MathDB

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Part of 2001 District Olympiad

Problems(6)

Romania District Olympiad 2001 - VII Grade

Source:

3/12/2011
A positive integer is called good if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Prove that:
a)2001 is good, but 3001 isn't good. b)the product of two good numbers is a good number. c)if the product of two numbers is good, then at least one of the numbers is good.
Bogdan Enescu
modular arithmeticnumber theory proposednumber theory
Romania District Olympiad 2001 - VIII Grade

Source:

3/12/2011
a) Find all the integers mm and nn such that
9m2+3n=n2+89m^2+3n=n^2+8
b) Let a,b∈N∗a,b\in \mathbb{N}^* . If x=aa+b+(a+b)ax=a^{a+b}+(a+b)^a and y=aa+(a+b)a+by=a^a+(a+b)^{a+b} which one is bigger?
Florin Nicoara, Valer Pop
number theory proposednumber theory
Romania District Olympiad 2001 - Grade IX

Source:

3/12/2011
Consider the equation x2+(a+b+c)x+λ(ab+bc+ca)=0x^2+(a+b+c)x+\lambda (ab+bc+ca)=0 with a,b,c>0a,b,c>0 and λ∈R\lambda\in \mathbb{R}. Prove that:
a)If λ≤34\lambda\le \frac{3}{4}, the equation has real roots. b)If a,b,ca,b,c are the side lengths of a triangle and λ≥1\lambda\ge 1, then the equation doesn't have real roots.
***
inequalitiesalgebra proposedalgebra
Romania District Olympiad 2001 - Grade XI

Source:

3/16/2011
Let A∈M2(R)A\in \mathcal{M}_2(\mathbb{R}) such that det⁡(A)=d≠0\det(A)=d\neq 0 and det⁡(A+dA∗)=0\det(A+dA^*)=0. Prove that det⁡(A−dA∗)=4\det(A-dA^*)=4.
Daniel Jinga
linear algebramatrixalgebrapolynomiallinear algebra unsolved
Re: Romania District Olympiad 2001 - Grade X

Source:

3/16/2011
Let (an)n≥1(a_n)_{n\ge 1} be a sequence of real numbers such that
a1(n1)+a2(n2)+…+an(nn)=2n−1an, (∀)n∈N∗a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*
Prove that (an)n≥1(a_n)_{n\ge 1} is an arithmetical progression.
Lucian Dragomir
inductionarithmetic sequencealgebra proposedalgebra
Romania District Olympiad 2001 - Grade XII

Source:

3/16/2011
For any n∈N∗n\in \mathbb{N}^*, let Hn={kn! ∣ k∈Z}H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}.
a) Prove that HnH_n is a subgroup of the group (Q,+)(Q,+) and that Q=⋃n∈N∗HnQ=\bigcup_{n\in \mathbb{N}^*} H_n;
b) Prove that if G1,G2,…,GmG_1,G_2,\ldots, G_m are subgroups of the group (Q,+)(Q,+) and Gi≠Q, (∀)1≤i≤mG_i\neq Q,\ (\forall) 1\le i\le m, then
G1∪G2∪…∪Gm≠QG_1\cup G_2\cup \ldots \cup G_m\neq Q
Marian Andronache & Ion Savu
group theoryabstract algebrapigeonhole principlesuperior algebrasuperior algebra unsolved