MathDB

1

Part of 2018 Tuymaada Olympiad

Problems(2)

Polynomial divisible by x^2+1

Source: Tuymaada 2018 Junior League/Problem 1

7/20/2018
Real numbers a≠0,b,ca \neq 0, b, c are given. Prove that there is a polynomial P(x)P(x) with real coefficients such that the polynomial x2+1x^2+1 divides the polynomial aP(x)2+bP(x)+caP(x)^2+bP(x)+c.
Proposed by A. Golovanov
algebrapolynomial
Roots of quadratic trinomials

Source: Tuymaada 2018 Senior League/Problem 1

7/20/2018
Do there exist three different quadratic trinomials f(x),g(x),h(x)f(x), g(x), h(x) such that the roots of the equation f(x)=g(x)f(x)=g(x) are 11 and 44, the roots of the equation g(x)=h(x)g(x)=h(x) are 22 and 55, and the roots of the equation h(x)=f(x)h(x)=f(x) are 33 and 66?
Proposed by A. Golovanov
quadraticsTuymaadaalgebra