MathDB

3

Part of 2008 Alexandru Myller

Problems(4)

sum of proportions of areas less than one in a convex pentagon

Source:

1/31/2024
For a convex pentagon, prove that area(ABC)area(ABCD)+area(CDE)area(BCDE)<1. \frac{\text{area} (ABC)}{\text{area} (ABCD)} +\frac{\text{area} (CDE)}{\text{area} (BCDE)} <1.
Dan Ismailescu
geometryarea
beautiful discrete-synthetic geometry with elimination of higher cases

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1/31/2024
Describe all convex, inscriptible polygons which have the property that however we choose three distinct vertexes of of one of them, those vertexes form an isosceles triangle.
Gheorghe Iurea
geometry
quick integer part squeeze theorem

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2/1/2024
Let be a β>1. \beta >1. Calculate lim⁡n→∞k(n)n, \lim_{n\to\infty} \frac{k(n)}{n} , where k(n) k(n) is the smallest natural number that satisfies the inequality (1+n)k≥nkβ. (1+n)^k\ge n^k\beta .
Neculai Hârţan
limitsqueeze theoremSequences
find all functions that satisfy an integral bounded by 2008

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2/2/2024
Find the nondecreasing functions f:[0,1]→R f:[0,1]\rightarrow\mathbb{R} that satisfy ∣∫01f(x)enxdx∣≤2008, \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , for any nonnegative integer n. n.
Mihai Piticari
functionintegralsreal analysisinequalitiescalculusintegration