MathDB

4

Part of 2000 District Olympiad (Hunedoara)

Problems(3)

pyramid problem

Source: Romanian District Olympiad 2000, Grade X, Problem 4

9/24/2018
Consider the pyramid VABCD, VABCD, where V V is the top and ABCD ABCD is a rectangular base. If ∠BVD=∠AVC, \angle BVD = \angle AVC, then prove that the triangles VAC VAC and VBD VBD share the same perimeter and area.
geometry3D geometrypyramidperimeterarea
vectorial geometry

Source: Romanian District Olympiad, Grade IX, Problem 4

9/24/2018
Let be a circle centeted at O, O, and A,B,C, A,B,C, points situated on this circle. Show that if ∣OA→+OB→∣=∣OB→+OC→∣=∣OC→+OA→∣, \left|\overrightarrow{OA} +\overrightarrow{OB}\right| = \left|\overrightarrow{OB} +\overrightarrow{OC}\right| = \left|\overrightarrow{OC} +\overrightarrow{OA}\right| , then A=B=C, A=B=C, or ABC ABC is an equilateral triangle.
geometrycontestsOlympiad
calculate limit using integral

Source: Romanian District Olympiad 2000, Grade XII, Problem 4

9/25/2018
Let f:[0,1]⟶R+∗ f:[0,1]\longrightarrow\mathbb{R}_+^* be a Riemann-integrable function. Calculate lim⁡n→∞(−n+∑i=1ne1n⋅f(in)). \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) .
functionslimitsreal analysiscontestsromaniacalculusintegration