MathDB

1

Part of 2024 Serbia Team Selection Test

Problems(2)

Classical invariant combo

Source: Serbia IMO TST 2024, P1

5/18/2024
Three coins are placed at the origin of a Cartesian coordinate system. On one move one removes a coin placed at some position (x,y)(x, y) and places three new coins at (x+1,y)(x+1, y), (x,y+1)(x, y+1) and (x+1,y+1)(x+1, y+1). Prove that after finitely many moves, there will exist two coins placed at the same point.
combinatorics
Product of a polynomial and its inverse is x^{2024}+1

Source: Serbia Additional IMO TST 2024, P1 (out of 4)

5/30/2024
Does there exist a positive integer nn and a) complex numbers a0,a1,…,an;a_0, a_1, \ldots, a_n; b) reals a0,a1,…,an,a_0, a_1, \ldots, a_n, such that P(x)Q(x)=x2024+1P(x) Q(x)=x^{2024}+1 where P(x)=anxn+…+a1x+a0P(x)=a_nx^n+\ldots +a_1x+a_0 and Q(x)=a0xn+a1xn−1+…+an?Q(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n?
algebra