mateforum.ro Forum Index mateforum.ro

 
 FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups   RegisterRegister 
 ProfileProfile   Log in to check your private messagesLog in to check your private messages   Log inLog in 

Doua perechi de numere inlantuite

 
Post new topic   Reply to topic    mateforum.ro Forum Index -> Juniori -> Teoria Numerelor
View previous topic :: View next topic  
Author Message
Virgil Nicula
Euler


Joined: 28 Sep 2007
Posts: 672

PostPosted: Mon Feb 18, 2008 11:24 pm    Post subject: Doua perechi de numere inlantuite Reply with quote

Fie numerele reale x\ ,\ y\ ,\ x_1\ ,\ y_1 pentru care 1<x<y si \left\{\begin{array}{c}
 x_1=x+\frac yx-1\\\\
 y_1=y+\frac xy-1\end{array} .

Sa se arate ca : \left\|\begin{array}{cc}
 1. & \{\ x_1\ ,\ y_1\ \}\subset (\ x\ ,\ y\ )\\\\ 
 2. & x_1=y_1\ \Longleftrightarrow\left\|\begin{array}{c}
 x=y=2\\\\
 \mathrm {\ sau\ }\\\\
 2\in (\ x\ ,\ y\ )\end{array}\end{array} .
Back to top
View user's profile Send private message
Laurian Filip
Site Admin


Joined: 25 Nov 2007
Posts: 473
Location: Bucuresti

PostPosted: Tue Apr 08, 2008 4:24 pm    Post subject: Reply with quote

1. y>x rezulta \frac{y}{x}>1 rezulta x_1=x+\frac{y}{y}-1>x
(x-y)(x-1)<0 echivalent cu x(x-y)+y-x<0 echivalent cu x^2+y-x<xy
x+\frac{y}{x}-1<y adica x_1<y

deci x_1\in\(x,y)
Analog si y_1\in\(x,y)


2. x_1=y_1
x+\frac{y}{x}-1=y+\frac{x}{y}-1
x^2y+y^2=y^2x+x^2
xy(x-y)=(x+y)(x-y)

I. x=y cu toate ca in enunt se dadea strict mai mare x ca y...
dar din x=y => x_1=y_1 oricare ar fi x real...

II. x\neq{y}
xy=x+y
(x-1)(y-1)=1 de unde rezulta 1\in\(x,y)
_________________
Bac Mat: Formule bacalaureat matematica
Triangle Solver: Rezolva orice triunghi
Gaseste Scrabble: Castiga la jocul cuvintelor
Resigilate PCG: Top oferte desigilate pcg + istoric pret
Back to top
View user's profile Send private message Visit poster's website
Display posts from previous:   
Post new topic   Reply to topic    mateforum.ro Forum Index -> Juniori -> Teoria Numerelor All times are GMT + 2 Hours
Page 1 of 1

 
Jump to:  
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum



Powered by phpBB © 2001, 2005 phpBB Group