MathDB

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Part of 2010 Laurențiu Panaitopol, Tulcea

Problems(4)

Parametric quadratic equation

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10/28/2019
Find the real numbers m m which have the property that the equation x2−2mx+2m2=25 x^2-2mx+2m^2=25 has two integer solutions.
quadraticsalgebraequations
Equation: arcsin(x)=[2x]

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10/28/2019
Solve in the real numbers the equation arcsin⁡x=⌊2x⌋. \arcsin x=\lfloor 2x \rfloor .
Petre Guțescu
floor functionequationsTrigonometric EquationsFlooralgebramonotone functions
Limit of stacked cubic roots

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10/28/2019
Show that if (sn)n≥0 \left( s_n \right)_{n\ge 0} is a sequence that tends to 6, 6, then, the sequence (sn+sn−1+sn−2+⋯+s033333)n≥0 \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} tends to 2. 2.
Mihai Bălună
Sequenceslimitsreal analysisWeierstrassNested Radicals
Relation between primitivableness and integrableness for certain functions

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10/28/2019
Let be two real numbers a<b a<b and a function f:[a,b]⟶R f:[a,b]\longrightarrow\mathbb{R} having the property that if the sequence (f(xn))n≥1 \left(f\left( x_n \right)\right)_{n\ge 1} is convergent, then the sequence (xn)n≥1 \left( x_n \right)_{n\ge 1} is convergent.
a) Prove that if f f admits antiderivatives, then f f is integrable. b) Is the converse of a) true?
Marcelina Popa
functionprimitivablenessantiderivativedifferentiabilityIntegralIndefinite integralreal analysis