3

Part of 2008 India Regional Mathematical Olympiad

Problems(2)

inequality on coefficients of cubic

Source: (Indian) RMO 2008 Problem 3

a

b

are real numbers such that the roots of the cubic equation ax^3\minus{}x^2\plus{}bx\minus{}1 are positive real numbers. Prove that:

(i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3}

[19 points out of 100 for the 6 problems]

inequalitiesVietacalculusderivativetrigonometryLaTeXalgebra

Source: Excursion

Prove that for every positive integer

n

and a non-negative real number

a

, the following inequality holds:

n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).

inequalitiesInequalitypositive integersreal number