MathDB

1

Part of 2008 District Olympiad

Problems(6)

Inequality with n

Source: Romanian district MO 7th grade P1

3/1/2008
Prove that for an integer n>\equal{}1 we have n(1\plus{}\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\plus{}\frac{1}{n})\geq (n\plus{}1)(\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\frac{1}{n\plus{}1})
inequalitiesalgebra proposedalgebra
plane cuts regular tetrahedron into rhombus (2008 Romania District VIII P1)

Source:

5/18/2020
A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.
geometryrhombusPlane3D geometrytetrahedronsquare
Sequence and sequence of arithmetic means of the terms

Source: Romanian District MO 2008, Grade 9, Problem 1

4/30/2008
Let {an}n≥1 \{a_n\}_{n\geq 1} be a sequence of real numbers such that |a_{n\plus{}1}\minus{}a_n|\leq 1, for all positive integers n n. Let {bn}n≥1 \{b_n\}_{n\geq 1} be the sequence defined by b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}. Prove that |b_{n\plus{}1}\minus{}b_n | \leq \frac 12, for all positive integers n n.
inequalitiestriangle inequality
Module expression

Source: RMO District Round, Bucharest 2008, Grade 10, Problem 1

1/27/2008
Let z∈C z \in \mathbb{C} such that for all k∈1,3‾ k \in \overline{1, 3}, |z^k \plus{} 1| \le 1. Prove that z \equal{} 0.
abstract algebraalgebra proposedalgebra
Romania District Olympiad 2008 - Grade XI

Source:

4/10/2011
If A∈M2(R)A\in \mathcal{M}_2(\mathbb{R}), prove that:
det⁡(A2+A+I2)≥34(1−det⁡A)2\det(A^2+A+I_2)\ge \frac{3}{4}(1-\det A)^2
linear algebralinear algebra unsolved
Apply to a modified form of <integral f = integral id.f> some sort of mvt

Source: Romanian District Olympiad 2008, Grade XII, Problem 1

10/7/2018
Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a countinuous function such that ∫01f(x)dx=∫01xf(x)dx. \int_0^1 f(x)dx=\int_0^1 xf(x)dx. Show that there is a c∈(0,1) c\in (0,1) such that f(c)=∫0cf(x)dx. f(c)=\int_0^c f(x)dx.
functionreal analysisFTCMVTIntegralcalculusintegration