Question

Numărul natural n=0,x(y)+0,y(z)+0,z(x) este pătrat perfect,iar x,y,z sunt cifre nenule distincte,Suma x+y+z este egală cu:​

Answer 1

$\displaystyle\sf\\0,x(y)+0,y(z)+0,z(x)=\frac{10x+y-x}{90}+\frac{10y+z-y}{90}+\frac{10z+x-z}{90}=\\\frac{(10x+y-x)+(10y+z-y)+(10z+x-z)}{90}=\frac{10x+10y+10z}{90}=\\\frac{x+y+z}{9}=n,~n~este~patrat~perfect,~fie~k\in\mathbb{N}~a.i.~n=k^2.\\\frac{x+y+z}{9}=k^2,~de~observat~ca~1\leq \frac{x+y+z}{9}\leq 3 ,~si~cum\\singurul~patrat~perfect~cuprins~intre~1~si~3~este~1 \implies\boxed{\sf k^2=n=1}.$

$\displaystyle\sf\\\implies x+y+z=9,~asadar,~suma~\boxed{\sf x+y+z~este~egala~cu~9}.$