2

Part of 2018 Middle European Mathematical Olympiad

Problems(2)

A polynomial with rational coefficients

Source: MEMO 2018 T2

P(x)

be a polynomial of degree

n\geq 2

with rational coefficients such that

P(x)

n

pairwise different reel roots forming an arithmetic progression .Prove that among the roots of

P(x)

there are two that are also the roots of some polynomial of degree

2

with rational coefficients .

algebrapolynomialroots

The rest of the board cannot be cut into staircases

Source: MEMO 2018 I2

The two figures depicted below consisting of

6

10

unit squares, respectively, are called staircases. Consider a

2018\times 2018

board consisting of

2018^2

cells, each being a unit square. Two arbitrary cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated).

memocombinatoricsinvariantColoring