Question

fie x,y,z nr naturale astfel incat x+y+z=11.Aratati ca suma xy+yz+zx(cu linie deasupra) este patrat perfect.Aratati ca nu exista x,y si z consecutive care sa indeplineasca aceasta conditie.

Answer 1

 [tex]Dac\u{a}\; x+y+z=11\;atunci,\\ (1)...\;\overline{xy}+\overline{yz}+\overline{zx}=p\u{a}trat\;perfect \\ \Leftrightarrow\;10x+y \;+\;10y+z\;+\;10z+x=11(\underbrace{x+y+z}_{=11})=11^2\\ (2)...\;Dac\u{a}\;x;y;z\; sunt\;nume\;consecutive\;atunci\;\\ y=x+1\;\c{s}i\;z=x+2 \;atunci\;,\\ \overline{xy}+\overline{yz}+\overline{zx}=\\ =10x+(x+1)\;+\;10(x+1)+(x+2)\;+\;10(x+2)+x=\\ =10x+10+10x+10+x+2+10x+20+x=32x+42\ne{p.p. [/tex]