Cristian Calude, proba pe echipe, R.IV, P.I

[Laurian Filip]

Sa se determine suma tuturor elementelor multimii

\( A=\left\{\overline{xyz} \in \mathbb{N} / x\neq 0, \frac{\overline{xy}}{\overline{zy}}=\overline{x,y} \right} \)

[Andi Brojbeanu]

xy/zy =x,y rezulta xy/zy=xy/10 rezulta zy=10, y=0, z=1.
Valorile lui xyz sunt (101; 201; 301; ……..; 901), in total 9 valori.
Sxyz=101+201+……+901=100(1+2+….+9)+9=100X9X10:2+9=4509.

[Claudiu Mindrila]

\( \frac{\overline{xy}}{\overline{zy}}=\overline{x,y}\Longrightarrow\frac{\overline{xy}}{\overline{zy}}=\frac{\overline{xy}}{10}\Longrightarrow10\cdot\overline{xy}=\overline{xy}\cdot\overline{zy}\Longrightarrow\overline{zy}=10\Longrightarrow z=1,\ y=0. \) Atunci \( A=\left\{ 101,\ 201,\dots,\ 901\right\} \) si suma elementelor multimii \( A \) este \( 101+201+\dots+901=100\left(1+2+\dots+9\right)+1\cdot9=100\cdot\frac{9\cdot10}{2}+9=45\cdot100+9=4509.




\)