point

N

is chosen inside the triangle

BMC

AM\parallel CN

BM\parallel DN

. Prove that triangles

ABN

CDM

have equal areas.