MathDB

3

Part of 2012 Bogdan Stan

Problems(4)

Find particular solutions to this equation

Source:

11/25/2019
Find the real numbers x,y,z x,y,z that satisfy the following:
(i)−2≤x≤y≤z \text{(i)} -2\le x\le y\le z (ii)x+y+z=2/3 \text{(ii)} x+y+z=2/3 (iii)1x2+1y2+1z2=1x+1y+1z+38 \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8}
Cristinel Mortici
algebraequations
Geometric progression of length 2011

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11/25/2019
Consider 2011 2011 positive real numbers a1,a2,…,a2011. a_1,a_2,\ldots ,a_{2011} . If they are in geometric progression, show that there exists a real number λ \lambda such that any i∈{1,2,…,1005} i\in\{ 1,2,\ldots , 1005 \} implies λ=ai⋅a2012−i. \lambda =a_i\cdot a_{2012-i} . Disprove the converse.
Teodor Radu
geometric sequencealgebra
Beautiful limit, in my opinion: floor of logarithm of sum

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11/25/2019
lim⁡n→∞1n!n⌊log⁡5∑k=21+5nk5n⌋ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor
Taclit Daniela Nadia
limitsFloorlimits of sequencesinequalitieslogarithmsBernoulli s Inequality
Characterization of f. that satisfy kind of FTC on themselves

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11/25/2019
Find all functions f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that verify the equality ∫abf(x)dx=f(b)−f(a), \int_a^b f(x)dx=f(b)-f(a), for any real numbers a,b. a,b.
Cosmin Nitu
functionFind all functionsreal analyssi