MathDB

1

Part of 2007 District Olympiad

Problems(6)

angles wanted, circumcenter O, OD = BD = 1/3 BC (2007 Romania District VII P1)

Source:

5/18/2020
Point OO is the intersection of the perpendicular bisectors of the sides of the triangle △ABC\vartriangle ABC . Let DD be the intersection of the line AOAO with the segment [BC][BC]. Knowing that OD=BD=13BCOD = BD = \frac 13 BC, find the measures of the angles of the triangle △ABC\vartriangle ABC.
geometryequal segmentsanglesAngle ChasingCircumcentercircumcircle
An integer is the arithmetic mean of the others

Source: Romanian DMO, 8th grade, problem 1

3/23/2007
Three positive reals x,y,zx,y,z are given so that xy=z−x+1y=z+12.xy=\frac{z-x+1}{y}=\frac{z+1}2. Prove that one of the numbers is the arithmetic mean of the other two.
algebra proposedalgebra
Function

Source: District MO, Romania, 9th grade, first problem

3/5/2007
We say that a function f:N→Nf: \mathbb{N}\rightarrow\mathbb{N} has the (P)(\mathcal{P}) property if, for any y∈Ny\in\mathbb{N}, the equation f(x)=yf(x)=y has exactly 3 solutions. a) Prove that there exist an infinity of functions with the (P)(\mathcal{P}) property ; b) Find all monotonously functions with the (P)(\mathcal{P}) property ; c) Do there exist monotonously functions f:Q→Qf: \mathbb{Q}\rightarrow\mathbb{Q} satisfying the (P)(\mathcal{P}) property ?
functionalgebra unsolvedalgebra
Romania District Olympiad 2007 - Grade XI

Source:

4/10/2011
Let a1∈(0,1)a_1\in (0,1) and (an)n≥1(a_n)_{n\ge 1} a sequence of real numbers defined by an+1=an(1−an2), (∀)n≥1a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1. Evaluate lim⁡n→∞ann\lim_{n\to \infty} a_n\sqrt{n}.
limitinductionreal analysisreal analysis unsolved
Cyclic, logarithmic inequality

Source: Romanian District Olympiad 2007, Grade X, Problem 1

10/7/2018
Let be three real numbers a,b,c, a,b,c, all in the interval (0,∞) (0,\infty ) or all in the interval (0,1). (0,1). Prove the following inequality: ∑cyclog⁡abc≥4⋅∑cyclog⁡abc. \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c .
inequalitieslogarithmalgebra
Groups and their subsets

Source: RMO 2007 - District Round - I

3/3/2007
For a group (G,⋆)\left( G, \star \right) and A,BA, B two non-void subsets of GG, we define A⋆B={a⋆b:a∈A and b∈B}A \star B = \left\{ a \star b : a \in A \ \text{and}\ b \in B \right\}. (a) Prove that if n∈N, n≥3n \in \mathbb N, \, n \geq 3, then the group \left( \mathbb Z \slash n \mathbb Z,+\right) can be writen as \mathbb Z \slash n \mathbb Z = A+B, where A,BA, B are two non-void subsets of \mathbb Z \slash n \mathbb Z and A \neq \mathbb Z \slash n \mathbb Z, \, B \neq \mathbb Z \slash n \mathbb Z, \, \left| A \cap B \right| = 1. (b) If (G,⋆)\left( G, \star \right) is a finite group, A,BA, B are two subsets of GG and a∈G∖(A⋆B)a \in G \setminus \left( A \star B \right), then prove that function f:A→G∖Bf : A \to G \setminus B given by f(x)=x−1⋆af(x) = x^{-1}\star a is well-defined and injective. Deduce that if ∣A∣+∣B∣>∣G∣|A|+|B| > |G|, then G=A⋆BG = A \star B. [hide="Question."]Does the last result have a name?
functionfloor functionsuperior algebrasuperior algebra unsolved