MathDB

5

Part of 2010 IMC

Problems(2)

IMC 2010 - Problem 5

Source:

7/26/2010
Suppose that a,b,ca,b,c are real numbers in the interval [−1,1][-1,1] such that 1+2abc≥a2+b2+c21 + 2abc \geq a^2+b^2+c^2. Prove that 1+2(abc)n≥a2n+b2n+c2n1+2(abc)^n \geq a^{2n} + b^{2n} + c^{2n} for all positive integers nn.
inequalities unsolvedinequalities
IMC 2010, Problem 5, Day 2

Source:

7/27/2010
Suppose that for a function f:R→Rf: \mathbb{R}\to \mathbb{R} and real numbers a<ba<b one has f(x)=0f(x)=0 for all x∈(a,b).x\in (a,b). Prove that f(x)=0f(x)=0 for all x∈Rx\in \mathbb{R} if ∑k=0p−1f(y+kp)=0\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0 for every prime number pp and every real number y.y.
functionalgebrapolynomialvectoranalytic geometryreal analysisnumber theory