MathDB

4

Part of 2012 Centers of Excellency of Suceava

Problems(4)

Logarithmic system

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10/31/2019
Solve in the reals the following system. {log⁡2∣x∣⋅log⁡2∣y∣=3/2x2+y2=12 \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right.
Gheorghe Marchitan
monotonyequationsCartesian circlealgebra
collinearity of three circumcenters and one incenter

Source:

10/31/2019
Let O O be the circumcenter of a triangle ABC ABC with ∠BAC=60∘ \angle BAC=60^{\circ } whose incenter is denoted by I. I. Let B1,C1 B_1,C_1 be the intersection of BI,CI BI,CI with the circumcircle of ABC, ABC, respectively. Denote by O1,O2 O_1,O_2 the circumcenters of BIC1,CIB1, BIC_1,CIB_1, respectively. Show that O1,I,O,O2 O_1,I,O,O_2 are collinear.
Cătălin Țigăeru
geometrycircumcircleinradiusincenter
Prove that f=ax+b

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10/31/2019
Let be two real numbers a<b a<b and a differentiable function f:[a,b]⟶R f:[a,b]\longrightarrow\mathbb{R} that has a bounded derivative. Show that if f(b)−f(a)b−a \frac{f(b)-f(a)}{b-a} is equal to the global supremum or infimum of f′, f', then f f is polynomial with degree 1. 1.
Cătălin Țigăeru
functioncalculusderivativereal analysislagrangeMVTConvexity
Another integral sequence (last)

Source:

10/31/2019
Let be the sequence (Jn)n≥1, \left( J_n \right)_{n\ge 1} , where Jn=∫(1+n)21+(1+n)2x−1−n−n2x−1dx. J_n=\int_{(1+n)^2}^{1+(1+n)^2} \sqrt{\frac{x-1-n-n^2}{x-1}} dx. a) Study its monotony. b) Calculate lim⁡n→∞Jnn. \lim_{n\to\infty } J_n\sqrt{n} .
Ion Bursuc
calculusintegrationreal analysissequence of integralsDefinite integral