MathDB

71

Part of 1992 IMO Longlists

Problems(1)

There exist integers A1,A2 - IMO LongList 1992 THA2

Source:

9/2/2010
Let P1(x,y)P_1(x, y) and P2(x,y)P_2(x, y) be two relatively prime polynomials with complex coefficients. Let Q(x,y)Q(x, y) and R(x,y)R(x, y) be polynomials with complex coefficients and each of degree not exceeding dd. Prove that there exist two integers A1,A2A_1, A_2 not simultaneously zero with ∣Ai∣≤d+1 (i=1,2)|A_i| \leq d + 1 \ (i = 1, 2) and such that the polynomial A1P1(x,y)+A2P2(x,y)A_1P_1(x, y) + A_2P_2(x, y) is coprime to Q(x,y)Q(x, y) and R(x,y).R(x, y).
algebrapolynomialnumber theoryrelatively primefunctional equationIMO ShortlistIMO Longlist