OLM 2008 BACAU
Moderators: Filip Chindea, Andrei Velicu, Radu Titiu
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Adriana Nistor
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- Joined: Thu Aug 07, 2008 10:07 pm
- Location: Drobeta Turnu Severin, Mehedinti
OLM 2008 BACAU
Fie \( z_{1},\ z_{2},\ z_{3} \) numere complexe de modul 1 cu suma egala cu 0. Aratati ca suma patratelor este egala cu 0.
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mihai miculita
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- Posts: 93
- Joined: Mon Nov 12, 2007 7:51 pm
- Location: Oradea, Romania
1) \( |z_1|=|z_2|=|z_3|=1\Leftrightarrow |z_1|^2=|z_2|^2=|z_3|^2=1\Leftrightarrow z_1\overline{z_1}=z_2\overline{z_2}=z_3\overline{z_3}=1\Rightarrow \left\{ \begin{\array} \overline{z_1}=\frac{1}{z_1}\\
\overline{z_2}=\frac{1}{z_2}\ \\
\overline{z_3}=\frac{1}{z_3}. \) (1)
2) \( z_1+z_2+z_3=0\Leftrightarrow \overline{z_1}+\overline{z_2}+\overline{z_3}=0\Leftrightarrow \frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}=0\Leftrightarrow z_1z_2+z_1z_3+z_2z_3=0; \) (2)
3) \( z_1+z_2+z_3=0\Rightarrow 0=(z_1+z_2+z_3)^2=z_1^2+z_2^2+z_3^2+2(z_1z_2+z_1z_3+z_2z_3). \) (3)
\( \mbox{In fine, tinand seama de relatiile (2) si (3), obtinem ca } z_1^2+z_2^2+z_3^2=0. \)
\overline{z_2}=\frac{1}{z_2}\ \\
\overline{z_3}=\frac{1}{z_3}. \) (1)
2) \( z_1+z_2+z_3=0\Leftrightarrow \overline{z_1}+\overline{z_2}+\overline{z_3}=0\Leftrightarrow \frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}=0\Leftrightarrow z_1z_2+z_1z_3+z_2z_3=0; \) (2)
3) \( z_1+z_2+z_3=0\Rightarrow 0=(z_1+z_2+z_3)^2=z_1^2+z_2^2+z_3^2+2(z_1z_2+z_1z_3+z_2z_3). \) (3)
\( \mbox{In fine, tinand seama de relatiile (2) si (3), obtinem ca } z_1^2+z_2^2+z_3^2=0. \)