Question

Numerele intregi x si y pentru care √5x-1 + √5y-1 = 5 sunt ................

Answer 1

conditii de existenta: 5x-1>=0 si 5y-1>=0, adica x>=1 si y>=1.

y>=1 => rad(5y-1)>=rad(5*1-1)=rad(4)=2. => rad(5x-1)<=3.
x>=1 => rad(5x-1)>=rad(5*1-1)=rad(4)=2.

Deci 2<=rad(5x-1)<=3.

Deosebim doua cazuri:
Caz I: rad(5x-1)=2 <=> 5x-1=4 <=> 5x=5 <=> x=1. In acest caz rad(5y-1)=3, de unde y=2.

Caz II: rad(5x-1)=3 <=> 5x-1=9 <=> 5x=10 <=> x=2. In acest caz rad(5y-1)=2, de unde y=1.

In concluzie, numerele sunt x=1,y=2 si x=2,y=1.