2

Part of 1980 Vietnam National Olympiad

Problems(2)

Sum (m_i+1/m_i)^2>=k(k/S+S/k) where Sum m_i=S.

Source: Vietnam MO 1980 P2

m_1, m_2, \cdots ,m_k

be positive numbers with the sum

S

. Prove that \displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2

functioninequalitiesinequalities proposed

Can x^3-2x^2-2x+m = 0 have three different rational roots?

Source: Vietnam MO 1980 P5

Can the equation

x^3-2x^2-2x+m = 0

have three different rational roots?

algebrapolynomialcalculusintegrationalgebra unsolved