MathDB

3

Part of 2011 Bogdan Stan

Problems(4)

Inequality of six variables (easy)

Source: Bogdan Stan 2011

10/7/2019
Prove that a+b+c>(α+β+γ)2, a+b+c>\left( \sqrt\alpha +\sqrt\beta +\sqrt\gamma \right)^2, for all positive real numbers a,b,c,α,β,γ a,b,c,\alpha ,\beta ,\gamma that are under the condition abc>αbc+βac+γab. abc>\alpha bc+\beta ac+\gamma ab.
Țuțescu Lucian and Chiriță Aurel
inequalitiescauchy schwarz
Easy equation

Source: Bogdan Stan 2011, based on GM nr.7,8,9(2010)

10/7/2019
Solve in R \mathbb{R} the equation 4x2−x=log⁡2x+x−1+14. 4^{x^2-x}=\log_2 x+\sqrt{x-1} +14.
Marin Tolosi
unique solutionfunctionalgebra
Limit of certain sequence (2011 is not an essential number..)

Source: Bogdan Stan 2011

10/7/2019
Let be a sequence of real numbers (xn)n≥1 \left( x_n \right)_{n\ge 1} chosen such that the limit of the sequence (xn+2011−xn)n≥1 \left( x_{n+2011}-x_n \right)_{n\ge 1} exists. Calculate lim⁡n→∞xnn. \lim_{n\to\infty } \frac{x_n}{n} .
Cosmin Nițu
real analysisSequenceslimit of sequencslimits
integral fog=integral gof

Source: Bogdan Stan 2011

10/7/2019
Find all Riemann integrable functions f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} which have the property that, for all nonconstant and continuous functions g:R⟶R, g:\mathbb{R}\longrightarrow\mathbb{R}, and all real numbers a,b a,b such that a<b, a<b, the following equality holds. ∫ab(f∘g)(x)dx=∫ab(g∘f)(x)dx \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx
Cosmin Nițu
functionintegrabilityriemann integralcalculusintegration