MathDB

2

Part of 2008 District Olympiad

Problems(6)

EF = AE + FC, square related (2008 Romania District VII P2)

Source:

5/18/2020
Consider the square ABCDABCD and E∈(AB)E \in (AB). The diagonal ACAC intersects the segment [DE][DE] at point PP. The perpendicular taken from point PP on DEDE intersects the side BCBC at point FF. Prove that EF=AE+FCEF = AE + FC.
geometrysquareperpendicular
Irrational x

Source: Romanian DMO 8th grade problem 2

3/1/2008
Determine x x irrational so that x^2\plus{}2x and x^3\minus{}6x are both rational.
algebrapolynomialnumber theory proposednumber theory
Reunion of three sets with the same number of elements

Source: Romanian District MO 2008, Grade 9, Problem 2

4/30/2008
Let S\equal{}\{1,2,\ldots,n\} be a set, where n≥6 n\geq 6 is an integer. Prove that S S is the reunion of 3 pairwise disjoint subsets, with the same number of elements and the same sum of their elements, if and only if n n is a multiple of 3.
induction
Romania District Olympiad 2008 - Grade XI

Source:

4/10/2011
Let A,B∈Mn(R)A,B\in \mathcal{M}_n(\mathbb{R}). Prove that rank A+rank B≤n\text{rank}\ A+\text{rank}\ B\le n if and only if there exists an invertible matrix X∈Mn(R)X\in \mathcal{M}_n(\mathbb{R}) such that AXB=OnAXB=O_n.
linear algebramatrixlinear algebra unsolved
Equivalence with arctan

Source: RMO District Round, Bucharest 2008, Grade 10, Problem 2

1/27/2008
Consider the positive reals x x, y y and z z. Prove that: a) \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2} iff xy<1 xy < 1. b) \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi iff xyz < x \plus{} y \plus{} z.
functionalgebradomainalgebra proposed
Riemann sums to periodic primitives

Source: Romanian District Olympiad 2008, Grade XII, Problem 2

10/7/2018
Let f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} be a countinuous and periodic function, of period T. T. If F F is a primitive of f, f, show that:
a) the function G:R⟶R,G(x)=F(x)−xT∫0Tf(t)dt G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt is periodic.
b) lim⁡n→∞∑i=1nF(i)n2+i2=ln⁡22T∫0Tf(x)dx. \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx.
functionreal analysisperiodicanalysisIntegralprimitives