MathDB

1

Part of 2011 Czech-Polish-Slovak Match

Problems(2)

Ineq: a^2 < bc implies b^3 + ac^2 > ab(a+c)

Source: Czech-Polish-Slovak Match, 2011

8/9/2011
Let aa, bb, cc be positive real numbers satisfying a2<bca^2<bc. Prove that b3+ac2>ab(a+c)b^3+ac^2>ab(a+c).
inequalitiesinequalities unsolved
Factoring a composition of two polynomials

Source: Czech-Polish-Slovak Match, 2011

8/9/2011
A polynomial P(x)P(x) with integer coefficients satisfies the following: if F(x)F(x), G(x)G(x), and Q(x)Q(x) are polynomials with integer coefficients satisfying P(Q(x))=F(x)ā‹…G(x)P\Big(Q(x)\Big)=F(x)\cdot G(x), then F(x)F(x) or G(x)G(x) is a constant polynomial. Prove that P(x)P(x) is a constant polynomial.
algebrapolynomialalgebra unsolved