MathDB

3

Part of 2000 District Olympiad (Hunedoara)

Problems(3)

find a particular set of points on a plane

Source: Romanian District Olympiad, Grade X, Problem 3

9/24/2018
Let α \alpha be a plane and let ABC ABC be an equilateral triangle situated on a parallel plane whose distance from α \alpha is h. h. Find the locus of the points M∈α M\in\alpha for which ∣MA∣2+h2=∣MB∣2+∣MC∣2. \left|MA\right| ^2 +h^2 = \left|MB\right|^2 +\left|MC\right|^2.
geometrycontestsOlympiadLocus problems
limit of an interesting sequence

Source: Romanian District Olympiad 2000, Grade XI, Problem 3

9/24/2018
Let be two distinct natural numbers k1 k_1 and k2 k_2 and a sequence (xn)n≥0 \left( x_n \right)_{n\ge 0} which satisfies x_nx_m +k_1k_2\le k_1x_n +k_2x_m, \forall m,n\in\{ 0\}\cup\mathbb{N}.
Calculate lim⁡n→∞n!⋅(−1)1+n⋅xn2nn. \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} .
limitsSequencescalculusreal analysiscontestsromania
primitivableness on contest

Source: Romanian District Olympiad 2000, Grade XII, Problem 3

9/25/2018
Let be a function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} such that: \text{(i)}  f(0)=0 \text{(ii)}  f'(x)\neq 0, \forall x\in\mathbb{R} \text{(iii)}  \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous}
Demonstrate that the function g:R⟶R g:\mathbb{R}\longrightarrow\mathbb{R} defined as g(x)={cos⁡1f(x),emsp;x≠00,emsp;x=0 g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},  x\neq 0\\ 0,  x=0\end{matrix}\right. is primitivable.
functionwilkoszprimitivesreal analysiscontestsromania