MathDB

4

Part of 2003 District Olympiad

Problems(6)

MP not the smallent side when <MNP>60, regular tetrahedron if VA=AB

Source: 2003 Romania District VIII P4

5/24/2020
a) Let MNPMNP be a triangle such that ∠MNP>60o\angle MNP> 60^o. Show that the side MPMP cannot be the smallest side of the triangle MNPMNP. b) In a plane the equilateral triangle ABCABC is considered. The point VV that does not belong to the plane (ABC)(ABC) is chosen so that ∠VAB=∠VBC=∠VCA\angle VAB = \angle VBC = \angle VCA. Show that if VA=ABVA = AB, the tetrahedron VABCVABC is regular.
Valentin Vornicu
geometry3D geometrytetrahedronanglesequal angles
triangle by symmetric points, area, construction (2003 Romania District VII P4)

Source:

5/24/2020
Let ABCABC be a triangle. Let B′B' be the symmetric of BB with respect to C,C′C, C' the symmetry of CC with respect to AA and A′A' the symmetry of AA with respect to BB.
a) Prove that the area of triangle AC′A′AC'A' is twice the area of triangle ABCABC. b) If we delete points A,B,CA, B, C, how can they be reconstituted? Justify your reasoning.
geometryconstructionarea of a triangleSymmetric
S-vectors

Source: RMO 2003, District Round

4/22/2006
We say that a set A\displaystyle A of non-zero vectors from the plane has the property (S)\displaystyle \left( \mathcal S \right) iff it has at least three elements and for all u→∈A\displaystyle \overrightarrow u \in A there are v→,w→∈A\displaystyle \overrightarrow v, \overrightarrow w \in A such that v→≠w→\displaystyle \overrightarrow v \neq \overrightarrow w and u→=v→+w→\displaystyle \overrightarrow u = \overrightarrow v + \overrightarrow w. (a) Prove that for all n≥6\displaystyle n \geq 6 there is a set of n\displaystyle n non-zero vectors, which has the property (S)\displaystyle \left( \mathcal S \right). (b) Prove that every finite set of non-zero vectors, which has the property (S)\displaystyle \left( \mathcal S \right), has at least 6\displaystyle 6 elements. Mihai Baluna
vectorcombinatorics unsolvedcombinatorics
Romania District Olympiad 2003 - Grade XI

Source:

3/18/2011
Let α>1\alpha>1 and f:[1α,α]→[1α,α]f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right], a bijective function. If f−1(x)=1f(x), ∀x∈[1α,α]f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right], prove that:
a)ff has at least one point of discontinuity; b)if ff is continuous in 11, then ff has an infinity points of discontinuity; c)there is a function ff which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity.
Radu Mortici
functionreal analysisreal analysis unsolved
Exponentials + increasing

Source: RMO 2003, District Round

5/29/2006
Let a,b,c,d∈R\displaystyle a,b,c,d \in \mathbb R such that a>c>d>b>1\displaystyle a>c>d>b>1 and ab>cd\displaystyle ab>cd. Prove that f:[0,∞)→R\displaystyle f : \left[ 0,\infty \right) \to \mathbb R, defined through f(x)=ax+bx−cx−dx, ∀x≥0, \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , is strictly increasing.
logarithmsfunctionalgebra unsolvedalgebra
Weird equality: integrals, limits, two functions and Riemann’s sums

Source: Romanian District Olympiad 2003, Grade XII, Problem 4

10/7/2018
Consider the continuous functions f:[0,∞)⟶R,g:[0,1]⟶R, f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , where f f has a finite limit at ∞. \infty . Show that: lim⁡n→∞1n∫0nf(x)g(xn)dx=∫01g(x)dx⋅lim⁡x→∞f(x). \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) .
functionlimitsIntegralcalculusintegrationreal analysis