MathDB

2

Part of 2012 Bogdan Stan

Problems(4)

Sets of multiples

Source:

11/25/2019
For any a∈Z≥0 a\in\mathbb{Z}_{\ge 0} make the notation aZ≥0={an∣n∈Z≥0}. a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . Prove that the following relations are equivalent:
(1)aZ≥0∖bZ≥0⊂cZ≥0∖dZ≥0 \text{(1)} a\mathbb{Z}_{\ge 0} \setminus b\mathbb{Z}_{\ge 0}\subset c\mathbb{Z}_{\ge 0} \setminus d\mathbb{Z}_{\ge 0} (2)b∣a or (c∣a and lcm(a,b)∣lcm(a,d)) \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d))
Marin Tolosi and Cosmin Nitu
GCDLCMDivisibilitynumber theory
General complex inequality

Source: G.M.

11/25/2019
Prove the complex inequality ∣x∣+∣y∣+∣z∣≤∣x+y+z∣+∣x−z∣+∣z−y∣+∣y−z∣. |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|.
inequalitiesalgebracomplex numbers
Convergence of a not consecutive recurrence

Source:

11/25/2019
Let be a bounded sequence of positive real numbers (xn)n≥1 \left( x_n \right)_{n\ge 1} satisfying the recurrence: xn+3=3xn−23. x_{n+3} =\sqrt[3]{3x_n-2} . Prove that (xn)n≥1 \left( x_n \right)_{n\ge 1} is convergent.
Cristinel Mortici
real analyssiSequences
Integral functional relation

Source:

11/25/2019
Find the continuous functions f:[0,13]⟶(0,∞) f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) that satisfy the functional relation 54∫01/3f(x)dx+32∫01/3dxx+f(x)=21. 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21.
Cristinel Mortici
functionFind all functionscalculusintegration