MathDB

4

Part of 2015 Belarus Team Selection Test

Problems(2)

(a+b+c)^5 >= 81 (a^2+b^2+c^2)abc for a,b,c>0

Source: 2015 Belarus TST 1.4

11/5/2020
Prove that (a+b+c)5≥81(a2+b2+c2)abc(a+b+c)^5 \ge 81 (a^2+b^2+c^2)abc for any positive real numbers a,b,ca,b,c
I.Gorodnin
algebrainequalities
p(x^2)=p(x)q(1-x)+p(1-x)q(x)

Source: 2015 Belarus TST 2.4

11/5/2020
Find all pairs of polynomials p(x),q(x)∈R[x]p(x),q(x)\in R[x] satisfying the equality p(x2)=p(x)q(1−x)+p(1−x)q(x)p(x^2)=p(x)q(1-x)+p(1-x)q(x) for all real xx.
I.Voronovich
polynomialalgebra