MathDB

3

Part of 2012 Centers of Excellency of Suceava

Problems(4)

R,Area,r in arithmetic progression

Source:

10/31/2019
Prove that the sum of the squares of the medians of a triangle is at least 9/4 9/4 if the circumradius of the triangle, the area of the triangle and the inradius of the triangle (in this order) are in arithmetic progression.
Dumitru Crăciun
geometrycircumcircleinradiusarithmetic sequence
Algebra....

Source:

10/31/2019
Let a,b,n a,b,n be three natural numbers. Prove that there exists a natural number c c satisfying: (a+b)n=c+(a−b)n+c \left( \sqrt{a} +\sqrt{b} \right)^n =\sqrt{ c+(a-b)^n} +\sqrt{c}
Dan Popescu
Newton s binomalgebra
A condition for boundedness

Source:

10/31/2019
Let be a continuous function f:R≥0⟶R f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} that has a root, and for which the line y=0 y=0 in the Cartesian plane is an horizontal asymptote. Show that f f is bounded and touches its boundaries.
Mihai Piticari and Vladimir Cerbu
functionreal analysis
Reciprocal of odd double factorial series

Source:

10/31/2019
Consider the sequence (In)n≥1, \left( I_n \right)_{n\ge 1} , where In=∫0π/4esin⁡xcos⁡x(cos⁡x−sin⁡x)2n(cos⁡x+sin⁡x)dx, I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, for any natural number n. n.
a) Find a relation between any two consecutive terms of In. I_n.
b) Calculate lim⁡n→∞nIn. \lim_{n\to\infty } nI_n.
c) Show that ∑i=1∞1(2i−1)!!=∫0π/4esin⁡xcos⁡x(cos⁡x+sin⁡x)dx. \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx.
Cătălin Țigăeru
factorialseriesDefinite integralreal analysisintegral sequence