MathDB

4

Part of 2021 European Mathematical Cup

Problems(2)

10th EMC - Colouring the set 1, 2, ..., n

Source: 10th European Mathematical Cup - Problem J4

12/22/2021
Let nn be a positive integer. Morgane has coloured the integers 1,2,…,n1,2,\ldots,n. Each of them is coloured in exactly one colour. It turned out that for all positive integers aa and bb such that a<ba<b and a+b⩽na+b \leqslant n, at least two of the integers among aa, bb and a+ba+b are of the same colour. Prove that there exists a colour that has been used for at least 2n/52n/5 integers. \\ \\ (Vincent Jugé)
emcEuropean Mathematical CupcombinatoricspartitionsColoringRamsey Theory
P(x)^2+1=(x^2+1)Q(x^2), der(P)=?

Source: 10th European Mathematical Cup - Problem S4

12/22/2021
Find all positive integers dd for which there exist polynomials P(x)P(x) and Q(x)Q(x) with real coefficients such that degree of PP equals dd and P(x)2+1=(x2+1)Q(x)2.P(x)^2+1=(x^2+1)Q(x)^2.
polynomialalgebraEuropean Mathematical Cup