Solve the ecuation: x=\frac{[x]+2012}{\{x\}+2013} where [x] is the whole part of the number x and {x} is the fractional part of the number x.

Question

Solve the ecuation:


`x=\frac{[x]+2012}{\{x\}+2013}`

where [x] is the whole part of the number x and {x} is the fractional part of the number x.

Answer 1

Scrii x = [ x ] + { x } ;
Folosesti proprietatea fundamentala a proportiilor;
Apoi, exprimi [ x ] in functie de { x }, si obtii ca :
[ x ] = ( 2012 - { x } ^2 - 2013{ x } ) / ( 2012 + { x } ) ;
Fie { x } = t ∈ [ 0 , 1 ) ;
Obtii ca [ x ] = - t - 1 + 4024 / ( t + 2012 ) ;
Stim ca [ x ] ∈ Z ;
Atunci t = 0 sau t = 2012 ;
Convine numai t = 0 => x = [ x ] ;
Ecuatia originala va avea solutia 1.

Bafta!